Minor in Mathematics Form

Applications for the Maths Minor are closed. Results will be out shortly!

Minor in Mathematics Announcement

Undergraduate students of Shiv Nadar 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!):

• Acquiring the academic background for higher studies in Mathematics.
• Acquiring modelling and computational skills for applications of Mathematics in other disciplines or in industry.

Academic Requirements:

You have to acquire a minimum of 27 credits from the University Wide Elective (UWE) courses offered by the Department of Mathematics. These credits must satisfy the following minimum requirements:

1. Three courses from Group A for a total of 12 credits: MAT 101 (Calculus I), MAT 240 (Algebra I), MAT 260 (Linear Algebra), MAT 280 (Numerical Analysis I), MAT 284 (Probability & Statistics).
2. One course from Group B (3 credits): MAT 199, 299, 399, 499 (Projects).
3. Remainder from any other UWE courses offered by the Department of Mathematics.
4. The above is subject to the further requirement that a course should not count towards both Major and Minor requirements.
5. The credit requirement may be lowered to 23 credits for majors which already have a significant 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:

1. For the 2013 session, there are 20 seats available for a Minor in Mathematics.
2. You are eligible to apply for the Minor if you have already earned or are currently enrolled in at least 3 credits from courses offered by the Department of Mathematics, and if your GPA from these courses is at least 6.
3. Eligible applicants will be interviewed, and admission to the Minor will be determined by the results of that interview. The probable interview dates are April 29 and 30.
4. You can apply by filling out the online form before 5pm on April 26. (Click to go to the online form.)
5. For further details please contact the UG Advisor for Mathematics, Amber Habib. You can meet him during his office hours or send an email to amber.habib@snu.edu.in to make an enquiry or seek an appointment.
6. Please note that you must take admission for the Minor as described here. It is not enough to merely take adequate credits on your own. 

Other Information:

1. Once admitted to the Minor in Mathematics, you will select courses for the Minor in consultation with the UG Advisor for Mathematics.
2. You must sign up for the Minor before the end of your 6th 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 3rd or 4th semesters.
3. 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.
4. You may of course enroll for UWE courses offered by the Department without being admitted to the Minor. However, students enrolled for the Minor will have priority while registering for these courses. The Department will also do its best to schedule courses so that Minor students will be able to complete their requirements.

What Maths Can Do - SIAM Reports

The Society for Industrial & Applied Mathematics (SIAM) is the preeminent body promoting applied mathematics. It "exists to ensure the strongest interactions between mathematics and other scientific and technological communities..."

SIAM has recently produced two reports for students and applied mathematicians:

Careers in Applied Mathematics - describes career possibilities outside academia for majors in mathematics or computing.

Mathematics in Industry - looks at how mathematics is used in industry today. It also analyzes the skills and abilities required by graduates for fruitful careers.

CCC101 Mathematics in Vedic India

If you can't see the pdf below, please sign in to your SNU email account and then try again. If you don't have an SNU account, try clicking on this link.

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.