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August 31, 2015

Recap of the 2015 World Championships in Athletics

Filed under: sport — ckrao @ 1:18 pm

The 15th IAAF World Championships in Athletics recently concluded in Beijing, with Kenya and Jamaica each winning 7 gold medals. Below are some of the many highlights that are worth mentioning.

  • Usain Bolt became the most decorated World Championship athlete of all time taking his tally to 11 gold and 2 silver. In the 100m his best time this year out of just three races leading into the meet was 9.87s. After recovering from an early stumble in the semi final to qualify, he came over the top in the final by 0.01s to end Justin Gatlin‘s 28-race winning streak.
  • Ashton Eaton broke his own world record in the decathlon, the only world record to fall at the meet. His time of 45.00s for the 400m was the fastest ever in a decathlon.
  • Christian Taylor had the second longest triple jump of all time of 18.21m, just 8cm off the world record set in 1995. He and Pedro Pichardo were threatening something this special after both have cleared 18m this year.
  • In the women’s hammer throw Anita Włodarczyk continued her stellar 2015 with another second throw beyond 80m this month. She now has the 9 of the top 13 throws of all time.
  • The women’s 200m was possibly the best such race ever with Dafne Schippers and Elaine Thompson beating their personal best times by around 0.4s and becoming the 3rd and 5th fastest over the distance of all time.
  • Almaz Ayana won the women’s 5000m by over 100 metres, with her last 3000m covered in 8:20.
  • In the men’s marathon Ghirmay Ghebreslassie became the youngest ever world champion in the event at just 19 years of age. He also gave Eritrea its first ever gold medal at the championships.
  • Allyson Felix won her 9th world championship gold medal with a personal best of 49.26s in the 400m. She also ran the third fastest 400m split ever (47.72s) in the 4x400m relay.
  • Mo Farah repeated his 5000-10000m double from the World Championships two years ago. His last 800m of the 5000 was completed in 1:48.6.
  • Ezekiel Kemboi won the 3000m steeplechase for the fourth time in a row.
  • Jesús Ángel Garcia made his 12th appearance at the World Championships (50km walk) dating back to 1993. The event was won by Matej Tóth by almost 2 minutes despite a bathroom break in the middle.
  • Only 0.05s separated the top five in the women’s 100m sprint with Shelly-Ann Fraser-Pryce triumphant again (she now has 7 gold meals at the Worlds).
  • The men’s 4x100m had plenty of average baton handovers (including by winners Jamaica), but China’s were flawless and were rewarded with silver from lane 9.
  • Kenya triumphed in the men’s 400m hurdles and javelin events (Nicholas Bett and Julius Yego respectively, not only the distance events.
  • The women’s javelin was one of the most exciting events with multiple lead changes and Katharina Molitor won with a personal best and 2015-best in the very last throw of the competition.
  • Asbel Kiprop became three-time champion in the men’s 1500m with just 0.41s separating the top 5.
  • Recent world record holder Genzebe Dibaba won the women’s 1500m with 1:57.3s for the final 800m, faster than the winning time in the women’s 800m.
  • The men’s 400m was very fast with the top three (led by Wayde van Niekerk) finishing under 44 seconds for the first time ever. Also an Asian record of 43.93s was set by Yousef Masrahi from Saudi Arabia in the heats.
  • Jessica Ennis-Hill won the hepathlon, just over a year after giving birth to a second child.
  • World record holder Aries Merritt won bronze in the 110m hurdles with his kidneys functioning at just 20%, just two days before a scheduled kidney transplant.
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August 28, 2015

The discriminant trick

Filed under: mathematics — ckrao @ 12:22 pm

Suppose you wish to find the maximum value of y =\frac{8}{2x+1} - \frac{1}{x} for x > 0. One way to do this without calculus is to massage the expression until one can apply an elementary inequality such as u + \frac{1}{u} \geq 2. To apply this particular result we aim to minimise 1/y and apply polynomial division.

\begin{aligned}  \frac{1}{y} &= \frac{1}{\frac{8}{2x+1} - \frac{1}{x}}\\  &= \frac{x(2x+1)}{8x - (2x+1)}\\  &=\frac{2x^2+x}{6x-1}\\  &= \frac{(6x-1)(x/3 + 2/9) + 2/9}{6x-1}\\  &= \frac{1}{9}\left( 3x+2 + \frac{2}{6x-1} \right)\\  &= \frac{1}{9}\left( \frac{5}{2} + \frac{6x-1}{2} + \frac{2}{6x-1} \right)\\  &\geq \frac{1}{9}\left(\frac{5}{2} + 2 \right)\\  &= \frac{1}{9}\left(\frac{9}{2}\right)\\  &= \frac{1}{2}\quad\quad \quad \quad \quad (1)  \end{aligned}

In the above steps we assume 6x-1 > 0, otherwise y is non-positive for x > 0. Hence y \leq 2 with equality when \frac{6x-1}{2} = 1 or x = 1/2.

Another elementary way that applies to quotients of quadratic polynomials is to re-write the expression as a quadratic in x:

\begin{aligned}  y &= \frac{8}{2x+1} - \frac{1}{x}\\  x(2x+1)y &= 8x - (2x+1)\\  2x^2y + (y-6)x + 1 &= 0.\quad \quad \quad \quad \quad \quad (2)  \end{aligned}

For fixed y this quadratic equation will have 0, 1 or 2 solutions in x depending on whether its discriminant is negative, zero, or positive respectively. At any maximum or minimum value y_0 of the function, the discriminant will be zero since on one side of y_0 the quadratic equation will have a solution (discriminant non-negative) while on the other it will not (discriminant negative). In the image below a maximum is reached at y_0 = 2 while it is of opposite signs either side of this.

discriminant

Hence setting the discriminant of the left side of (2) to 0, (y-6)^2 - 4(2y) = 0 from which y^2 - 20y + 36 = (y-18)(y-2) = 0. Hence extrema are at y=2 and y = 18. We can solve (y-18)(y-2) \geq 0 to find that y \leq 2 or y \geq 18 is the range of the function y = \frac{8}{2x+1} - \frac{1}{x}. This tells us that y = 2 is a local maximum (illustrated above) and y = 18 is a local minimum (occurring when x < 0).

fullplot

One advantage of this method is that unlike elementary calculus, one bypasses the step of finding the corresponding x value (i.e. by solving dy/dx = 0) before substituting this into the function to find the extremum value for y.

Another advantage is that the equation need not be polynomial in y. For example below is a plot of x^2 + 4x\sin y + 1 = 0. Using the above-mentioned discriminant trick we solve 16 \sin^2 y - 4 \geq 0 and find the range of the function is when \displaystyle \sin^2 y \geq 1/4, or y \in \bigcup_{k \in \mathbb{Z}} [k\pi + \pi/6, k\pi + 5\pi/6] . Below is a plot confirming this using WolframAlpha.

x^2+4xsiny

The reader is encouraged to try out other examples, for example this method should work for any equation of the form \displaystyle h(y) = \frac{ax^2 + bx + c}{dx^2 + ex + f} where h is a continuous function of y. Of course one should also take care in noting when the function is defined before cross-multiplying.

Further Reading

[1] Finding the range of rational functions – Mathematics Stack Exchange

[2] Find Range of Rational Functions – analyzemath.com

[3] Critical point (mathematics): Use of the discriminant – Wikipedia, the free encyclopedia

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