For instance, if one city is 10 times as populous as another one, does it need 10 times as many gas stations? No. Bigger cities have more gas stations than smaller ones (of course), but not nearly in direct proportion to their size. The number of gas stations grows only in proportion to the 0.77 power of population. The crucial thing is that 0.77 is less than 1. This implies that the bigger a city is, the fewer gas stations it has per person. Put simply, bigger cities enjoy economies of scale. In this sense, bigger is greener.

In recent years much of my research has been directed towards providing mathematical analysis for popular machine learning and data mining paradigms that seem to lack clear theoretical justification. I have looked into the performance guarantees one can provide for Support Vector Machines (with pessimistic conclusions), at Semi-Supervised Learning (once again, coming up with some inherent limitations of that approach), at the Learning-To-Learn paradigm (providing some theoretical justifications to common practices), at the Stability method for determining the number of clusters in a data set (see my COLT06, COLT07 papers on that and a recent submission), and quite a few more topics.

Clustering is a really large area that also suffers from the lack of mathematical foundations, and I have been working extensively trying to address the theoretical challenges of developing such a theory (see my recent submissions with Ackerman as well as my ’05 paper with von Luxburg)

Hama (means a hippopotamus in Korean) is a parallel matrix computation package currently in incubation with Apache. It is a library of matrix operations for large-scale processing and development environments as well as a Map/Reduce framework for a large-scale numerical analysis and data mining, that need the intensive computation power of matrix inversion, e.g., linear regression, PCA, SVM and etc. It will be useful for many scientific applications, e.g., physics computations, linear algebra, computational fluid dynamics, statistics, graphic rendering and many more.

The International String Figure Association (ISFA) was founded in 1978 by Hiroshi Noguchi, a Japanese mathematician, and Philip Noble, an Anglican missionary stationed in Papua New Guinea. ISFA is a small, not-for-profit organization funded solely by membership dues and private donations. The primary goal of ISFA is to gather, preserve, and distribute string figure knowledge so that future generations will continue to enjoy this ancient pastime. We also encourage the invention of new string figures, and enjoy sharing them with one another.

Here is one of my very favorite Math tricks that’s sure to impress your friends (and yourself the first time you try it). I learned this trick over 30 years ago and amazingly enough I still remember it.Binary sorting trick

To help students master the facts, you may use Fact Triangles. These triangular cards have three numbers on each corner, an asterisk, or solid dot in the corner, which contains the sum or product, and the operation (addition, subtraction, multiplication, division) in the center.

It is hard to exaggerate the importance and influence of these books. Many mathematicians of my generation (translation: old fogeys like me) grew up reading either the columns in Scientific American or the books that collected them. These books played a role in getting us interested in mathematics, and provided solace and entertainment when mathematics turned out to be hard. As a mathematician, I am delighted to have them back in print. (Yes, a complete edition on CD-ROM was — and is — available, but while electronic form is great for searching, it is terrible for reading.) As a bibliophile, I am delighted to have these in book form, especially as books that look so good; the fifteen volumes will eventually have an honored place in my shelf. Can a Folio Society edition be far behind?

An Excel spreadsheet, named Distance calculator, is available for the foregoing calculation. It is contained in the distcalcnew.zip file. In this calculator, one enters the coordinates of each of the two points (beginning in row eight) as the pair (longitude, latitude) of quintets of sign (negative for east meridian of longitude or south parallel of latitude), name, degrees, minutes, and seconds, for each point. Then, the calculator computes and displays the solution of the navigation problem to the right, on the same row. For use on an other sphere (than the Earth), the radius may be entered near the upper-right corner of the spreadsheet. Please observe that there are many hidden columns, which may be unhidden; if you are curious.

That is, while I wish I could say that my students now perform at the level of the advanced books, I cannot. However, as a teacher I can work steadily to bring them up to it over the course of our undergraduate program. This means stepping back from focussing on rote computations in favor of focussing on an understanding of the mathematics. It means proving things and having students understand, e.g., that matrix multiplication is the application of a linear function. But it means also avoiding an approach that is too advanced for the students: the presentation must emphasize motivation, must have many illustrative examples, and must include exercises with many of the medium-difficult questions that are a challenge to a learner without being overwhelming. And, it means communicating to my students that the change of focus is what we are up to, right from the start.

One other contemporary artist, Robert Irwin, has explored the splitting of focus. Writing about Irwin in his book Seeing Is Forgetting the Name of the Thing One Sees, Lawrence Weschler quotes him explaining that he trained himself by “placing a dot on a window and gazing both at and beyond it, thus allowing two planes of focus, one for each eye.” Irwin played with perception in a famous series of paintings of dots and discs, but the Oakes twins are alone in using split focus to draw perspective. They didn’t know this until after they’d done it.

Rating collateralised debt obligations (CDOs), which are based on tranched pools of credit risk exposures, does not only require attributing a probability of default to each obligor within the portfolio. It also involves assumptions concerning recovery rates and correlated defaults of pool assets, thus combining credit risk assessments of individual collateral assets with estimates about default correlations and other modelling assumptions. In this paper, we explain one of the most well-known models for rating CDOs, the so-called binomial expansion technique (BET). Comparing this approach with an alternative methodology based on Monte Carlo simulation, we then highlight the potential importance of correlation assumptions for the ratings of senior CDO tranches and explore what differences in methodologies across rating agencies may mean for senior tranche rating outcomes.

COLOR SUDOKU contains 180 colorful, hand-designed Sudoku variation puzzles with additional rules that will challenge you to rethink your old Sudoku strategies and devise new ones. "we turn coffee into puzzles"

more math papers on Sudoku; the relationship between sudoku and Hamming codes.

talk by Peter Cameron on Sudoku from a combinatorics math perspective, and the relationship between it and the game of Set

anamorphic art is art that has to be distorted to be viewed; this is a deep deep link into a collection. the next link will be to the ontology browser that helped me find this. first saw stuff like this in Martin Gardner's Mathematical Games column long ago.

tremendous collection of heuristics and rules of thumb for deciding how to visualize data on a map; many different ways to put the same data into a color or texture scheme, and depending on what you are doing you get various levels of truthiness

A musical chord can be represented as a point in a geometrical space called an orbifold. Line segments represent mappings from the notes of one chord to those of another. Composers in a wide range of styles have exploited the non-Euclidean geometry of these spaces, typically by using short line segments between structurally similar chords. Such line segments exist only when chords are nearly symmetrical under translation, reflection, or permutation. Paradigmatically consonant and dissonant chords possess different near-symmetries and suggest different musical uses.

Department of Music, Princeton University, Princeton, NJ 08544, USA, and Radcliffe Institute for Advanced Study, 34 Concord Avenue, Cambridge, MA 02138, USA.

The monster group is also called the friendly giant group. It was constructed in 1982 by Robert Griess as a group of rotations in 198633-dimensional space.

"A Rare Blend of Monster Raving Egomania and Utter Batshit Insanity". cosma holds no punches

mayan numerical system is base 20; 1, 20, 400, 8000; for calendars the 400 is replaced by 365

UM has a good stat dept

The slide rule began to go extinct in 1972 when its planet was shattered by an immense asteroid in the form of the digital calculator. I was there when it struck.

I learned multiplication in 12th grade math on a slide rule and with log tables; we had an enormous slide rule mounted above the blackboard. awesome.

This family event, typically held on an evening in March, provides opportunities for students and parents to explore activities related to a particular topic, such as biology, chemistry, earth science, or math.

It is important that most students have mastery of basic facts. It is equally important that they make sense of number combinations as they are learning these facts. Here are some strategies to help with this understanding.

This Catalogue Raisonne describes every slide rule ever made by the Hemmi Slide Rule Company of Japan--or at least all those I have been able to identify. It is the most comprehensive catalogue of Hemmi slide rules ever published.

Pick-your-own carving pumpkins are placed in a special pricing box and are priced on size. An average of the width and height is used to compute the price. The exact formula is: Price = ( .003 X S³·²³ - .22 X S) X 1.05

In Super Paper Mario, life begins as a regular 2-D game -- your Mario is a flat, "paper" cutout in a world of paper cutouts -- until he suddenly acquires the power to shift into a 3-D perspective. Suddenly you can see that all your 2-D enemies are wafer-t

In the Saxon land-measuring system, 40 rods make a furlong (fuhrlang), the length of the traditional furrow (fuhr) as plowed by ox teams on Saxon farms.

@jhritz: some amazing precision engineering here

a number m of the form m = a^2 * b^3

on the golden ratio and its mythology

John, this is the finger counting that I do - I'll have to take a video of it some time. It goes pretty fast if you get a rhythm.

speed addition, multiplication etc

all the columns on one cd-rom

big 4-d model made of zome toy building tools

American college football faces a conflict created by the desire to stage national championship games between the best teams of a season when there is no conventional playoff system to decide which those teams are. Instead, ranking of teams is based on their record of wins and losses during the season, but each team plays only a small fraction of eligible opponents, making the system underdetermined or contradictory or both. It is an interesting challenge to create a ranking system that at once is mathematically well-founded, gives results in general accord with received wisdom concerning the relative strengths of the teams, and is based upon intuitive principles, allowing it to be accepted readily by fans and experts alike. Here we introduce a one-parameter ranking method that satisfies all of these requirements and is based on a network representation of college football schedules.

A complex index to mathematical puzzles and games explained and analyzed by Martin Gardner in his books.