CCC101 Mathematics in Vedic India

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SNU VC welcomes Ramanujan Mathematical Society delegates

Dr Nikhil Sinha welcomes the delegates to the 27th Annual Conference of the Ramanujan Mathematical Society. The conference was hosted by SNU in Delhi and attended by over 200 delegates. See the conference website for full details.

Brahmagupta's Sine Interpolation

In a previous post we saw a very accurate approximation to the Sine function provided by the scientist Bhaskara who lived in the 7^th century AD. In this post, we present an alternative formula given by his contemporary Brahmagupta. This is based on starting with a table of Sine values. The standard tables presented by Aryabhata and others preceding Brahmagupta used intervals of 3^o45'. Intermediate values were then found by linear interpolation. Let's express this in our language:

We assume intervals of fixed width $\delta$. We are thus looking at the angles 0, $\delta$, $2\delta$, ..., $i\delta$ ,... Suppose a table is available for the Sines of these angles: $\sin(0)$, $\sin(\delta)$, $\sin(2\delta)$, ..., $\sin(i\delta)$, ... Our task is to estimate the value of $\sin(x)$ when $x$ is not a tabulated angle. Suppose $x$ is between $x_{i-1}=(i-1)\delta$ and $x_i=i\delta$. The total change of the Sine function over this interval is $\sin(x_i)-\sin(x_{i-1})$ and we denote this by $\triangle\sin_i$. If this change were happening linearly then every unit change in the angle would produce a change of
$$\frac{\triangle\sin_i}{\delta}$$ in the Sine function. In moving from  $x_{i-1}$ to $x$ we would therefore create a change of
$$\frac{\triangle\sin_i}{\delta}(x-x_{i-1})$$ This gives the following approximation, which is called linear interpolation:
$$\sin(x)-\sin(x_{i-1}) \approx \frac{\triangle\sin_i}{\delta}(x-x_{i-1}) \quad\mbox{or}\quad \sin(x) \approx \sin(x_{i-1}) + \frac{\triangle\sin_i}{\delta}(x-x_{i-1})$$
In linear interpolation we match the function which a straight line which meets it at 2 points: $x_{i-1}$ and $x_i$. Brahmagupta's approach amounts to improving this by matching the function with a quadratic which meets it at 3 points: $x_{i-2}$, $x_{i-1}$ and $x_i$. To derive this formula (and we don't know the steps Brahmagupta took) we start with a form similar to the linear interpolation formula: $$\sin(x)\approx \sin(x_{i-1}) + (x-x_{i-1})p(x)$$ where $p(x)$ is a linear function $Ax+B$. This already does the right thing at $x_{i-1}$. We have to choose $p(x)$ so that it also does the right thing at $x_{i-2}$ and $x_i$. In other words, we need $p(x)$ to satisfy the following: $$\sin(x_{i-2})=\sin(x_{i-1}) + (x_{i-2}-x_{i-1})p(x_{i-2})$$ $$\sin(x_{i})=\sin(x_{i-1}) + (x_{i}-x_{i-1})p(x_{i})$$ These can be rearranged into $$p(x_{i-2}) = \frac{\triangle\sin_{i-1}}{\delta}$$ $$p(x_{i}) = \frac{\triangle\sin_{i}}{\delta}$$
This is again a linear interpolation problem and we have already seen how to solve it: \begin{eqnarray*}
p(x) &=&p(x_{i-2}) + \frac{x-x_{i-2}}{x_i-x_{i-2}}(p(x_i)-p(x_{i-2}))= \frac{\triangle\sin_{i-1}}{\delta}+\frac{x-x_{i-2}}{2\delta}\left(\frac{\triangle\sin_{i}-\triangle\sin_{i-1}}{\delta}\right)\\
&=& \frac{\triangle\sin_{i-1}}{\delta}+\frac{x-x_{i-1}+\delta}{2\delta}\left(\frac{\triangle\sin_{i}-\triangle\sin_{i-1}}{\delta}\right) = \frac{\triangle\sin_i+\triangle\sin_{i-1}}{2\delta}+\frac{x-x_{i-1}}{\delta^2}\left(\frac{\triangle\sin_{i}-\triangle\sin_{i-1}}{2}\right)
\end{eqnarray*}

We bring things to a close by putting this expression for $p(x)$ back into our initial quadratic approximation. We get:
$$\sin(x) \approx \sin(x_{i-1}) + \frac{x-x_{i-1}}{\delta}\left(\frac{\triangle\sin_i+\triangle\sin_{i-1}}{2}+\frac{x-x_{i-1}}{\delta}\left(\frac{\triangle\sin_{i}-\triangle\sin_{i-1}}{2}\right)\right)$$ Brahmagupta used $\delta=15^o =900'$. So his base trigonometric table needed only five values (ignoring the trivial cases of 0 and 90 degrees). And here are graphs showing the accuracy of his formula between 60 and 75 degrees:

Linear Interpolation

Brahmagupta's Quadratic Interpolation

A 1400 Year Old Formula for Sine

Bhaskara was an Indian mathematician who lived in the 7^th century AD. One of his claims to fame was his commentary Aryabhatiyabhasya, which is our main source for understanding Aryabhata's cryptic verses. Another was a formula describing the Sine function through a rational approximation. In our terminology, measuring angles in radians, his formula becomes:
$$\sin(x) \approx \frac{16x(\pi-x)}{5\pi^2-4x(\pi-x)},\qquad (0\le x\le \frac{\pi}{2})$$

In this post, we will take a graphical approach to "discovering" this formula. Of course, we are not claiming Bhaskara thought like this. Perhaps he did not need to, as his skill and judgement in direct computation would have been far beyond ours.

Let's start by looking at the Sine function from 0 to $\pi$ (Again, Bhaskara would have only gone up to $\pi/2$ or 90 degrees.) The graph looks like part of an inverted parabola:

 

A parabola that is 0 at origin and $\pi$ must have the form $y=Cx(\pi-x)$, for some constant $C$. Now we want the central value of $y$, at $x=\pi/2$, to be 1. The actual value is $C\pi^2/4$ and so we set $C\pi^2/4=1$ and solve to obtain $C=4/\pi^2$. In Bhaskara's time, a popular estimate for $\pi$ was $\sqrt{10}\approx 3.16$. If we substitute that, we get $C=0.4$. We have obtained an estimate

$$\sin(x) \approx 0.4\, x(\pi-x)$$

How good is this? Let's compare the graphs:

 

Not bad! We could easily be satisfied with this. But Bhaskara was not, so let's take a closer look.  There are two ways of testing the closeness of quantities: their difference could be close to zero, or their ratio could be close to 1. Correspondingly, there are two ways of adjusting a quantity so that it becomes closer to another - by shifting or scaling. Let's first look at the difference between $\sin(x)$ and the quadratic approximation:

 

This kind of shape can be generated by a 4^th degree polynomial. But adjusting the coefficients of that polynomial so that it has zeroes and peaks at the right locations calls for quite a bit of fiddling. So let's look at the ratio:

This looks much simpler - a quadratic again! To match this, we need a quadratic that is 1.25 at the ends and 1 in the middle. The following one does the trick:
 $$1.25-0.1\,x(\pi-x)$$
keeping in mind that $\pi^2\approx 10$. So we have
$$\frac{0.4\,x(\pi-x)}{\sin(x)} \approx 1.25-0.1\,x(\pi-x)$$
or
 $$\sin(x)\approx \frac{0.4\,x(\pi-x)}{1.25-0.1x(\pi-x)} = \frac{16\,x(\pi-x)}{50-4\,x(\pi-x)} \approx \frac{16\,x(\pi-x)}{5\pi^2-4\,x(\pi-x)}$$
Let us ask, one last time, how good is our approximation? And answer again with a graph:

 

The red and blue curves representing the two functions overlap too perfectly for the eye to distinguish them. 

The Importance of Zero

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Listening to Mathematics

The BBC's "In Our Time" radio program has had many fascinating episodes on mathematics and mathematicians. Here are some examples that you can access online:

Archimedes Calculus Cryptography Game Theory Godel's Incompleteness Theorems Imaginary Numbers Indian Mathematics Infinity Mathematics Negative Numbers

Pi Prime Numbers Probability Pythagoras Renaissance Maths Symmetry The Fibonacci Sequence The Observatory at Jaipur The Poincare Conjecture Zero

MAT240 Algebra-1. First Assignment.

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CCC 101 - Mesopotamia & Indus Valley

Notes for Lecture 3 and parts of Lecture 4: Mesopotamia and Indus Civilizations.
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CCC 101 - Course Intro & Ancient Egypt

• Lecture 1 - Course Introduction
• Lecture 2 - Ancient Egypt

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Email list for CCC "Maths in India"

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The SNU Maths Library

The SNU Library now has a very nice collection of Maths books. Not just the usual textbooks, but books that make the subject accessible, and enjoyable, to a wide audience. Here are some that will show you new aspects of Mathematics, or explain things more lucidly than you might have seen before, or simply entertain. Clicking on the covers will take you to readers' reviews on Amazon.

Writing Maths Online

We have just integrated MathJax into this blog. This allows the use of TeX to represent mathematics. For example, we can display the Laplace transform
\[ \mathcal{L}(f) = \int_0^\infty f(t) e^{-st}\,dt \]
by typing \(\mbox{\[ \mathcal{L}(f)  =  \int_0^\infty f(t) e^{-st}\,dt \]}\)

MAT 684 - Statistics I

Syllabus for MAT 684 – Statistics I

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 201 (Probability & Statistics) or equivalent

Overview: This course builds on a standard undergraduate probability and statistics course in two ways. First, it makes probability more rigourous by using the concept of measure. Second, it discusses more advanced topics such as multivariate regression, ANOVA and Markov Chains.

Detailed Syllabus:

1. Probability: Axiomatic approach, conditional probability and independent events
2. Random Variables – Discrete and continuous. Expectation, moments, moment generating function
3. Joint distributions, transformations, multivariate normal distribution
4. Convergence theorems: convergence in probability, Weak law of numbers, Borel- Cantelli lemmas, Strong law of large numbers, Central Limit Theorem
5. Random Sampling & Estimators: Point Estimation, maximum likelihood, sampling distributions
6. Hypothesis Testing
7. Linear Regression, Multivariate Regression
8. ANOVA
9. Introduction to Markov Chains

References:

• Statistical Inference by Casella and Berger. Brooks/Cole, 2007. (India Edition)
• An Intermediate Course in Probability by Allan Gut. Springer, 1995.
• Probability: A Graduate Course by Allan Gut. Springer India.
• Measure, Integral and Probability by Capinski and Kopp. 2^nd edition, Springer, 2007.

MAT 660 - Linear Algebra

Syllabus for MAT 660 - Linear Algebra

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 240 and 260, or an undergraduate algebra course with basics of groups and fields.

Overview: The theory of vector spaces is an indispensible tool for Mathematics, Physics, Economics and many other subjects. This course aims at providing a basic understanding and some immediate applications of the language of vector spaces and morphisms among such spaces.

Detailed Syllabus:

1. Familiarity with sets: Finite and infinite sets; cardinality; Schroeder-Bernstein Theorem; statements of various versions of Axiom of Choice.
2. Vector spaces: Fields; vector spaces; subspaces; linear independence; bases and dimension; existence of basis; direct sums; quotients.
3. Linear Transformations: Linear transformations; null spaces; matrix representations of linear transformations; composition; invertibility and isomorphisms; change of co-ordinates; dual spaces.
4. Systems of linear equations: Elementary matrix operations and systems of linear equations.
5. Determinants: Definition, existence, properties, characterization.
6. Diagonalization: Eigenvalues and eigenvectors; diagonalizability; invariant subspaces; Cayley-Hamilton Theorem.
7. Canonical Forms: The Jordan canonical form; minimal polynomial; rational canonical form.

Main References:

• Friedberg, Insel and Spence: Linear Algebra, 4^th edition, Prentice Hall India
• Hoffman and Kunze: Linear Algebra, 2^nd edition, Prentice Hall India

Other references:

• Paul Halmos: Finite Dimensional Vector Spaces, Springer India
• Sheldon Axler: Linear Algebra Done Right, 2^nd edition, Springer International Edition
• S. Kumaresan: Linear Algebra: A Geometric Approach, Prentice Hall India

MAT 622 - Topology

Syllabus for MAT 622 - Topology

Credits (Lec:Tut:Lab): 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 220 (Real Analysis) or equivalent

Overview: This course concerns 'General Topology' which can be characterized as the abstract framework in which the notion of continuity can be framed and studied. Thus topology provides the basic language and structure for a large part of pure and applied mathematics.

We will take up the following topics: Open and closed sets, continuous functions, subspaces, product and quotient topologies, connected and path connected spaces, compact and locally compact spaces, Baire category theorem, separability axioms.

Detailed Syllabus:

1. Review: Operations with infinite collections of sets, axiom of choice, Zorn's lemma, real line, metric spaces.
2. Topological Spaces: Definition and examples of topological spaces, Hausdorff property, fine and coarse topologies, subspace topology, closed sets, continuous functions, homeomorphisms, pasting lemma, product topology, quotient topology.
3. Connectedness and Compactness: Connected spaces and subsets, path connectedness, compact spaces and subsets, tube lemma, Tychonoff theorem, local compactness, one-point compactification, Baire category theorem.
4. Separation Axioms: First and second countability, separability, separation axioms (T₁ etc.), normal spaces, Urysohn lemma, Tietze extension theorem.
5. Topics for Student Presentations: Order topology, quotients of the square, locally (path) connected spaces, sequential and limit point compactness, topological groups, nets, applications of Baire category theorem.

Main Reference:

• Topology by James R. Munkres, 2nd Edition. Pearson Education, Indian Reprint, 2001.

Other References:

• Basic Topology by M. A. Armstrong. Springer-Verlag, Indian Reprint, 2004.
• Topology by K. Jänich. Undergraduate Texts in Mathematics, Springer-Verlag, 1984.
• Introduction to Topology and Modern Analysis by G. F. Simmons. International Student Edition. McGraw-Hill, Singapore, 1963.
• Topology of Metric Spaces by S. Kumaresan. 2^nd edition, Narosa, 2011.

MAT 601 - Mathematical Computing

In the next few posts, we will put up the Master's level courses being taught by the SNU maths department.

Syllabus for MAT 601 – Mathematical Computing

Credits(Lec:Tut:Lab): 1:0:1 (1 lecture and 2 lab hours weekly)

Prerequisites:

Overview: In this course we introduce MATLAB as a platform for scientific computation and simulations; and follow with a brief introduction to C++ as a formal programming language. We also demonstrate how MATLAB and C++ can be integrated to build powerful applications. The course complements other graduate courses like Linear Algebra, Numerical Analysis and Optimization.

Detailed Syllabus:

1. MATLAB:

• Arithmetic expressions, assignment, input and output, Boolean expressions, conditional statements.
• For loop, while loop, nested loops, nested conditionals, vectors, elementary graphics, color schemes.
• Elementary mathematical functions, functions with multiple input parameters, graphics functions.
• Two dimensional arrays, contour plotting, sorting, searching, cell arrays, cell arrays of matrices, functions as parameters, structures.
• Working with image files, acoustic file processing, recursive functions, solving linear programming problems.

2. C++ Programming:

• Fundamental data types, operators, control structures, user defined functions
• Arrays, pointers, function pointers, multi- dimensional arrays
• Classes, constructors & destructors, bitwise operators
• Integrating C++ with MATLAB – calling MATLAB functions within a C++ program.

Main References:

1. Programming in Matlab for Engineers by Stephen J. Chapman. Cengage, 2011.
2. A Guide to Matlab by Brian R. Hunt, Ronald L. Lipsman and others. 2^nd edition, Cambridge, 2011.
3. Insight Through Computing: A MATLAB introduction to Computational Science and Engineering by Charles F. Van Loan and K. Y. Daisy Fan. SIAM, 2009.
4. Introducing C++ for Scientists, Engineers and Mathematicians by D. M. Capper. Springer India, 2001.

Other references:

1. Mastering Matlab 7 by Duane C. Hanselman and Bruce L. Littlefield. Pearson Education, 2005.
2. C++ Programming Language by Bjarne Stroustrup. 3^rd edition, Pearson, 2002.
3. Object Oriented Programming in C++ by R. Lafore. 3^rd edition, Galgotia, 2006.

Ramanujan Mathematical Society Conference

SNU is hosting the 27^th Annual Conference of the Ramanujan Mathematical Society during October 20-23, 2012. The conference venue is the Radisson Blu hotel in Paschim Vihar, Delhi. The main academic programme - twelve plenary talks and five symposia (about 40 invited talks in all!) - is already set. There will also be several sessions of contributed 10-minute talks.

Details of how to register to participate, and apply to give a short talk, will soon be up on the conference website.

Minor in Mathematics

Undergraduate students of the university who are not majoring in Mathematics have the option to take a Minor in Mathematics. A Minor in Mathematics can serve two distinct functions (apart from enjoying the beauty of the subject!):

1. Acquiring the academic background for higher studies in mathematics.
2. Acquiring modelling and computational skills for applications of mathematics in other disciplines or in industry.

Academic Requirements:

You have to acquire a minimum of 21 lecture credits from the University Wide Elective (UWE) courses offered by the Department of Mathematics. These 21 lecture credits must satisfy the following:

1. At least 9 lecture credits from Group A: MAT 101 (Calculus I), MAT 260 (Linear Algebra), MAT 280 (Numerical Analysis I), MAT 284 (Probability & Statistics)

2. 3 to 6 lecture credits from Group B: MAT 199, 299, 399, 499 (Projects)

3. Remainder from any other UWEs offered by Department of Mathematics
   
   
4. The above is subject to the further requirement that a course should not count towards both Major and Minor requirements. This may be partially waived for majors which already have a large component of compulsory mathematics courses.

The Undergraduate Advisor for Mathematics will help you work out an appropriate choice of courses depending on your interests and background.

How to Apply and Select Courses:

1. You have to register for the minor – the first step is to obtain permission from the UG Advisor of the Department of Mathematics. The current UG Advisor for Mathematics is Prof. Amber Habib. Start by contacting him during his office hours or send an email to amber.habib@snu.edu.in to make an appointment.
   
   
2. You will select courses for the minor in consultation with the UG Advisor for Mathematics.
   
   
3. You must sign up for the Minor before the end of your 6^th semester. However, it is advisable to do so earlier so that there is sufficient time to plan your courses. The best time is during your 3^rd or 4^th semesters.
   
   
4. Please note that you must register for the minor as described here. It is not enough to merely take adequate credits on your own. 
    
5. If you fail to complete the minor during your first 4 years, you may have to spend an extra semester to complete it. If you do so, any scholarship or fee waiver you were granted for your regular course of study will lapse and you will have to pay the full fees for the extra period.

MAT 110 - Computing

Syllabus for MAT 110 – Computing

This is a compulsory course for BS Mathematics students in their 1^st semester.

Credits (Lec:Tut:Lab)= 1:0:1 (One lecture hour and two lab hours weekly)

Prerequisites: None

Overview: This course provides an introduction to the programs Matlab and Microsoft Excel as tools for mathematical computing. The focus is on their use in applications from the fields of Statistics, Finance, Image Processing etc. Student presentations of assignment solutions will be a major component of the course.

Detailed Syllabus:

1. MATLAB:

• Arithmetic expressions, assignment, input and output, Boolean expressions, conditional statements
• For loop, while loop, nested loops, nested conditionals, vectors, elementary graphics, color schemes in Matlab
• Elementary math functions, Functions with multiple input parameters, plotting
• Two dimensional arrays, sorting, searching, cell arrays, cell arrays of matrices
• Working with image files

2. EXCEL:

• Charts
• Lookup, Match, Index, Offset functions
• Embedding form controls in a spreadsheet
• Array functions, Goal Seek, Solver
• Descriptive statistics with Analysis Toolpak

Main References:

1. Programming in Matlab for Engineers by Stephen J. Chapman, Cengage, 2011.
2. Guide to Matlab by Brian R. Hunt, Cambridge, 2001.
3. Microsoft Excel 2010: Data analysis and Business Modeling by Wayne L. Winston, Prentice Hall India.

Other References:

1. Mastering Matlab 7 by Duane C Hanselman and Bruce L Littlefield, Pearson Education, 2005.
2. Excel 2010 Formulas by John Walkenbach, Wiley India, 2011.
3. Favourite Excel 2010 Tips & Tricks by John Walkenbach, Wiley India, 2011.