Hilbert's Question and Proof Systems

     In the year 1900 David Hilbert asked (among a set of 23 questions/problems) if there is a program that takes as input any given statement and decides whether it is true or not, i.e, it should output either a true or false. This will immediately bring to our mind the questions about automated reasoning,artificial intelligence pattern recognition and such other themes. However the above mentioned question came up in the context of mathematical logic and one should keep in mind the kind of computational power we had back then in the early 1900’s. In Mathematics we generally deduce statements from a set of axioms or priorly ascertained facts. So one can imagine feeding a database of mathematical facts and their interrelations and from this create a system to check whether a certain mathematical statement or conjecture is true or false. So we shall call this elusive procedure demanded by David Hilbert as ‘Hilbert’s program’. In fact a famous result in Mathematics namely the the four color problem was approached in a similar fashion.

It was K. Godel and A. Turing who showed the way toward solving Hilbert’s question and finally it was known that there were logical obstructions to get the Hilbert’s program. Godel showed that the axioms of arithmetic on which proof systems depend sometimes lead to paradoxes thus asserting that there is the so called Incompleteness in logical reasoning. Well Godel and Turing worked independently under different contexts but essentially came to the same conclusion. For more on this one can read [1].

However an Indian origin scientist Madhu Sudan developed the so called probabilistically checkable proof system which uses probability theory to check the correctness of a given proof. The crux of the matter lies in his PCP Theorem (probabilistically checkable proof theorem) for which he was awarded the Rolf Nevanlinna prize in 2002.The PCP theorem says that for some universal constant K, for every n, any mathematical proof of length n can be rewritten as a different proof of length poly(n) that is formally verifiable with 99% accuracy using a randomized algorithm.

[1] H. Ramesh, V. Vinay “Who will win the toss?” Resonance Journal of Science Education, April 1998.
ALSO AVAILABLE ON MY CONNECTRONIX BLOG

On the Abel Prize to L. Nirenberg and John Nash

On the Abel Prize to L. Nirenberg and John Nash

Many in the Math-community feel let down that there is no Nobel for Mathematicians. But in fact there are several prizes to encourage budding mathematicians. One such is the Abel Prize which people say is an answer to the question of ‘no Nobel for math’. The prize money is also incredibly attractive!
On May 19 2015, Louis Nirenberg and John Nash received the famed Abel prize which includes a citation, gold medal and 6 Million NOK (Norwegian Currency Norway Kroner) which approximately amount to Rs 5,00,00,000. The citation says “For striking and seminal contributions to the theory of non-linear PDE’s and it’s applications to geometric analysis”
Solutions to certain differential equations arising in nature were hard to find and then one
George de Rham (a Swiss mathematician) gave the so called weak solution. The works of Louis Nirenberg and John Nash show that these solutions can be rendered ‘regular’. John Nash(of the Beautiful Mind fame) besides the work on game theory also worked on Holder estimates for the solutions of linear elliptic equations in general dimension without any regularity assumptions. This led to the solution of Hilbert’s 19^th problem. Similarly Nirenberg made beautiful application of the so called maximum principles which refer to certain analytic results on the attainment of maximum bounds on some small domains in the context of non-linear elliptic partial differential equations.
Applications of these works include the solution of the prescribed curvature problem in geometry, the Navier-Stokes problems, stability of the GNS inequalities(named after Gagliardo-Nirenberg-Sobolev) and problems involving GR(General Relativity) in cosmology.
Other recipients include S. R. S. Varadhan, Yakov Sinai, and Peter Lax. The latest recipient of the Abel Prize is a leading women mathematician- Karen Uhlenbeck of the University of Texas at Austin.

Squares and Arithmetic Progressions

Squares and Arithmetic Progressions
Observe the Squares 1², 5², 7².. these numbers viz . . 1, 25 and 49 are in an Arithmetic progression. So one wonders if there are more such examples. Consider one more such triple 7², 13² and 17²… looks like there can be infinitely many. We shall try proving such a statement .. and what if we look for 4 such squares?

let a², b²,  and c² be in arithmetic progression so that b²-a²=c²-b²,

or 1-(a/b)²= (c/b)²-1.

For convenience we write x=a/b, y=c/b.

This implies 1-x²= y²-1. Factorising we get

(1+x)/(y+1)=(y-1)/(1-x).........(1)

let t be the common value of (1) which gives us

1+x=t(y+1)

 AND y-1=t(1-x)

OR

x-ty=t-1.................[2]

tx+y=t+1..............[3]

solving for x and y we get the following parametrization

x=(t²+2t-1)/(t²+1)

y=(-t²+2t+1)/(t²+1).
For t=1/2 we get x=1/5, y=7/5 so that we get 1, 1/25 and 49/25. Clearing denominators we get 1, 25 and 49...the triple we started with. 

Similarly taking some other value for 't' we can arrive at other triples of squares that form an A.P. For instance t=2/3 leads us to 49, 169 and 289..the common difference being 120. The parametrization we have arrived at implies that we can get an infinitude of such triples!!.

Next we ask the question if there are 4 such square numbers which form an A.P. This question was asked by Fermat and the answer is NO. One cannt get hold of 4 squares satisfying such a property. The proof for this assertion was first given by Leonhard Euler!!

ADAPTED FROM THE BOOK-Numbers and the beginning of Algebra By Shailesh Shirali [Little Mathematical Treasures]

Life of a Mathematician..

By the title which sounds like "Life of Pi"you may be wondering if you are here to read a biography or may be my own...I mean Autobiography.But no I do not intend to chronicle a single mathematician in this article although I have done it earlier.
Several people want to know what exactly a mathematician is involved in and this write-up is to clarify this point. Some of you may be thinking we go on cooking up formulae and some others may be thinking that we supply accurate measures to engineers and other technocrats and so on. People have their own imagination. Well one single answer will not represent the entire class of mathematicians. For instance for some cases the first guess written above may be true...yes there are mathematicians who try to discover new formulae or equations. For instance an Indian mathematician S.Minakshisundaram (collaborating with A.Pleijel in the 1950's approximately) gave formulae for trace of a certain operator defined on a Riemannian manifold...This is something to do with harmonics of a vibrating membrane at least for some special cases. There are some other mathematicians who develop a certain philosophy leading to a paradigm shift in the way a class of problems can be looked at. An example for this is one Professor Alexander Grothendieck's work.
Anyways the information to be passed on here is that Mathematics is a huge landscape and there are several varieties of mathematicians one can find. Their work typically involves pen and paper and a lots of discussion and of course coffee. One may be remembering this joke that Mathematicians are the ones who turn coffee cups into theorems!
I recently saw this discussion as to what is a typical day of a mathematician like...? Yes this was a question raised by the commoners to a group of professional mathematicians. He works just as any employee say a software technocrat or a banker. Typically they work for about 8 hours a day but there is more than that actually.Well their talk and their behavior is just like anybody else's except that the mathematician thinks even while he is driving or may be even while he is purchasing vegetables and so on. But the most important thing is that once this thought gets more and more crystallized he gets back to his cabin (either at home or at his office) and writes it out and then refines the thought process and rewrites the stuff and often turns it up into a theorem, a proposition, a calculation or just philosophy!! Now it is not that this is the case every single day. It may so happen that for many days or sometimes even for many months there is no such pure thought coming up. But then some flashes occur suddenly may be in a shopping mall while gazing at a horizon and the excitement comes back...

Legendary Mathematician Donald Knuth

LEGENDARY DONALD KNUTH Computers can be quite useful for a Mathematician. This is exemplified by the legendary Mathematician and computer Scientist Don Knuth (pronounced KANNOOTH) Donald Knuth is a Mathematician who is more popular among computer Science students for his contributions to sementics and algorithms. For mathematicians anybody who uses the “Tex” environment for typing mathematical manuscripts owes a lot to Knuth.(I should mention that there is a lot of geometry in action when one looks at the way “fonts” are designed) As a student Knuth showed great enthusiasm for puzzles and problems. When confectionary manufacturer Ziegler organised a competition to form maximum number of words using the letters from the phrase 'Ziegler's Giant Bar', Don Knuth was the obvious winner making up 4500 words while the judges of the competition had only 2500. While at Case Institute of Technology, Cleveland Ohio, Don Knuth already started using the IBM-150. He was analysing graphs of various surfaces described in several dimensions. He also used the IBM-150 for analysing the performance of his Basket ball team. So IBM started using his name and picture to advertise the capabilities of the machine. His brilliance was such that his college awarded him an MS degree while he had just cleared his BS. Don Knuth wrote a Research paper while still a student “On methods of constructing sets of mutually orthogonal latin Squares”. He thus entered the Caltech with a distinction of having research experience well in advance. He joined Caltech in 1960 and he got his PhD in 1963 working on the thesis “ On Finite Semifields and Projective Planes” and joined Burroughs Corporation. He had a fascination for numerical analysis. He computed euler's constant correct to 1271 decimal places. He was also interested in evaluation of polynomials using computers. In computer Science he is famous for his work on compiler design and syntax development, He is the joint collaborator for the Knuth-Morris-Pratl string searching algorithm. He also contributed to the LR(k) grammer and syntax semantics in VLSI design. He thus earned the titles : “Euler of Computer Science” and “The Father of Analysis of Algorithms” His book “Art of Computer Programming” was named among the 12 best Physical Sciences Monographs along with Dirac's Quantum Mechanics and Einstein's Relativity. He also wrote a Science fiction novel “ Surreal Numbers”. He is currently Professor Emeritus at Stanford University.

Selected Expositions of Great Mathematicians-Arnold Ross

Selected Expositions of Great Mathematicians-Arnold Ross
Arnold E.Ross born in Chicago (1906, a year after the Annus Mirabilis), only child of Jewish emigrants from the Ukraine, was an extraordinary human being and a researcher in Mathematics. In a feature article that appeared in the Notices of the American Mathematical Society (Vol-48, #7) August 2001, he was described to be a mathematician always a step ahead organising a program to improve a teacher's mathematical knowledge. In all his endeavours his aim was to kindle a passion for intellectual challenges.
When once asked about the early influences in his life he refersed to his mother and said that she made him feel the mystery of language as a tool for communication. His father wanted him to be an engineer, but he wanted to become a mathematician. At the tender age of 17 he was confronted with a difficult situation. “ If you study engineering I will help you, if you want to be a mathematician you can starve on your own”said his father....
Thus at that tender age he went on to earn his tuition fees by working in a book binding shop and thus landed into the Mathematics department at University of Chicago. Luckily for him Prof.E.H.Moore took special care and attention to hone his skills in basic concepts and thus the young boy later turned into a great number theorist. Among the achievements of Ross, one of his students became the first black woman to receive an M.S in Mathematics. Arnold Ross is most famously remembered for the Ross Summer Mathematics Program funded by the Ohio state University and the National Science foundation of the US. This was a program aimed at deepening the mathematical understanding of high school and junior college teachers.
This program of Ross was so influential that there was a spill over effect leading to several similar programmes aimed at fostering Mathematical talent in the nation. Several Mathematicians of a very high calibre were inspired by his energy and enthusiasm to make mathematical exposition more penetrating.

Opportunities in Mathematics and allied areas.

This article is in response to queries raised by some of my students. Yes Mathematics has come of age and there are lots of opportunities if you have a problem solving bent of mind, creativity and curiosity in patterns. Provided one takes all of these qualities seriously, pursuing mathematics can be quite rewarding. Here is a list of different careers (alphabetical order) one can think of after being graduated in “Mathematics”
1.Actuaries 2.Aerospace modeling 3.Bioinformatics 4.Computational Finance 5.Computer related modeling6.Computer Animation and Digital Imaging 7.Communication service providers 8.Cryptology 9.Engineering Research 10.Electronic related analysis 11.Financial Investment Services 12.International government and non-government agencies 13.Robotics
I can explain some of the above terms. In Actuarial Sciences one requires an understanding of the dynamics of funds that get accumulated by way of premia and then if the distribution of claims is analysed on a probabilistic basis then you can contribute toward the progress of insurance firms. Bioinformatics requires the modeling of various objects that life scientists study. Even the delivery of molecules to the targeted organ is studied via mathematical models. This is just one aspect I can relate to. Finance has a big role to offer to Math enthusiasts. In fact there is this fancy term making rounds these days ..”Financial Engineering”. Many processes involved in Investments and their returns involve dynamical systems requiring mathematical analysis. For example Wall Street firms employ a large number of Mathematicians. Even RBI has several mathematicians and working. Engineering firms like Hewlett Packard, AT-and T Bell Labs have been seeking services of Mathematicians. One can see so many names related to the mathematical world in new algorithms invented in the computer world.For example Google Inc, Microsoft Research and Yahoo Research teams have good math experts. By International agencies I mean those which serve the world community at large with constructive research and applications. The NASA for example is one among this category. The NSA is another organization which has a large number of mathematicians working. Likewise the Defence Research organizations, Aerospace modelers require huge inputs from Mathematicians. Cryptology is a term that includes a set of processes required to make possible reliable communication of digital data. This has as its main components coding schemes and encryption/decryption algorithms.
All these are apart from the mainstream occupations that a majority of Mathematicians love to do –Teaching and Research.
For further reading check out the following:
BLOOMBERG BUSINESS WEEK REPORT:
Math will rock your world http://www.businessweek.com/magazine/content/06_04/b3968001.htm
WALL STREET REPORT:
Mathematicians take top spot in job ranking study-Wall street journal online
Association for Women in Mathematics:
Read their career advice: http://sites.google.com/site/awmmath/awm-resources/career