# Category Archives: Personal Experiences

My mathematics related experiences

# Bye

Standard

I won’t be writing new blog posts here anymore.

All the past blog posts and static pages will be available in read-only format (i.e. no new comments can be made on this website).

# Thank you, Dr. Majumder

Standard

Dr. Jaydeep Majumder (07 June 1972 – 22 July 2009)

Recently I finished the first part of my master’s thesis related to (complex) algebraic geometry. There are not many (useful) books available on this topic, and most of them are very costly. In fact, my college library couldn’t buy enough copies of books in this topic. However, fortunately,  Dr. Jaydeep Majumder‘s books were donated to the library and they will make my thesis possible:

Principles of Algebraic Geometry by Joseph Harris and Phillip Griffiths

Algebraic Curves and Riemann Surfaces by Rick Miranda

Hodge Theory ans Complex Algebraic Geometry – I by Claire Voisin

While reading the books, I assumed that that these books were donated after the death of some old geometer. But I was wrong. He was a young physicist, who barely spent a month at NISER. A heart breaking reason for the books essential for my thesis to exist in the college library.

Dr. Majumder was a theoretical high energy physicist who did research in String Theory. He obtained his Ph.D. under the supervision of Prof. Ashoke Sen at HRI. He joined NISER as Reader-F in June 2009, and was palnning to teach quantum mechanics during the coming semester. Unfortunately, on 22 July 2009 at the young age of 37 he suffered an untimely death due to brain tumor.

I just wanted to say that Dr. Majumder has been of great help even after his death. The knowldege and good deeds never die. I really wish he was still alive and we could discuss the amazing mathematics written in these books.

# Enclosing closed curves in squares

Standard

Let’s look at the following innocent looking question:

Is it possible to circumscribe a square about every closed curve?

The answer is YES! I found an unexpected and interesting proof in the book “Intuitive Combinatorial Topology ” by V.G. Boltyanskii and V.A. Efremovich . Let’s now look at the outline of proof for our claim:

1. Let any closed curve K be given. Draw any line l and the line l’ such that line l’ is parallel to l as shown in the fig 1.

2. Move the lines l and l’ closer to K till they just touch the curve K as shown in fig 2. Let the new lines be line m and line m’. Call these lines as the support lines of curve K with respect to line l.

3. Draw a line l* perpendicular to l and the line (l*)’ parallel to l* . Draw support lines with respect to line l* to the curve K as shown in the fig 3. Let the rectangle formed be ABCD .

4. The rectangle corresponding to a line will become square when AB and AD are equal . Let the length of line parallel to l (which is AB)  be $h_1(\mathbf{l})$ and line perpendicular to l (which is AD) be $h_2(\mathbf{l})$. For a given line n, define a real valued function $f(\mathbf{n}) = h_1(\mathbf{n})-h_2(\mathbf{n})$ on the set of lines lying outside the curve .  Now rotate the line l in an anti-clockwise direction till l coincides with l’. The rectangle corresponding to l* will also be ABCD (same as that with respect to l). When l coincides with l’, we can say that  $AB = h_2(\mathbf{l^*})$ and $AD = h_1(\mathbf{l^*})$.

5. We can see that when the line is l$f(\mathbf{l}) = h_1(\mathbf{l})-h_2(\mathbf{l})$. When we rotate l in an anti-clockwise direction the value of the function f changes continuously i.e. f is a continuous function (I do not know how to “prove” this is a continuous function but it’s intuitively clear to me; if you can have a proof please mention it in the comments). When l coincides with l’ the value of $f(\mathbf{l^*}) = h_1(\mathbf{l^*})-h_2(\mathbf{l^*})$. Since $h_1(\mathbf{l^*}) = h_2(\mathbf{l})$ and $h_2(\mathbf{l^*}) = h_1(\mathbf{l})$. Hence $f(\mathbf{l^*}) = -(h_1(\mathbf{l}) - h_2(\mathbf{l}))$. So f is a continuous function which changes sign when line is moved from l to l’. Since f is a continuous function, using the generalization of intermediate value theorem we can show that there exists a line p between l and l* such that f(p) = 0 i.e. AB = AD.  So the rectangle corresponding to line p will be a square.

Hence every curve K can be circumscribed by a square.

# Evolution of Language

Standard

We know that statistics (which is different from mathematics) plays an important role in various other sciences (mathematics is not a science, it’s an art). But still I would like to discuss one very interesting application to linguistics. Consider the following two excerpts from an article by Bob Holmes:

1. ….The researchers were able to mathematically predict the likely “mutation rate” for each word, based on its frequency. The most frequently used words, they predict, are likely to remain stable for over 10,000 years, making these cultural artifacts, or “memes”, more stable than some genes…..

2. ….The most frequently used verbs (such as “be”, “have”, “come”, “go” and “take”) remained irregular. The less often a verb is used, the more likely it was to have been regularised. Of the rarest verbs in their list, including “bide”, “delve”, “hew”, “snip” and “wreak”, 91% have regularised over the past 1200 years…….

The first paragraph refers to  the work done by evolutionary biologist Mark Pagel and his colleagues at the University of Reading, UK. Also, “mathematically predicted” refers to the results of the statistical model analysing the frequency of use of words used to express 200 different meanings in 87 different languages. They found the more frequently the meaning is used in speech, the less change in the words used to express it.

The second paragraph refers to the work done by Erez Lieberman, Jean-Baptiste Michel and others at Harvard University, USA.  All people in this group have mathematical training.

I found this article interesting since I never expected biologists and mathematicians spending time on understanding evolution of language and publishing the findings in Nature journal. But this reminds me of the frequency analysis technique used in cryptanalysis:

# Allostery

Standard

Cosider the folowing definiton:

Allostery is the process by which biological macromolecules (mostly proteins) transmit the effect of binding at one site to another, often distal, functional site, allowing for regulation of activity.

Many allosteric effects can be explained by the concerted MWC model put forth by Monod, Wyman, and Changeux, or by the sequential model described by Koshland, Nemethy, and Filmer.  The concerted model of allostery, also referred to as the symmetry model or MWC model, postulates that enzyme subunits are connected in such a way that a conformational change in one subunit is necessarily conferred to all other subunits. Thus, all subunits must exist in the same conformation. The model further holds that, in the absence of any ligand (substrate or otherwise), the equilibrium favors one of the conformational states, T (tensed) or R (relaxed). The equilibrium can be shifted to the R or T state through the binding of one ligand (the allosteric effector or ligand) to a site that is different from the active site (the allosteric site). [Wikipedia]

In this post, I want to draw attention towards application of mathematics in understanding biological process, allostery. Consider the following equation which relates the difference between $n$, the number of binding sites, and $n'$, the Hill coefficient, to the ratio of the ligand binding function, $\overline{Y}$, for oligomers with $n-1$ and $n$ ligand binding sites

$\displaystyle{\boxed{n-n' = (n-1) \frac{\overline{Y}_{n-1}}{\overline{Y}_n}}}$

This is known as Crick-Wyman Equation  in enzymology, where $\displaystyle{\overline{Y}_n = \frac{\alpha(1+\alpha)^{n-1}+ Lc\alpha(1+c\alpha)^{n-1}}{(1+\alpha)^n+L(1+c\alpha)^n}}$ and $\displaystyle{n' =\frac{d( \ln(\overline{Y}_n) - \ln(1-\overline{Y}_n))}{d\ln\alpha}}$; $L$ is allosteric constant and $\alpha$ is the concentration of ligand under some normalizaton conditions.

For derivation, see this article by Frédéric Poitevin and Stuart J. Edelstein. Also, you can read about history of this equation here.

It’s not uncommon to find simple differential equations in biochemistry (like Michaelis-Menten kinetics), but the above equation stated above is not a kinetics equation but rather a mathematical model for a biological phenomina. Comparable to the Hardy-Weinberg Equation discussed earlier.

# New Proofs on YouTube

Standard

Earlier, YouTube maths channels focused mainly on giving nice expositions of non-trivial math ideas. But recently, two brand new theorems were presented on YouTube instead of being published in a journal.

• Proofs of the fact that $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$ and $\sqrt{6}$ are irrational numbers – Burkard Polster (13 April 2018)

This is an extension of the idea discussed in this paper by Steven J. Miller and David Montague.

• A new proof of the Wallis formula for π – Sridhar Ramesh and Grant Sanderson (20 Apr 2018):

This is an extension of Donald Knuth‘s idea documented here by Adrian Petrescu.

It’s nice to see how the publishing in maths is evolving to be accessible to everyone.

# Sum of squares

Standard

In the past few posts, I have talked about representing integers as a sum of squares:

In this post, I would like to state Lagrange’s four-square theorem following section 6.4 of Niven-Zuckerman-Montgomery’s An introduction to the theory of number.

Firstly, by applying Hensel’s lemma to the result from the earlier post we get (Theorem 5.14):

Proposition: Let $a,b,c$ be arbitrary integers. Then the congruence $ax^2+by^2+cz^2\equiv 0\pmod{p}$ has a non-trivial solution modulo any prime $p$.

The theorem stated in the earlier post establishes that there is no need for any condition modulo primes p not dividing abc. The above proposition, application of Hensel’s lemma, just demonstrates it more explicitly by telling that the equation is solvable everywhere locally (i.e. modulo every prime).

Secondly, we need following result from Geometry of numbers (Theorem 6.21):

Minkowski’s Convex Body Theorem for general lattices: Let $A$ be a non-singular $n\times n$ matrix with real elements, and let $\Lambda = A\mathbb{Z}^n=\{A\mathbf{s}\in \mathbb{R}^n: \mathbf{s}\in \mathbb{Z}^n\}$ be a lattice. If $\mathcal{C}$ is a set in $\mathbb{R}^n$ that is convex, symmetric about origin $\mathbf{0}$, and if $\text{vol}(\mathcal{C})> 2^n |\det(A)|$, then there exists a lattice point $\mathbf{x}\in\Lambda$ such  that $\mathbf{x}\neq 0$ and $\mathbf{x}\in \mathcal{C}$.

Now we are ready to state the theorem (for the proof see Theorem 6.26):

Lagrange’s four-square theorem: Every positive integer $n$ can be expressed as the sum of four squares, $n=x_1^2+x_2^2+x_3^2+x_4^2$, where $x_i$ are non-negative integers.