Category Archives: My Research

Something interesting which I believe to have discovered

Childhood Maths – II

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I found two documents that I was very proud of as a child. Both were the result of trying to understand the kind of things Ramanujan did in free time, a result of the little AMTI books I read as a child. I will share the second document in this post and the other one was in the previous post.

Following document was created in MS Word on my old Windows XP desktop. The calculations were done using some Microsoft advanced calculator:

primes

I was not happy with the result though since the pattern didn’t continue which was supposed to continue according to Ramanujan.

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Childhood Maths – I

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I found two documents that I was very proud of as a child. Both were the result of trying to understand the kind of things Ramanujan did in free time, a result of the little AMTI books I read as a child. I will share one document in this post and another one in the next since both are not related to each other in substance.

Following is a calculation table generated using some spreadsheet software:

caylay

As you can see, it was motivated by the famous Taxicab number story.

Understanding Geometry – 4

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Aleksej Ivanovič Markuševič’s book, “Remarkable Curves” discusses the properties of ellipses, parabolas, hyperbolas, lemniscates, cycloids, brachistochrone, spirals and catenaries.  Among these “lemniscates” are the ones that I encountered only once before starting undergraduate education (all other curves appeared frequently in physics textbooks) and that too just to calculate the area enclosed by this curve. So I will discuss the properties of lemniscates in this post.

Let’s begin with the well-known curve, ellipse. An ellipse is the locus of points whose sum of distances from two fixed points (called foci) is constant. My favourite fact about ellipses is that we can’t find a general formula for the perimeter of an ellipse, and this little fact leads to the magical world of elliptic integrals. This, in turn, leads to the mysterious elliptic functions, which were discovered as inverse functions of elliptic integrals. Further, these functions are needed in the parameterization of certain curves, now called elliptic curves. For more details about this story, read the paper by Adrian Rice and Ezra Brown, “Why Ellipses are not Elliptic curves“.

Lemniscate is defined as the locus of points whose product of distances from two fixed points F_1 and F_2 (called foci) is constant. Lemniscate means, “with hanging ribbons” in Latin.  If the length of the segment \overline{F_1F_2} is c then for the midpoint of this line segment will lie on the curve if the product constant is c^2/4. In this case we get a figure-eight lying on its side.

Lemniskate_bernoulli2.svg

Lemniscate of Bernoulli; By Kmhkmh (Own work) [CC BY 4.0], via Wikimedia Commons

The attempt to calculate the perimeter of the above curve leads to elliptic integral, hence can’t derive a general formula for its perimeter. Just like an ellipse!

If we equate the value of the constant product not to c^2/4 but to another value, the lemniscate will change its shape. When the constant is less than c^2/4, the lemniscate consists of two ovals, one of which contains inside it the point F_1, and the other the point F_2.

oval

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700b.gif

When the product constant is greater than c^2/4 but less than c^2/2, the lemniscate has the form of a biscuit. If the constant is close to c^2/4, the “waist” of the biscuit is very narrow and the shape of the curve is very close to the figure-eight shape.

biscuit

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700b.gif

If the constant differs little from c^2/2, the waist is hardly noticeable, and if the constant is equal or greater than c^2/2 the waist disappears completely, and the lemniscate takes the form of an oval.

oval.gif

Cassini oval (x^2+y^2)^2−2c^2(x^2−y^2)=a^4−c^4; Source: https://www.encyclopediaofmath.org/legacyimages/common_img/c020700a.gif

We can further generalize this whole argument to get lemniscate with an arbitrary number of foci, called polynomial lemniscate.

Polynomials and Commutativity

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In high school, I came to know about the statement of the fundamental theorem of algebra:

Every polynomial of degree n with integer coefficients have exactly n complex roots (with appropriate multiplicity).

In high school, a polynomial = a polynomial in one variable. Then last year I learned 3 different proofs of the following statement of the fundamental theorem of algebra [involving, topology, complex analysis and Galois theory]:

Every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.

A more general statement about the number of roots of a polynomial in one variable is the Factor Theorem:

Let R be a commutative ring with identity and let p(x)\in R[x] be a polynomial with coefficients in R. The element a\in R is a root of p(x) if and only if (x-a) divides p(x).

A corollary of above theorem is that:

A polynomial f of degree n over a field F has at most n roots in F.

(In case you know undergraduate level algebra, recall that R[x] is a Principal Ideal Domain if and only if R is a field.)

The key fact that many times go unnoticed regarding the number of roots of a given polynomial (in one variable) is that the coefficients/solutions belong to a commutative ring (and \mathbb{C} is a field hence a commutative ring). The key step in the proof of all above theorems is the fact that the division algorithm holds only in some special commutative rings (like fields). I would like to illustrate my point with the following fact:

The equation X^2 + X + 1 has only 2 complex roots, namely \omega = \frac{-1+i\sqrt{3}}{2} and \omega^2 = \frac{-1-i\sqrt{3}}{2}. But if we want solutions over 2×2 matrices (non-commutative set) then we have at least  3 solutions (consider 1 as 2×2 identity matrix and 0 as the 2×2 zero matrix.)

\displaystyle{A=\begin{bmatrix} 0 & -1 \\1 & -1 \end{bmatrix}, B=\begin{bmatrix} \omega & 0 \\0 & \omega^2 \end{bmatrix}, C=\begin{bmatrix} \omega^2 & 0 \\0 & \omega \end{bmatrix}}

if we allow complex entries. This phenominona can also be illusttrated using a non-commutative number system, like quaternions. For more details refer to this Math.SE discussion.

Four Examples

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Following are the four examples of sequences (along with their properties) which can be helpful to gain a better understanding of theorems about sequences (real analysis):

  • \langle n\rangle_{n=1}^{\infty} : unbounded, strictly increasing, diverging
  • \langle \frac{1}{n}\rangle_{n=1}^{\infty} : bounded, strictly decreasing, converging
  • \langle \frac{n}{1+n}\rangle_{n=1}^{\infty} : bounded, strictly increasing, converging
  • \langle (-1)^{n+1}\rangle_{n=1}^{\infty} : bounded, not converging (oscillating)

I was really amazed to found that x_n=\frac{n}{n+1} is a strictly increasing sequence, and in general, the function f(x)=\frac{x}{1+x} defined for all positive real numbers is an increasing function bounded by 1:

 

Thre graph of x/(1+x) for x>0. Plotted using SageMath 7.5.1

The graph of x/(1+x) for x>0, plotted using SageMath 7.5.1

 

Also, just a passing remark, since \log(x)< x+1 for all x>0, and as seen in prime number theorem we get an unbounded increasing function \frac{x}{\log(x)} for x>1

dort

The plot of x/log(x) for x>2. The dashed line is y=x for the comparison of growth rate. Plotted using SageMath 7.5.1

 

Prime Polynomial Theorem

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I just wanted to point towards a nice theorem, analogous to the Prime Number Theorem, which is not talked about much:

# irreducible monic polynomials with coefficients in \mathbb{F}_q and of degree n \sim \frac{q^n}{n}, for a prime power q.

The proof of this theorem follows from Gauss’ formula:

# monic irreducible polynomialswith coefficients in \mathbb{F}_q and of degree n = \displaystyle{\frac{1}{n}\sum_{d|n}\mu\left(\frac{n}{d}\right)q^d}, by taking d=n.

 

For details, see first section of this: http://alpha.math.uga.edu/~pollack/thesis/thesis-final.pdf

Recursion and Periodicity

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One of the simplest recursive formula that I can think of is the one which generates the Fibonacci sequence, F_{n+1} = F_n +F_{n-1}, n\geq 1 with F_0 = F_1=1. So, I will illustrate a following general concept about recursions using Fibonacci sequence.

A sequence generated by a recursive formula is periodic modulo k, for any positive integer k greater than 1.

I find this fact very interesting because it means that a sequence which is strictly increasing when made bounded using the modulo operation (since it will allow only limited numbers as the output of recursion relation), will lead to a periodic cycle.

Following are the first 25 terms of the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025.

And here are few examples modulo k, for k=2,3,4,5,6,7,8

table110

As you can see, the sequence repeats as soon as 1,0 appears. And from here actually one can see why there should be a periodicity.

For the sequence to repeat, what we need is a repetition of two consecutive values (i.e. the number of terms involved in the recursive formula) in the sequence of successive pairs. And for mod k, the choices are limited, namely k^2.  Now, all we have to ensure is that “1,0” should repeat. But since consecutive pairs can’ repeat (as per recursive formula) the repetition of “1,0” must occur within the first k^2.

For rigorous proofs and its relation to number theory, see: http://math.stanford.edu/~brianrl/notes/fibonacci.pdf