Abel Prize

John Tate of the United States wins the Abel Prize in 2010 “for his vast and lasting impact on the theory of numbers”. So, what is this Abel Prize?

The Abel Prize is an international prize presented annually by the King of Norway to one or more outstanding mathematicians. The prize is named after Norwegian mathematician Niels Henrik Abel (1802–1829). It has often been described as the “mathematician’s Nobel” prize and is among the most prestigious awards in mathematics. It comes with a monetary award of six million kroner, which in 2009 was €684,000 or US$929,000.

The only Indian by origin to win it so far was S. R. Srinivasa Varadhan in 2007 “for his fundamental contributions to probability theory and in particular for creating a unified theory of large deviation”.

Millennium Prize

For all the mathematicians, here is news! There are currently 6 unsolved problems in Mathematics that were posted by the Clay Mathematics Institute in 2000. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute.

Originally 7 problems were stated, of which only one has been solved by Grigori Perelman.

Click here for a list of these problems.

Catalan Number

The Catalan numbers on nonnegative integers are a set of numbers that arise in tree enumeration problems of the type, “In how many ways can a regular -gon be divided into triangles if different orientations are counted separately?” (Euler’s polygon division problem). The solution is the Catalan number

A few catalan numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020, 91482563640, 343059613650, 1289904147324

Number of sides	3	4	5	6	7	8	9
Number of way to partition it into triangles	1	2	5	14	42	132	429

Fibonacci Series

Most of us who have pursued serious math at some point would have encountered the famous Fibonacci Series.

For those who haven’t, fibonacci series starts with 0 and 1 and each number in the series is a sum of the previous two numbers.

So the series goes… 0, 1, 1, 2, 3, 5, 8, 13, 21….

The Fibonacci sequence was well known in ancient India, where it was applied to the metrical sciences (prosody), before it was known in Europe. Developments have been attributed toPingala (200 BC).

What appeals to me most about fibonacci series is its uncanny closeness to nature

• arrangement of leaves on a stem is a fibonacci number
• the fruitlets of a pineapple follow the fibonacci series
• The arrangement of a pinecone is fibonacci in nature
• relating to the breeding of rabbits, the spirals of shells, and the curve of waves are all fibonacci in nature.

Well Curve

The Well curve, as opposed to the standard normal distribution or the bell curve is “bi-modal”, i.e. low in the center and high on the sides.

Some examples where the well curve distribution is followed are:

• Organization size: For a variety of reasons, big companies are growing bigger (witness the number of mergers) and the number of small businesses is multiplying. But there are fewer midsize companies.
• Geopolitics: The past decade has seen the rise of huge multinational federations (NAFTA, the European Union, etc.) At the same time, small independent states and secessionist movements are also multiplying. And the mid-sized countries, like Italy and Spain, are losing population, power and status.
• Consumer culture: Tiny screens (on cell-phones, PDAs and now wrist-PDAs) are multiplying, while large-screen TVs are also proliferating. Mid-sized seems to be going out of style.
• Retail: Wal-Mart has become the largest company in the Fortune 500, and big retail chains are expanding. At the same time, the number of small, specialty boutiques is soaring. But mid-sized companies are struggling.
• Drooping middle class: The Federal Reserve Board’s latest analysis of family finances shows that, in the past decade, American incomes were up across the board. But when the population is divided into five equal segments, a well curve emerges – there was faster growth at the top and bottom, with less growth in the middle.
• School Grades: In the last decade, the percentage of students scoring in the highest ranges increased, the percentage scoring at the bottom level increased, but the number with mid-scores dropped.
• Purchasing patterns: The Wall Street Journal noted last year, “consumers are flocking to the most expensive products and the cheapest products, fleeing the middle ground.”

The implications are huge: insurers, marketers, policy makers and pollsters may be basing decisions on faulty premises about what is “normal”. They are assuming “middle-America”, “middlebrow” tastes. But, the importance seems to be shifting to the edges.

Read more here.

Algorithmic Trading

Aloha!

We are back after a loooong break. A lot of water has flown in Ganga since the last post. For starters, both authors changed have changed their workplaces, locations et al. Lots of brainstorming as to post topics etc (we also wanted to change the looks a bit, add some features, but I guess you’ll have to wait for sometime for that to happen) … So here we start with our first post on algorithmic trading.

Ok, I was not aware of this concept till Akshay briefed me about it. I was blown away when I got to know that more than 50% of trading in US and UK markets are automated. Yes, human programming and a computer execute over 50% of the trade in these markets (read this). So thought I should get a primer on how this works.

Market makers and some hedge funds, provide liquidity to the market, generating and executing orders automatically. In this “high frequency trading” (HFT) computers make the decision to initiate orders based on information that is received electronically, before human traders are even aware of the information.

Many different algorithms have been developed to implement different trading strategies. These algorithms or techniques are commonly given names such as “Iceberg”, “Dagger”, “Guerrilla”, “Sniper” and “Sniffer”.

Might just go in depth of each of these strategies in future posts.

An example of Algorithmic Trading in Indian markets

A program could be to sell the stock futures of a particular company and buy the stock if the futures price is x% higher than the stock price. Also, it could be to compare a set of variables — if rupee is more than 45 to the dollar, and crude oil is less than $60 per barrel — then the software would sell Infosys futures and buy HPCL shares.

Read this economic times article to see how it effects the Indian markets.

Four Color Theorem

How many different colors are sufficient to color the countries on a map in such a way that no two adjacent countries have the same color?

After examining a wide variety of different planar graphs, one discovers the apparent fact that every graph, regardless of size or complexity, can be coloured with just four distinct colors. This “four-color conjecture” was first noted by August Ferdinand Mobius in 1840.

Three colors are adequate for simpler maps, but an additional fourth color is required for some maps, such as a map in which one region is surrounded by an odd number of other regions that touch each other in a cycle.

Read mathpages for more.

Golden Ratio

Two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to (=) the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.6180339887.

Golden ratio is found in everything from Da Vinci’s Monalisa to Salvador Dali’s Last Supper to Fibonacci series to Egyptian Pyramids to the spirals in the sunflower head.

You could visit this page for detailed explanations

But why such stress on the Golden Ratio?

The Golden Rectangle has been said to be one of the most visually satisfying of all geometric forms.

Golden Rectangle is a rectangle with proportions that are two consecutive numbers from the Fibonacci sequence.

Mandelbrot set

In mathematics the Mandelbrot set, named after Benoît Mandelbrot, is a set of points in the complex plane, the boundary of which forms a fractal. When computed and graphed on the complex plane the Mandelbrot Set is seen to have an elaborate boundary which does not simplify at any given magnification. This qualifies the boundary as a fractal.

The Mandelbrot set has become popular outside mathematics both for its aesthetic appeal and for being a complicated structure arising from a simple definition.

Regardless of the extent to which one zooms in on a Mandelbrot set, there is always additional detail to see. During the twelve-second zoom in the animation Mandelbrot zoom, the set becomes magnified eleven-million fold. Thus, assuming the first frame is life-size at 45 mm across, a carbon atom would comprise 36 pixels in the final frame!

Click here for more details.

Sierpinski’s Triangle

In the last post on fractals, I gave examples to describe the concept of fractals. Fractals are figures with fractional dimesions.

One of the simplest of them is the Sierpinski’s Triangle.

Sierpinski's Triangle

Constructing a triangle

1. Start with any triangle in a plane (any closed, bounded region in the plane will actually work). The canonical Sierpinski triangle uses an equilateral triangle with a base parallel to the horizontal axis
2. Shrink the triangle to ½ height and ½ width, make three copies, and position the three shrunken triangles so that each triangle touches the two other triangles at a corner . Note the emergence of the central hole – because the three shrunken triangles can between them cover only 3/4 of the area of the original. (Holes are an important feature of Sierpinski’s triangle.)
3. Repeat step 2 with each of the smaller triangles.

The Sierpinski triangle has dimension log(3)/log(2) ≈ 1.585, which follows from the fact that it is a union of three copies of itself, each scaled by a factor of 1/2. (and hence a fractal)

Leaving you with an applet on Sierpinski’s triangle. Try it and you can see the triangle build step by step.