Monthly Archives: June 2016

Galton Board

Standard

Like previous post, in this post I will discuss another contribution of Jacob (Jacques) Bernoulli. The motivation for this post came from Cédric Villain’s recent TED talk. Though I am not a fan of probability theory, but this “toy”, which I am going to discuss, is really interesting.  Consider following illustration from a journal’s cover:

“Galton Board” was invented by Francis Galton in 1894.  It provided a remarkable way to visualize the distribution obtained by performing several Bernoulli Trials in pre-digital computer era.  Bernoulli trial is the simplest possible random experiment with exactly two possible outcomes, “success” and “failure”, in which the probability of success (say, p) is the same every time the experiment is conducted.  If we perform these Bernoulli trials more than one time (say, n times) we get, what we call, Binomial Distribution. We get a discrete distribution like this:

And  when the number of Bernoulli trials is very large (theoretically what we would call infinite number of trials), this Binomial Distribution can be approximated to Normal Distribution, which is a continuous distribution.

The Normal Distribution is important because of the Central Limit Theorem. This theorem implies that if you have many independent variables that may be generated by all kinds of distributions, assuming that nothing too crazy happens, the aggregate of those variables will tend toward a normal distribution. This universality across different domains of science makes the normal distribution one of the centerpieces of applied mathematics and statistics.

Here is a video in which James Grime demonstrates how Galton Board can be used to visualize Normal Distribution approximation of Binomial Distribution for very large number of Bernoulli trials. The trial outcome are represented graphically as a path in the Galton board: success corresponds to a bounce to the right and failure to a bounce to the left.

 

 

 

 

Bernoulli Numbers

Standard

I have referred to them twice (here and here) so far. Also, the Euler-Maclaurin formula I discussed in second post, explains a lot about their occurrences (for example). Now I think it’s time to dive deeper and try to understand them.

In 1631, Johann Faulhaber   published Academia Algebra (it was a German text despite the Latin title). This text contains a generalisation of sums of powers, which in modern notations reads:

\displaystyle{\sum_{m=1}^{n} m^{k-1} = \frac{1}{k}\left[n^k + \binom{k}{1} n^{k-1} \times \frac{1}{2} + \binom{k}{2}n^{k-2} \times \frac{1}{6} +\binom{k}{3}n^{k-3} \times 0+\binom{k}{4}n^{k-4} \times \frac{-1}{30}+ \ldots\right]}

Observe that the expression on the right hand side in square brackets appears like binomial expansion, but there are some constant terms multiplied to them (which can also be 0). These constant terms were named “Bernoulli Numbers” by Abraham de Moivre, since they were intensively discussed by Jacob (Jacques) Bernoulli in Ars Conjectandi published in Basel in 1713, eight years after his death.

I will follow the notation from The book of numbers (published  in 1996). So we will denote n^{th} Bernoulli number by B^{n} where

\displaystyle{B^0 = 1, B^1 = \frac{1}{2}, B^2 = \frac{1}{6}, B^3= B^{5} = \ldots = B^{odd} = 0, B^4 = B^8 = \frac{-1}{30}, B^6 = \frac{1}{42}, B^{10} = \frac{5}{66}, \ldots}

This notation enables us to calculate sum of k^{th} power of first n natural numbers quickly. We can re-write above summation formula as:

\displaystyle{\sum_{m=1}^n m^{k-1} = \frac{(n+B)^k - B^k}{k}}

To illustrate, how to use this formula, let’s calculate sum of 5^{th} powers of first 1000 natural numbers:

\displaystyle{\sum_{m=1}^{1000} m^{5} = \frac{(1000+B)^6 - B^6}{6}}

\displaystyle{ = \frac{1}{6}\left[1000^6 + 6B^1 1000^5 + 15B^2 1000^4 + 15 B^4 1000^2\right]}

So, we have done binomial expansion of the right hand side and used the fact that B^{odd>1} = 0. Now we will replace corresponding values of Bernoulli Numbers to get:

\displaystyle{\sum_{m=1}^{1000} m^{5} =\frac{1}{6}\left[10^{18} + 3\times 10^{15} + 2.5 \times 10^{12} - 0.5 \times 10^6\right]=\frac{1003002499995\times 10^5}{6}}

1^5 + 2^5 + \cdots + 1000^5 = 16716708333250000

(This answer was cross-checked  using SageMath)

There are many ways to calculate the value of Binomial numbers, but the simplest one is to using the recursive definition:

(B - 1)^k = B^k for k>1, gives value of B^{k-1}

There is another definition of Bernoulli  Numbers using power series:

\displaystyle{\frac{z}{e^z-1} = \sum_{k=0}^{\infty} \frac{B^k z^k}{k!}}

This gives slightly different sequence of Bernoulli numbers, since in this B^{1}=\frac{-1}{2}, and the recursive definition is

(B+1)^{k} = B^{k} for k>1, gives value of B^{k-1}

This definition can be used to calculate val value of  \tan(z),  since its infinite series expression has Bernoulli numbers in coefficients.

\displaystyle{\tan(z)=\sum_{n=0}^{\infty}\frac{B^{2n}(-4)^n(1-4^n)}{2n!}z^{2n-1}}

 

135th Carnival of Mathematics

Standard
image

Welcome to the carnival…

Before we begin, let’s know our number-friend of this month. 135 is the smallest three-digit number that is the sum of its first digit and the square of its second digit and the cube of its third digit: \displaystyle{135 = 1 + 3^2 + 5^3}.

Now, get hold of pen and paper, the carnival begins…

carnivalofmathematics_nonwhite-background

 

Cross-Number Puzzle – Numbrcise.com

This is a “medium” brainteaser to help you warm up. To play, you must fill in all the blank squares in the grid with numbers ranging from 1 to 9 in order to find a synchronized solution to a series of horizontal and vertical equations – all at once.


Three dimensional tessellation of crosses – circlesandtriangles (Dan)

Day after tomorrow is 118th birthday of Maurits Cornelis Escher and will be celebrated as first “Tessellation Day” [details]. And in this article, Dan investigates the three dimensional analogue of Greek Cross tessellation using animations. He elegantly explains the mathematics involved behind tessellation by reducing  the 3D problem to a 2D one.

Snell and Escher – Joshua Bowman

When we study the concept of refraction, one of the central law is “Snell’s Law”. Motivated by Johann Bernoulli’s (a mathematician) clever application of Snell’s Law to solve brachistochrone problem,  Joshua introduces reader to hyperbolic geometry via works of Maurits Cornelis Escher  (an artist) using Snell’s Law! I have many times discussed hyperbolic geometry with young  minds, but this introduction is really illuminating.

5 Mistakes to Avoid when Drawing a Soccer Ball – David Swart

At some point of time, you must have tried your hand at drawing various objects around you. Moreover, soccer is claimed to be most played sports in the world. But still many of us can’t draw a soccer ball properly! In this lovely article, David helps us to mathematically identify and correct these mistakes…


Boolean Algebra – mathsbyagirl

We all use electronic devices and they  will not work without the transistors linked together on a silicon chip. These transistors interact with each other using “on” or “off” state. This interaction can be mathematically modelled using “Boolean Algebra”. In this article, the author has successfully transmitted her passion and excitement for the subject to other people. This article is accessible to anyone curious enough to learn about this subject.

Banach-Tarski Paradox  –  Austin Lawson

If you try to take a stroll through “mathematical logic”, you will encounter the statements that contradict themselves and yet might be true.  Such statements are called paradoxes. In this article, Austin gives a sketch of the Banach-Tarski theorem and paradox.  It is intended for mathematicians of intermediate skills though the first few paragraphs are accessible to more people.


Programming is a piece of cake  – Paula Rowinska

This is an article discussing the importance of computing in the modern maths, especially the applied areas. There’s a common misconception that mathematicians work only on very pure and useless for the society problems – Paula explains that it’s a myth.

How to assign partial credit on an exam of true-false questions? Terence Tao

Many of us would have appeared for exams in May, and hence this article by a gifted mathematician becomes oddly relevant! In this article Prof. Terence does a serious analysis of grading system using techniques from Probability and Statistics. In case you happen to be a person associated with academia, this is worth reading.


Float like butterfly and Sting like a Mathematician!  – Nira Chamberlain

Muhammed Ali has been regarded as the most influential sportsman of the century. A week ago, he died aged 74. This is intriguing story about how a heavyweight boxing champion influenced Dr. Nira to become a mathematician.

Interview with a mathematician: Maria Chudnovsky  – Anthony Bonato

Graph Theory is one of those branches of Mathematics which consist of simple to state but difficult to solve problems. In case you are not familiar with it, just spend some time on this “Math for seven-year-olds” kit. Prof. Anthony interviewed Maria Chudnovsky, a leading mathematician specializing in graph theory. Maria is famous for proving the “Strong Perfect Graph theorem”, which was open for forty years. Maria is engaging and gives great advice to young mathematicians. She also talks about her upbringing, including facts you can’t find on her wiki page or other interviews.


Prime After Prime  – Brian Hayes

There’s lots about the prime numbers that seems random; you can even play a good game of dice with them. But in March Robert J. Lemke Oliver and Kannan Soundararajan discovered some remarkable biases or correlations between pairs of consecutive primes. Brain explores this discovery in computer code and “very illuminating” pictures.

Chi-square goodness of fit test example and prime numbers – John Cook

This article builds upon Brian Hayes‘ article about the distribution of primes, which we just discussed.  Motivated by the argument that “Primes aren’t random, but sometimes it can be useful to treat them as if they were.”, John writes about the connection between statistics and number theory. Mathematically mature audience will surely enjoy reading this.


Maximal density subsquare-free arrangements – Peter Karpov

I will end this Carnival with an open problem. In this article, Peter discusses some computational results for the “no subsquares problem”. The statement of problem is as follows

What is the largest number of points that can be placed on a N × N grid so that no four of them form a square?

Get your hands dirty and try to find an efficient algorithm to calculate answer for N>16!

 


 

But, before we say good bye to this edition of monthly roundup, let me remind you to visit the previous edition which was hosted by Kartik at Comfortably Numbered. Join us next time for the 136th edition, hosted by Manan at Math Misery. You can find all the other Carnivals, and submit articles for future carnivals at: http://aperiodical.com/carnival-of-mathematics/ 

Why I want to be a Mathematician?

Standard

A few weeks ago I revised my views about “Why I love Mathematics?“. It has been two years since I have been trying to get into mainstream mathematics research. It will take another four to five years for me to start contributing to mainstream mathematics research. In this post I will try to add a bit more to what I wrote an year ago.

Though mathematics is the main thing my life revolves around,  it’s not the only thing. I love doing few other non-mathematical things. Few months ago I watched the movie “Ship Of Theseus” by Anand Gandhi , and it is one of the few movies I have agreed to watch twice. In case you are curious to know about Theseus’ paradox, I will suggest this ted-ed video

As you may be aware of, there are a lot of people who are not mainstream mathematicians (i.e. working in a Research Organization in a specific research area) but still claim to love mathematics. They are called recreational-mathematicians (like Dattaraya R. Kaprekar, Tanya Khovanova,…), maths-popularizer (like Eric Temple Bell, Constance Reid,
Simon Singh,… ), maths-historians (like Bartel Leendert van der Waerden, Jacqueline Anne Stedall,… ) etc.  But I want to become a mainstream-mathematician (like full time professors in research organizations). Why?

I want to become immortal (i.e. to exist as long as humans exist).

You may be think that I have lost my mind, but please continue reading…

I hope we agree that our wish to live (as opposed to fear of death) motivates us to live. Paul Erdős used to tell following story about his second discovery as a child (first being that of negative numbers):

I knew I would die. From then on, I’ve always wanted to be younger. In 1970, I preached in Los Angeles on ‘my first two and a half billion years in mathematics.’ When I was a child, the Earth was said to be two billion years old. Now scientists say it’s four and a half billion. So that makes me two and a half billion….

I believe that Paul Erdős has indeed gained immortality, we just keep listing about his conjecture being proved now and then (recent one: Erdős-Rado sunflower problem).

From a biologist’s point of view (by the way, I also study a bit of Biology), our body along with our consciousness defines us as an individual. So, for a human to become  immortal his/her body as a whole must be preserved as it is. But, biologists have faced a dilemma of  conserving body versus consciousness. If you preserve body, by eliminating defects from our body at DNA level (since not every organ can be transplanted) , the you lose the distinctiveness in personalities (like clones) since all will be perfect and thus identical (causing threat to evolution). If you preserve consciousness, by transferring it to an artificial body (which will become reality with advent of quantum computing), then you lose your body (a major part of your personality).

chappie17

Consciousness transferred to a robotic body in the sci-fi movie Chappie (© 2015, Sony Pictures Entertainment)

But, I believe that the only way to become immortal is by publishing (articles, books, movies, songs,…) and propagating (lectures, discussions,…) your ideas among others. As we know that our body is immortal in a sense that all its atoms remain as such (law of conservation of mass, energy…), and form various molecules like the molecules of life. For example, say an animal dies in a forest (without human intervention). After few days the microorganisms inside & outside the body (which are much more than the number of cells of that animal) will start decaying the body and release different chemicals. These chemicals will attract different insects which will start consuming the body and finally scavengers will completely clean the flesh part. The bones will take longer time depending on environmental conditions. Now these insects and scavengers will be consumed by bigger animals (and eventually, may be, by humans). In this way the atoms from the dead animal will disperse among various life-forms but will never cease existence. In case of humans, we make this process to take longer time by doing various rituals. So, if I am able to propagate all of my ideas (which will also evolve over time), then I am immortal.

Screenshot_20160520-211451-2

Goddess Saraswati (symbol of knowledge in Hindu Mythology) used to popularize a mathematics conference in Belgium (url: http://www.mathconf.org/app-gvl-summer2016)

The  immortality “ideology” which I want to adopt is actually what various civilizations have done to make their gods/goddesses immortal! You make a story about a character in written form (which they called “sacred texts”) and propagate among others (which they called “religions”). After some time the stories become part of our life and characters of that story become immortal. I view the “library of Alexandria” as power house of  Egyptian civilization, since it was a major center of scholarship. Same is true for eminent people (the people about whom biographies are written and movies are made) in modern society.

In my opinion, only mainstream researchers like scientists, psychologists, economists, etc. have an opportunity to gain immortality. Whereas people like non-innovative-teachers, librarians, science-popularizers, non-research physicians, non-research engineers etc. ensure immortality of others, just like the craftsman reproducing work  of ancestors again and again thus helping to keep the work alive.  So, all professions are about “collecting knowledge” but what makes researchers stand apart from other professions is their ability to “create knowledge“. So all professions are important but in different prospective. For example, if you want to become powerful, become politician and so on….

I admit that my thoughts may be very childish and I in future I may change  my opinion…