Trials with triangles

Outline: The article is about some sessions that I had with students of classes 5-8 during the course of some workshops. Most of the students had never ‘seen’ mathematics outside of the school curriculum. Our intention was to make them think mathematically and at the same time, to see how different students look at a problem and come up with solutions

A mathematical pilgrimage : Ramanujan yatra

All of us have heard anecdotes from the life of Srinivasa Ramanujan through books, friends and teachers. Robert Kanigel presents very beautifully the life of Ramanujan in his book The Man who Knew Infinity. A lot about the life of Ramanujan became known after Hollywood brought out the biopic on Ramanujan. Despite this, we felt that it would be worth taking some Math enthusiasts for a tour where they could get immersed in the life of Ramanujan for a few days. The idea took seed sometime in September 2017 while I was talking to Rajith who heads a culture and heritage tour company – The Traveling Gecko – and we thought, why not do an experiential tour on the life of Ramanujan? A detailed account of the 5-day Yatra (Chennai – Thanjavur – Kumbakonam – Chennai) from 12th to 16th November 2018, which I called A Mathematical Pilgrimage | Ramanujan Yatra, is put up on my blog https://vinayrnair.wordpress.com/2018/11/19/ramanujan-yatradiaries- day-1/ . Hence, I am not writing a travelogue again. What I would like to share in this article are some stories, discussions and thoughts about Ramanujan and the lives of people that revolved around him during and after his time

Algebracadabra magic squares ; with prime entries

A magic square is an array of numbers in an n × n square grid arranged in such a way that the sum of the numbers in each row, each column and each of the two main diagonals is the same number (called the ‘Magic Sum’ of that square). There are various ways of classifying magic squares; one is by the number of rows and columns. Thus we have Odd-order Magic Squares, with an odd number of rows, and Even-order Magic Squares, with an even number of rows. Even-order Magic Squares can be further classified into Singly-even Magic Squares, for which the number of rows is even but not a multiple of 4, and Doubly-even Magic Squares, for which the number of rows is a multiple of 4

Cutting a square into equal parts

The article talks about a simple activity which can be performed with students of primary, middle and high school. The shape that is used to discuss here is a square and hence it is expected that students know the basic properties of a square. The article also talks about using lines. Even if students don't have a Euclidean notion of definition of a line, that idea can be instilled as the teacher executes this activity. Similarly, students might have a notion of the words equal, area, congruence, similar, etc., and this activity is very useful in clarifying the subtle difference between terminologies as well as the rigour of mathematical language

Divisibility by primes

I n school we generally study divisibility by divisors from 2 to 12 (except 7 in some syllabi). In the case of composite divisors beyond 12, all we need to do is express the divisor as a product of coprime factors and then check divisibility by each of those factors. For example, take the case of 20; since 20 = 4 × 5 (note that 4 and 5 are coprime), it follows that a number is divisible by 20 if and only if it is divisible by 4 as well as 5. It is crucial that the factors are coprime. For example, though 10 × 2 = 20, since 10 and 2 are not coprime, it cannot be asserted that if a number is divisible by both 10 and 2, then it will be divisible by 20 as well. You should be able to find a counterexample to this statement. Divisibility tests by primes such as 7, 13, 17 and 19 are not generally discussed in the school curriculum. However, in Vedic Mathematics (also known by the name “High Speed Mathematics”; see Box 1), techniques for testing divisibility by such primes are discussed, but without giving any proofs. In this article, proofs of these techniques are discussed

Fractional triplets

Consider this puzzle. and Oldest – each receive some money in the form of some inheritance. Youngest keeps half the amount he receives and Three brothers – Youngest, Middle divides the balance equally among Middle and Oldest. After Middle receives his share, he too keeps half the amount and distributes the balance equally among Youngest and Oldest. Oldest, in turn, keeps half of the amount he now has (after receiving the shares from his brothers) and divides the remaining equally among Youngest and Middle. At the end, if all three of them have equal amounts, can we determine how much each of them had at the start? If so, how do we do so? If not, why not

The Good, the Bad and the Lucky: CEO Pay and Skill

CEO compensation varies widely, even within industries. In this paper, we investigate whether differences in skill explain these differences in CEO pay. Using the idea that skilled CEOs should be more likely to continue prior good performance and more likely to reverse prior poor performance, we develop a new methodology to detect whether skill is related to pay. We find that highly paid CEOs are more skilled than their less well paid peers when pay is performancebased and when there is a large shareholder. This detected link between pay and skill is strong even when we examine industry-wide declines: highly paid CEOs are more likely to reverse the firm’s fortunes. We also examine CEO turnovers and show that the firm’s post-turnover performance is related to differences between the two CEO’s pay levels. These results highlight conditions where pay and skill are linked, and hence identify firms where high pay appears to have no justification

Takeovers, Governance and The Cross-Section of Returns

This paper considers the impact of the takeover channel on firm valuation. We use the idea that takeover activity responds to investor expectations of future rate of return and hence to state variable(s) related to the time variation in risk premia. Thus firms with higher exposure to takeovers, due to higher expectations of receiving a takeover premium, have a higher exposure to the state variable that dictates time variation in risk premia. Consequently, the difference in the returns between firms that differ in their takeover vulnerabilities can be used to used to proxy these state variables. To do so, we create a takeover-spread portfolio that buys firms with low cash-adjusted-leverage (cheaper targets) and shorts firms with high cash-adjusted-leverage and show that such a portfolio generates annualized abnormal returns of up to 11.20% between 1980 and 2003. Also, abnormal returns associated with governance-spread portfolios (Gompers, Ishii and Metrick, 2003 and Cremers and Nair, 2004) decrease significantly once the asset pricing model includes this ’cash-adjusted-leverage’ factor. Finally, we propose a new ‘takeover’ factor to proxy for the risk due to changes in these risk-premia related state variables, which is shown to be important in explaining cross-sectional differences in equity returns. The paper shows why investors require a higher rate of return on firms exposed to takeovers and yet value them higher than firms protected from takeovers

The Impact of Shareholder Control on Bondholders

This paper investigates the effect of shareholder control on bondholder wealth. While stronger shareholder control can benefit bondholders by disciplining managers, it also increases the likelihood of events that can hurt bondholders, e.g. hostile takeovers. We hypothesize that shareholder control can have contrasting effects on bond yields depending on the takeover vulnerability of a firm. Using the presence of an institutional blockholder to proxy for shareholder control and firm-level anti-takeover provisions to proxy for takeover vulnerability, we find that shareholder control is associated with lower yields if the firm is protected from takeovers. We also find that shareholder control is associated with higher yields if the firm is exposed to takeovers. The contrasting effects of shareholder control on yields are the strongest for firms that are small and have low leverage. In the presence of shareholder control, the difference in bond yields due to differences in takeover vulnerability can be as high as 93 basis points. Further, the results are insignificant for a sub-sample of firms where the bondholders are protected from takeovers through the poison put covenant. Bond ratings also appear to incorporate a similar effect of shareholder control on bondholders Finally, we find that a bond pricing model that does not account for shareholder control generates an annualized abnormal return of 1% to 1.4% for portfolios that long firms with both strong shareholder control and high takeover vulnerability and short firms without either shareholder control or takeover vulnerability. Combined, these results suggest that the use of different governance mechanisms, such as shareholder monitoring and takeover vulnerability, depends on a firm’s capital structure and that bond-pricing models should account for shareholder control