Evolution of Language

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:

Source: Enigma Cryptanalysis, Gaurish Korpal

Allostery

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 , the number of binding sites, and , the Hill coefficient, to the ratio of the ligand binding function, , for oligomers with and ligand binding sites

This is known as Crick-Wyman Equation  in enzymology, where and ; is allosteric constant and 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

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 , , and 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.

 

Probability Musing

Please let me know if you know the solution to the following problem:

What is the probability of me waking up at 10am?

What additional information should be supplied so as to determine the probability? What do you exactly mean by the probability of this event? Which kind of conditional probability will make sense?

Consider the following comment by Timothy Gowers regarding the model for calculating the probability of an event involving a pair of dice:

Rolling a pair of dice (pp. 6), Mathematics: A very short introduction © Timothy Gowers, 2002 [Source]

I find probability very confusing, for example, this old post.

Sum of squares

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

• Fermat’s two-square theorem
• Legendre’s three-square theorem
• Algorithms for finding the sum of squares representation

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 be arbitrary integers. Then the congruence has a non-trivial solution modulo any prime .

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 be a non-singular matrix with real elements, and let be a lattice. If is a set in that is convex, symmetric about origin , and if , then there exists a lattice point such  that and .

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

Lagrange’s four-square theorem: Every positive integer can be expressed as the sum of four squares, , where are non-negative integers.

Conway’s Prime Producing Machine

Primes are not randomly arranged (since their position is predetermined) but we can’t find an equation which directly gives us nth prime number. However, we can ask for a function (which surely can’t be a polynomial) which will give only the prime numbers as output. For example, the following one is used for MRDP theorem:

Source: Andrew Granville (http://www.dms.umontreal.ca/~andrew/PDF/PrimesLecture2014.pdf)

 

But it’s useless to use this to find bigger primes because the computations are much more difficult than the primality tests.

Conway’s PRIMEGAME takes whole numbers as inputs and outputs if and only if is prime.

Source: https://cstheory.stackexchange.com/a/14727 [Richard Guy, © 1983 Mathematical Association of America]

 PRIMEGAME is based on a Turing-complete esoteric programming language called FRACTRAN, invented by John Conway. A FRACTRAN program is an ordered list of positive fractions together with an initial positive integer input n. The program is run by updating the integer as follows:

1. for the first fraction in the list for which is an integer, replace by ;
2. repeat this rule until no fraction in the list produces an integer when multiplied by n, then halt.

PRIMEGAME is an algorithm devised to generate primes using a sequence of 14 rational numbers:

Starting with 2, one finds the first number in the machine that multiplied by 2 gives an integer; then for that integer we find the first number in the machine that generates another integer. Except for the initial 2, each number output have an integer for a binary logarithm is a prime number, which is to say that powers of 2 with composite exponents don’t show up.

If you have some knowledge of computability and unsolvability theory, you can try to understand the working of this Turing machine. There is a nice exposition on OeisWiki  to begin with.

“Hilbert’s 10th Problem” by Martin Davis and Reuben Hersh [© 1973 Scientific American, doi: 10.1038/scientificamerican1173-84] Illustrating the basic idea of machines from unsoilvability theory.

Following is an online program by Prof. Andrew Granville illustrating the working of PRIMEGAME:

Source: http://www.dms.umontreal.ca/~andrew/conwaymachine.html

Motivation for this post came from Andrew Granville’s Math Mornings at Yale.

Generalization of Pythagoras equation

About 3 years ago I discussed following two Diophantine equations of degree 2:

• A special case of negative Pell’s equation (determining solvability conditions for a special degree 2 equations in 2 variables)
• A slight modification of Pythagorean equation (finding explicit solutions for a particular degree 2 equations in 3 variables, using method used for finding Pythagorean triplets)

In this post, we will see a slight generalization of the result involving Pythagorean triplets. Unlike Pythagoras equation, , we will work with a little bit more general equation, namely: , where . For proofs, one can refer to section 5.5 of Niven-Zuckerman-Montgomery’s An introduction to the theory of numbers.

Theorem: Let be non-zero integers such that the product is square free. Then have a non-trivial solution in integers if and only if do not have same sign, and that are quadratic residues modulo respectively.

In fact, this result helps us determine the existence of a non-trivial solution of any degree 2 homogeneous equation in three variables, due to the following lemma:

Lemma: There exists a sequence of changes of variables (linear transformations) so that can be written as an equation of the form with .

Now let’s consider the example. Let , and we want to determine whether this has a non-trivial solution. Firstly, we will do change of variables:

where , and . Thus

where , and . Thus

where , and . Now we apply the theorem to . Since all the coefficients are prime numbers, we can use quadratic reciprocity to conclude that the given equation has non-trivial solution (only non trivial thing to note that is quadratic residue mod -3, is same as is quadratic residue mod 3).

Academia is not pious

Many young people (me included) might get attracted to academia due to the outward appearance of “freedom” of working hours and being “paid” for what you enjoy doing the most in your free time.

But in reality, academia is just like any other profession. There is politics, drama, ….. I will illustrate using Reddit examples:

That one student…
byu/jon110334 inProfessors

Grad Life Woes: An Indian PhD student’s Starry Night
byu/General-Ising inindia

Postdocs who were depressed/anxious in grad school: does it get better?
byu/slutty_R2 inAskAcademia

Failed out of college 9 years ago. Today I was accepted to a PhD at Harvard!
byu/gradadmitthrow321 ingradadmissions

Counting Cards – II

Since we have introduced the game of blackjack at the end of last year, we can now talk about the winning strategies. A hand with an ace valued as 11 is called “soft”, meaning that the hand will not bust by taking an additional card; the value of the ace will become one to prevent the hand from exceeding 21. Otherwise, the hand is “hard”.  Each blackjack game has a basic strategy, which is playing a hand of any total value against any dealer’s up-card, which loses the least money to the house in the long term. For example:

For details, like when to double-hard or double-soft, see: https://www.blackjackarmy.com/basic-blackjack-strategy

Blackjack’s house edge is usually between 0.5%–1% when players use basic strategy. Card counting can give the player an edge of up to 2% over the house.

A card counting system assigns a point score to each rank of a card. When a card is exposed, a counter adds the score of that card to a running total, the ‘count’. A card counter uses this count to make betting and playing decisions according to a table which they have learned. The count starts at 0 for a freshly shuffled deck for “balanced” counting systems. Unbalanced counts are often started at a value which depends on the number of decks used in the game.

The most common variations of card counting in blackjack are based on statistical evidence that high cards (especially aces and 10s) benefit the player more than the dealer, while the low cards, (3s, 4s, 6s, and especially 5s) help the dealer while hurting the player.

Basic card counting assigns a positive, negative, or zero value to each card value available. When a card of that value is dealt, the count is adjusted by that card’s counting value. Low cards increase the count as they increase the percentage of high cards in the remaining set of cards, while high cards decrease it for the opposite reason. For instance, the Hi-Lo system subtracts one for each dealt 10, Jack, Queen, King or Ace, and adds one for any value 2-6. Values 7-9 are assigned a value of zero and therefore do not affect the count. Here is a quick explanation of this system:

 

A lot of content for this post was shamelessly copied from other articles. In case of copyright violation, please ask me to delete this.

Number Devil

If you enjoyed reading Lewis Carroll’s Alice’s Adventures in Wonderland, George Gamow’s Mr Tompkins, Abbott’s Flatland, Malba Tahan’s The Man Who Counted, Imre Lakatos’s Proofs and Refutations or Tarasov’s Calculus, then you will enjoy reading Enzensberger’s The Number Devil. But that is not an if and only if statement.

Originally written in German and published as Der Zahlenteufel, so far it has been translated into 26 languages (as per the back cover).

After reading this book one will have some knowledge of infinity, infinitesimal, zero, decimal number system, prime numbers (sieve of eratothenes, Bertrand’s postulate, Goldbach conjecture), rational numbers (0.999… = 1.0, fractions with 7 in denominator), irrational numbers (√2 = 1.4142…, uncountable), triangular numbers, square numbers, Fibonacci numbers, Pascal’s triangle (glimpse of Sierpinski triangle in it), combinatorics (permutations and combinations, role of Pascal’s triangle), cardinality of sets (countable sets like even numbers, prime numbers,…), infinite series (geometric series, harmonic series), golden ratio (recursive relations, continued fractions..), Euler characteristic (polyhedra and planar graphs), how to prove (11111111111^2 does not give numerical palindrome, Principia Mathematica), travelling salesman problem, Klein bottle, types of infinities (Cantor’s work), Euler product formula, imaginary numbers (Gaussian integer), Pythagoras theorem, lack of women mathematicians  and pi.

Since this is a translation of original work into English, you might not be happy with the language.  Though the author is not a mathematician, he is a well-known and respected European intellectual and author with wide-ranging interests. He gave a speech on mathematics and culture, “Zugbrücke außer Betrieb, oder die Mathematik im Jenseits der Kultur—eine Außenansicht” (“Drawbridge out of order, or mathematics outside of culture—a view from the outside”), in the program for the general public at  the International Congress of Mathematicians in Berlin in 1998. The speech was published under the joint sponsorship of the American Mathematical Society and the Deutsche Mathematiker Vereinigung as a pamphlet in German with facing English translation under the title Drawbridge Up: Mathematics—A Cultural Anathema, with an introduction by David Mumford.