Matrix Inversion: Partition Method

Introduction

Today we will discuss a not-so-famous method of inverting matrices. This method is recursive in the sense that given a method to find inverse of square matrix of order $n$ it can be applied to find the inverse of a matrix of order $(n + 1)$. This method is named Partition Method or the Escalator Method. The idea is to partition a matrix into smaller sub-matrices and then calculate the inverse from the given inverse of one of the smaller sub-matrices.

Two Problems not from IIT-JEE

After the widely read post Two Problems from IIT-JEE, I am going to discuss two problems which are not from IIT-JEE (as far as I am aware). They are taken from the masterpiece "A Course of Pure Mathematics" by G. H. Hardy. The first one is a tough limit problem (at least I have not been able to find a simpler solution till now) and the second one is an instructive example which deals with the behavior of derivatives for large values of the argument.

The Riemann Integral: Part 3

Oscillation of a Function

In a previous post we obtained the Riemann's condition of integrability using the concept of upper and lower Darboux sums. We observed that in order that a function be Riemann integrable on interval $[a, b]$ it was necessary (and sufficient) to make the sum $$U(P, f) - L(P, f) = \sum_{k = 1}^{n}(M_{k} - m_{k})(x_{k} - x_{k - 1})$$ arbitrarily small for some partition $P = \{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\}$ of $[a, b]$.

The Riemann Integral: Part 2

In the last post we defined the Riemann integral of a function on a closed interval and discussed some of the conditions for the integrability of a function. Here we develop the full machinery of the Riemann integral starting with the basic properties first.

The Riemann Integral: Part 1

Introduction

The theory of integration forms an important part of mathematical analysis. Historically integration was used to find areas of plane figures. Archimedes used the very same process to find areas of parabola but he called it the method of exhaustion. The idea used by Archimedes was to divide the desired area in terms of smaller and smaller areas so that the sum of the areas of these smaller parts tended to a finite limit. It was the genius of Newton (and Leibniz too) to recognize that the process of integration could be viewed as the inverse process of differentiation. This greatly helped in finding areas of curves for which summing the areas of smaller parts was difficult. After Newton people started thinking of integration as the inverse of differentiation and the older approach based on summation was put at the back front.

Functions of Bounded Variation: Part 2

Continuity and Bounded Variation

In the last post we saw that continuity is not essential to the property of being a function of bounded variation. However monotonicity is absolutely essential in the sense that every function of bounded variation can be expressed as a sum or difference of monotone functions. But does that mean that continuity is not at all required? Can we have a function which is discontinuous everywhere and still be of bounded variation? The answer is NO! As an example the function $f(x) = 0$ when $x$ is irrational and $f(x) = 1$ when $x$ is rational is not of bounded variation. We can choose a partition to consists of equal number of rational and irrational points lying alternately and then the variation can be seen as a linear function of the number of points of subdivision so that the variation is not bounded.

Functions of Bounded Variation: Part 1

Introduction

In the last two posts we studied monotone functions which vary in the same direction in a given interval. Here we will study functions which do not vary too much. In a sense continuous functions also don't vary too much (for example they are bounded on closed intervals). But here we need to discuss variation in a different sense. More technically we try to measure variation in smaller parts of an interval and then add up these variations to form total variation. We formalize these concepts now.

Monotone Functions: Part 2

In the last post we established certain conditions for the monotonicity of a function in an interval. In this post we will establish the same results via a different approach. This is based on the standard theorems of differential calculus, namely the Rolle's Theorem and the Langrange's Mean Value Theorem. We first need to establish these theorems.

Monotone Functions: Part 1

Introduction

Few posts ago we discussed continuous functions and their properties. In this series of posts we discuss another class of functions namely the monotone functions and their extensions. The word monotone crudely suggests that these functions should have a single tone which translates properly to "variation in a single direction". In other words such functions either increase all the time or decrease all the time.

Logarithms using Square Roots

Introduction

In the last post we discussed that a proper theory of logarithmic function can not be developed without using the analytical approach (or methods of calculus in simpler language). Here we discuss more about the common logarithms (i.e. logarithms to the base 10) which are normally introduced to students in secondary classes without using any calculus. The basic idea is simple enough: if $y = 10^{x}$ then we write $x = \log_{10}y$ and say that $x$ is the common logarithm of $y$. The difficulty with this approach is that the meaning of $10^{x}$ cannot be explained properly without using calculus if $x$ is irrational. However in the examples and exercises given the exponent is either a symbol (like $a, b, c, x$) or is a rational number so there is no confusion when the concept of logarithm is presented via this approach. And the student is able to learn the fundamental properties of logarithms and their practical usage in computation with relative ease.

Definitions in Mathematics

Introduction

Definitions per se are not something specific to the field of mathematics, but are rather prevalent in almost all subjects (for example English Grammar). An obvious need for definitions is to express a new term using already existing vocabulary. In mathematics however they have much more importance than in any other subject. Definitions in mathematics are rather precise and clearly express what a thing is or is not. From now on we will restrict ourselves to the form of definition used in mathematics.

Conics and the Cone: Part 3

After having dealt with the case of circular sections of an oblique cone in previous post, let's now focus on the conics. We will first treat the case of a parabola as this is the simplest case after the case of a circular section.

Conics and the Cone: Part 2

In the previous post we established that sections of a right circular cone are the familiar curves (ellipse, parabola, hyperbola) having the focus directrix property. Now we will have a look at the more general case when the cone is not right but oblique. Our approach will be identical to the one followed by the great Greek geometer Apollonius. However we will not be developing a systematic theory of conics as described by Apollonius, but rather focus on the interesting results which will help us to connect them with the modern definition of conics. In doing so we will observe that the main tool used by Apollonius is the similarity of triangles.

Conics and the Cone: Part 1

Introduction

While studying co-ordinate geometry (aka analytic geometry) in intermediate classes we normally arrive at the study of conic sections or in short "conics". Three new curves namely "ellipse", "parabola", and "hyperbola" come into picture and their theory is quite unlike those of the elementary geometrical objects (line, triangle, circle etc) studied in secondary classes. In case of the elementary geometrical objects like points, lines, triangles, circles we have two approaches: 1) using the axioms of Euclid and then deducing the properties of these objects logically from Euclid's axioms and, 2) using the language of coordinate geometry which transforms the subject of geometry into algebra. Unfortunately the beautiful approach of using Euclid's axioms is discarded in higher secondary classes in favor of the approach using coordinate geometry which makes the subject dull with huge amount of laborious algebraical manipulations. In fact most students think that the only way to study new curves like ellipses, parabola and hyperbola is through coordinate geometry.

Modular Equations and Approximations to π(PI): Part 3

Series Based on Alternative Theories

In the previous post we established certain series for $1/\pi$ following Ramanujan's technique. These were based on formulas in the classical theory of elliptic functions and integrals. In the field of elliptic functions, Ramanujan surpassed all his predecessors and developed alternative theories which bore striking resemblance to the classical theory and thus provided a grand generalization of the theory of elliptic functions.

Modular Equations and Approximations to π(PI): Part 2

Ramanujan's Series for $\pi$

Using the values of the function $P(q)$ for $q = e^{-\pi\sqrt{n}}$ (see previous post for the definition of $P(q)$) Ramanujan was able to derive many beautiful series for $\pi$. He did this in very clever way. The fundamental idea he used was the fact that the function $\phi^{4}(q) = (2K/\pi)^{2}$ could be expressed in the form of a generalized hypergeometric series.

Modular Equations and Approximations to π(PI): Part 1

In this post we will discuss Ramanujan's classic paper "Modular Equations and Approximations to $\pi$" where Ramanujan offered many amazing formulas and approximations for $\pi$ and showed us the way to create new theories of elliptic and theta functions. However the paper as written in his classic style is devoid of proofs of the most important results. The post would try to elaborate on some of the results mentioned therein.

Ramanujan's Class Invariants

After a heavy discussion on the modular equations found by Ramanujan, we will now focus on another significant discovery of his namely "Class Invariants".

Elementary Approach to Modular Equations: Ramanujan's Theory 7

Continuing from previous post we proceed to derive further modular equations of degree $5$ in this post. Clearly in order to establish such equation we need to establish further theta function identities. This time we establish an identity concerning Ramanujan's $\psi$ function.

Identity Concerning $\psi(q)$ of Degree $5$

We will establish the following identity $$\psi^{2}(q^{2}) - q^{2}\psi^{2}(q^{10}) = \frac{\phi(-q^{10})f(-q^{10})}{\chi(-q^{2})}\tag{1}$$

Elementary Approach to Modular Equations: Ramanujan's Theory 6

The Fundamental Formulas

In this post we will continue our journey of modular equations and derive a host of these mostly by using Lambert series for various theta functions. The following formula (see equation $(14)$ of this post) will be of great help here: $$\phi^{2}(-ab)\,\frac{f(a, b)}{f(-a, -b)} = 1 + 2\sum_{n = 1}^{\infty}\frac{a^{n} + b^{n}}{1 + a^{n}b^{n}}\tag{1}$$

Elementary Approach to Modular Equations: Ramanujan's Theory 5

Quintuple Product Identity

We first establish an identity similar to Jacobi's Triple Product which involves five factors and is quite useful in establishing various other identities involving q-series and products. This was first introduced in the mathematical literature by G. N. Watson in order to prove some of Ramanujan's theorems. The quintuple product identity is given by \begin{align}&\prod_{n = 1}^{\infty}(1 - q^{n})(1 - q^{n}z)(1 - q^{n - 1}z^{-1})(1 - q^{2n - 1}z^{2})(1 - q^{2n - 1}z^{-2})\notag\\ &\,\,\,\,\,\,\,\,= \sum_{n = -\infty}^{\infty}q^{n(3n + 1)/2}(z^{3n} - z^{-3n - 1})\notag\end{align}

Elementary Approach to Modular Equations: Ramanujan's Theory 4

Lambert Series

In this post we will focus our attention on series of the form: $$\sum_{n = 0}^{\infty}a_{n}\cdot\frac{q^{b_n}}{1 \pm q^{c_n}}$$ which are more popularly known as Lambert Series. We will not deal with the general theorems concerning such series but will restrict ourselves to the Lambert series for the theta functions and study some identities involving these series.