Chaitanya's Random Pages

November 30, 2013

Filed under: cricket,sport — ckrao @ 11:54 pm

Recently retired Sachin Tendulkar is probably the sportsman who gave me the most total enjoyment over my lifetime. His peak batting period was probably from 1993 to the 2003 World Cup but he came back from injury well to become ICC Player of the year in 2010 at age 37 and after more than 20 years of playing internationally. He has been adored by millions and through all the pressure and expectations over such a long period managed to remain a likeable humble figure. His solid compact batting technique allowed him to flourish in both longer and shorter formats of the game better than almost all players during his time (he played with or against 982 others in international matches!). He also chipped in handily with the ball taking over 150 ODI wickets – Steve Waugh said Tendulkar could spin the ball more than regular spinners and could have taken 100 wickets had he put his mind to it [ref].

Here are his test and ODI career summarised by two graphs. The last 30 innings averages are intended his “form” at the time.

Below are some links collected about his career highlights and statistics, I may add to these over time.

HowSTAT! player profile for tests, ODIs (including career graphs for tests and ODIs)

Cricbuzz:

Video:

November 29, 2013

Critical points of polynomials with respect to the roots

Filed under: mathematics — ckrao @ 1:28 pm

Recall that a critical point of a polynomial $p:\mathbb{C} \rightarrow \mathbb{C}$ is a point $w$ for which $p'(w) = 0$. The Gauss-Lucas theorem states that the critical points of $p$ all lie within the convex hull of its set of zeros and we shall explain why this holds below. The real-number analogue of this is that the stationary points of a polynomial with real roots all lie between the smallest and largest root.

Denote the roots of the polynomial by $z_1, z_2, \ldots, z_n$ (we’ll assume they are distinct) and the critical points by $w_1, w_2, \ldots, w_{n-1}$. We then may write our polynomial as $p(z) = k \prod_{i=1}^n (z - z_i)$ for some constant $k$ and its derivative by the product rule is

$\displaystyle p'(z) = k \sum_{j=1}^n \prod_{i=1}^n \frac{z-z_i}{z-z_j} = p(z) \sum_{j=1}^n \frac{1}{z-z_j}.\quad \quad ...(1)$

If $p(w_i) \neq 0$ then as $w_i$ is a critical point of $p$, $p'(w_i) = 0$ and so

$\displaystyle 0 = \sum_{j=1}^n \frac{1}{w_i-z_j} = \sum_{j=1}^n \frac{\bar{w_i} - \bar{z_j}}{|w_i - z_j|^2},$

from which

$\displaystyle w_i \sum_{j=1}^n \frac{1}{|w_i - z_j|^2} = \sum_{j=1}^n \frac{z_j}{|w_i - z_j|^2}.\quad\quad...(2)$

In other words,

$\displaystyle w_i = \sum_{k=1}^n \alpha_k z_k,\quad\quad...(3)$

where $\alpha_k = \frac{1/|w_i - z_k|^2}{\sum_{j=1}^n 1/|w_i - z_j|^2}$.

Since $\sum_{k=1}^n \alpha_k = \frac{\sum_{k=1}^n 1/|w_i - z_k|^2}{\sum_{j=1}^n 1/|w_i - z_j|^2} = 1$, $w_i$ is a convex combination of the roots $z_k$ of $p(z)$. In other words, $w_i$ is in the convex hull of the set of roots $z_k$.

In the other case of $p(w_i) = 0$, $w_i$ is equal to one of the roots and so is also in the convex hull of the set of roots. This completes the proof of the theorem.

However as mentioned in [1] an interpretation of (3) by Cesàro in 1885 is that if we fix unit masses at the roots $z_i$ (if there are repeated roots place mass equal to the multiplicity), the critical points will be at precisely those points experiencing zero net force, assuming particles repel with a force proportional to the inverse of the distance between them (as opposed to an inverse square law). To see why this is so, if the force at a point $w$ due to a mass at $z$ is proportional to the vector $\frac{1}{|z-w|^2}(z-w)$ (an inverse law, as this has magnitude $1/|z-w|$), then the net force is the sum of these over the masses at all the roots. In terms of any critical point $w_i$, this is the following:

\begin{aligned} \sum_{k=1}^n \frac{1}{|z_k-w|^2}(z_k-w) &= \sum_{k=1}^n \frac{1}{|z_k-w|^2}z_k- w\sum_{k=1}^n \frac{1}{|z_k-w|^2}\\&= w_i\sum_{k=1}^n \frac{1}{|z_k-w|^2}- w\sum_{k=1}^n \frac{1}{|z_k-w|^2} \quad \text{by (2)}\\&=(w_i-w)\sum_{k=1}^n \frac{1}{|z_k-w|^2}, \end{aligned}

which is zero if and only if $w=w_i$, i.e. the point having zero net force is a critical point of $p$.

Bôcher’s theorem generalises this result, replacing polynomials with rational functions (ratios of polynomials) and simply applying negative unit masses at poles.

Another interesting result, mentioned in [2], is that the product of the distances from a root $z_1$ to the other roots (assuming all distinct roots) is precisely $n$ times the product of the distances from $z_1$ to the critical points $w_j$. That is,

$\prod_{i=2}^n |z_1 - z_i| = n \prod_{j=1}^{n-1} |z_1 - w_j|.\quad\quad ...(4)$

To see this, if $p(z) = k \prod_{i=1}^n (z-z_i)$, $p'(z)$ will have the form $kn \prod_{j=1}^{n-1} (z-w_j)$ (since it has leading term $nz^{n-1}$. Note that $z_1$ is not a critical point since we assume the roots are distinct. Hence we may consider the quotient

\begin{aligned} \frac{\prod_{i=2}^n (z_1 - z_i)}{\prod_{j=1}^{n-1} (z_1 - w_j)} &= \frac{\lim_{z \rightarrow z_1} p(z)/(z-z_1)}{p'(z_1)/n}\\&= \lim_{z \rightarrow z_1} \frac{np(z)}{(z-z_1)p'(z)}\\ &= \lim_{z \rightarrow z_1} \frac{np(z)}{(z-z_1)p(z) \sum_{j=1}^n 1/(z-z_j)} \quad \text{from (1)} \\&= \lim_{z \rightarrow z_1} \frac{n}{\sum_{j=1}^n (z-z_1)/(z-z_j)} \\ &= \lim_{z \rightarrow z_1} \frac{n}{1 + \sum_{j=2}^n (z-z_1)/(z-z_j)}\\ &= n.\end{aligned}

Taking magnitudes of both sides leads to the desired result (4). One also can obtain an angular relationship by taking the argument of both sides.

The critical points for the case $n = 3$ have a beautiful geometric interpretation as described in [3]: from the triangle formed by the roots, the critical points are the foci of the largest ellipse that is inscribed in the triangle (this ellipse also happens to pass through the triangle’s midpoints) – this is now known as Marden’s theorem. A generalisation proved by Marden in 1945 and mentioned in [1] is that the critical points of an $n$-degree polynomial are the foci of a degree $n-1$ plane curve tangent to each of the $n(n-1)/2$ line segments formed by taking pairs of the $n$ roots. The points of tangency divide each line segment into a ratio corresponding to the multiplicities of the roots at the endpoints. Also, if the $n$-dimensional polygon is a linear (affine) transform of a regular polygon, it will have an inscribed ellipse passing through its midpoints (called the Steiner inellipse) and the foci of that ellipse will correspond to critical points.

Below are some other nice results about the critical points of a polynomial, gathered from [4] (also see references therein):

• The centre of mass of the roots of $p$ coincides with the centre of mass of the critical points of $p$ (a nice exercise to prove).
• (Anderson) The polynomial root dragging theorem for polynomials with real distinct roots: as roots are dragged to the right by at most $\epsilon$ (so as not to coincide), the critical points also move to the right, each by less than $\epsilon$.
• (Boelkins et al., later Frayer) The polynomial root squeezing theorem for polynomials with real distinct roots: if two roots $z_i, z_j$ are squeezed together by the same amount without passing other roots, the critical points move towards $(z_i + z_j)/2$ or stay fixed.
• (Peyser) Critical points are not too close to the roots: if a polynomial has only real roots $z_1 < z_2 < \ldots < z_n$ and critical points $w_1 < w_2 < \ldots w_{n-1}$ the critical points satisfy
$\displaystyle z_k + \frac{z_{k+1} - z_k}{n - k + 1} \leq w_k \leq z_{k+1} - \frac{z_{k+1} - z_k}{k+1}, \quad k= 1, \ldots, n-1.$
Equality cases are satisfied by Bernstein polynomials. Replacing $1/(n-k+1)$ and $1/(k+1)$ by $m_k/(n-k+m_k)$ and $m_{k+1}/(k + m_{k+1}$ respectively gives a corresponding result (due to Melman) taking into account multiplicities $m_k$ for root $z_k$.

References

[1] Qazi Ibadur Rahman, Gerhard Schmeisser, Analytic Theory of Polynomials, Oxford University Press, 2002.

[2] Maxime Bocher, Some Propositions Concerning the Geometric Representation of Imaginaries, Annals of Mathematics, Vol. 7, No. 1/5 (1892 – 1893), pp. 70-72.

[3] Dan Kalman, The Most Marvelous Theorem in Mathematics,  The Journal of Online Mathematics and Its Applications, Vol. 8, March 2008, Article ID 1663.

[4] Neil Biegalle, Investigations in the Geometry of Polynomials, McNair Scholars Journal, Volume 13, Issue 1, Article 3. Available at: http://scholarworks.gvsu.edu/mcnair/vol13/iss1/3

Blog at WordPress.com.