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Archive for July, 2016

I need to update this, which I will do in the next week

thecuriousastronomer

With the men’s and women’s finals of Wimbledon coming up this weekend, I thought I would share this interesting chart that I came across recently in “The Economist” magazine. It shows the number of Grand Slam (major) titles won by different players since tennis became “open” in 1968. It also breaks it down into which of the four majors each player has won.

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If we look at the top five grand slam winnners in both the men’s and women’s games in more detail it is interesting to see some of the differences between the men’s and women’s all-time best. Note: I have re-ordered the table so that it is in the order the tournaments are played during the year, with the Australian Open (January), followed by the French Open (June), then Wimbledon (July) and finally the US Open (September). I have also worked out the percentage of each player’s total…

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Serena Williams has finally done it, she has equalled Steffi Graf’s record of 22 Major titles. It has taken her longer than most people expected, given that she had 21 after last year’s Wimbledon, but after three successive defeats in the intervening majors (one at the semi-final stage and two in the final), she is now equal to the great Steffi Graf.

Surely she will go on and win more; although she is now 34 she shows little sign of losing her appetite for tennis and for winning. Also, she is yet to win a Grand Slam, and I am under little doubt that she will try in 2017 (and 2018?) to do this, before retiring. Even if she goes on to win 24 or 25 or 26 Major titles, I think she would not feel satisfied of her legacy unless she can also hold all four titles in the same year.


There has been mention of Margaret Court’s haul of 24 Major titles. But, to my mind, this record does not count. Margaret Court did not turn professional until tennis had become ‘open’, and so from 1963 to 1968 she was not competing against some of the best players of her day, who had decided to turn professional. If Graf or Williams had many of the best players removed from the Majors in which they were playing, who knows how many titles they could have won. So yes, technically, Court won 24 Majors, but many of them were hollow ones, as she surely would now admit.

Serena Williams is a truly remarkable athlete. I love her passion, her energy, her commitment and her drive. I really do hope that in 2017 (or 2018) she can get her Grand Slam, which would not only cap a remarkable career, but ensure her at least 26 Major titles and seal her place as the greatest female tennis player in history.

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Today I thought I would share this poem by Wales’ most famous anglo-welsh poet, Dylan Thomas. I have blogged about Thomas before; in this blog I shared the opening passage of his radio play for voices, Under Milk Wood. The poem I am sharing today is one of his most famous – “Do not go gentle into that good night”, which he wrote in 1947 when he was 33 years old.

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Dylan Thomas wrote “Do not go gentle into that good night” in 1947. He would be dead himself just 6 years later, at the age of 39.

The poem deals with death, or rather the refusal to fade away in old age. “Rage, rage against the dying of the light.” Profound words for a 33-year old to write, and ironic that Thomas himself should never live to see old age. He drank himself to death just a few years after composing this poem, when he was only 39 years old.

Do not go gentle into that good night.
Old age should burn and rave at close of day;
Rage, rage against the dying of the light.

Though wise men at their end know dark is right,
Because their words had forked no lightning they
Do not go gentle into that good night.

Good men, the last wave by, crying how bright
Their frail deeds might have danced in a green bay,
Rage, rage against the dying of the light.

Wild men who caught and sang the sun in flight,
And learn, too late, they grieved it on its way,
Do not go gentle into that good night.

Grave men, near death, who see with blinding sight
Blind eyes could blaze like meteors and be gay,
Rage, rage against the dying of the light.

And you, my father, there on the sad height,
Curse, bless, me now with your fierce tears, I pray.
Do not go gentle into that good night.
Rage, rage against the dying of the light.

Here is a video of Thomas reading his poem. What a beautiful voice he had. Enjoy!

Which is your favourite Dylan Thomas poem?

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After last weekend’s matches we now know the line-up for the Euro 2016 semi-finals. This evening (Wednesday 6 July) at 20:00 BST (19:00 GMT) Wales play Portugal, then tomorrow evening (Thursday 7 July) the hosts France take on the World champions Germany.

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Wales play Portugal this evening (Wednesday), France play Germany on tomorrow (Thursday). Both matches start at 20:00 BST (19:00 GMT)

Wales’ match against Portugal is in Lyon, whilst the second semi-final is in Marseille. We in Wales are incredulous that we’ve got to the semi-finals, but we are also grateful to be facing Portugal and not France or Germany.

Of the teams to reach the semi-finals, Portugal have been by far the least impressive. They qualified 3rd in their group with three drawn matches. Then, in the round of 16, they only beat Croatia 1-0 in extra time. In the quarter finals they beat Poland, but this time only on penalties with it being 1-1 after extra time.

Of course, any team that has Cristiano Ronaldo in its ranks cannot be written off. With his brilliance he can turn a game around in a flash. But the contrast with Gareth Bale is interesting. Both play for Real Madrid, both are the stars of their teams, but there the similarities end. Bale is a team player, down to earth and a very popular member of the squad. The very opposite of a prima donna. Ronaldo is petulant, greedy and, one gets the impression, does not consider himself ‘one of the lads’. He is like a preening peacock; if he weren’t so brilliant he would surely be universally disliked.

Wales are, of course, riding on a wave of euphoria. The team will be bursting with confidence after our thrilling 3-1 win against Belgium on Friday evening. But, I am also confident that we will not take Portugal for granted. Chris Coleman is far too wily a coach to allow his team to do that. We will be missing midfielder Aaron Ramsey and defender Ben Davies, both suspended after picking up second yellow cards in the match against Belgium. But, we have such a structure  and game plan, which the team all believe in, that I am sure their replacements will step up admirably to to the high standard Ramsey and Davies have set.

It is going to be a very tense countdown to the big match in Wales! Dere ‘mlaen Cymru!!! (come on Wales!!!)

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Last week, in this blogpost here, I tried to explain why the scale height of the atmosphere H is defined as the altitude one has to go up for the pressure to drop by a factor of 1/e \;(\approx 37\%) of its value at sea level. After posting that blog I decided I would write a blog to say a little bit more about the mathematical constant e, a very important number in mathematics.

Probably the best known mathematical constant is \pi, which is defined as the ratio of the circumference of a circle to its diameter. Pretty much every school child comes across \pi; but it is only people who study maths at a more advanced level who come across e, so let me try in this blogpost to explain what e is and why it is so important.

Euler’s number

e is also know as Euler’s number, named after the Swiss mathematician Leonhard Euler who lived from 1707 to 1783. Euler was one of the great mathematicians, but it was not he who discovered e. The constant was discovered by Jacob Bernoulli (from the same family as the name attached to “the Bernoulli effect” which causes lift in the wings of an aeroplane), but it was Euler who started to use the letter ‘e’ to represent the constant, in 1727 or 1728.

The number itself is defined as the solution to the following sum

\displaystyle e \equiv \sum_{n=0}^{\infty} \frac{1}{n!} = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \frac{1}{5!} + .... = 2.718281828...

where 2! \text{ (called "two factorial") } = 2 \times 1, 3! = 3 \times 2 \times 1, \; 4! = 4 \times 3 \times 2 \times 1 etc. 0! \equiv 1 by definition. This is an example of a converging series. It is summed to infinity, but each term is smaller than the one before; but it never ends. So, as you go to more and more terms in the series you only affect numbers which are maybe 100 or even 1,000 places to the right of the decimal point. Calculators usually display numbers to 7 or 8 decimal places, so you would not need to go very far in this series to get the number displayed by your calculator (try it to find out how many!)

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The mathematical constant e is the sum of 1/n! where n goes from zero to infinity (\displaystyle \sum_{n=0}^{\infty} \frac{1}{n!}). It is equal to 2.718281828…….

Just like with \pi, e is a transcendental number (I will explain what that is in more detail in a future blogpost). Briefly, this means that it carries on forever and does not repeat, but it is slightly more complicated than that. Unlike with \pi, where people have competitions to remember it to thousands of decimal places (the current world record is 70,000 decimal places achieved by Rajveer Meena on 21 March 2015!!), no one seems that concerned in remembering e.

Why is e important in mathematics?

Mathematicians often love numbers and formulae for their own sake, sometimes just for their beauty. So, for example, the solution to something like

\displaystyle a = \sum_{n=0}^{\infty} \left( 1 + \frac{1}{n^{3}} \right)

(which I just made up) may not have any importance mathematically, but still mathematicians may enjoy playing and exploring such a series. But, in the case of

\displaystyle e \equiv \sum_{n=0}^{\infty} \frac{1}{n!}

the number which comes form this, 2.718281828……., is important mathematically (and physically), and here I will try to explain why.

Compound interest

I mentioned above that e was discovered by Jacob Bernoulli. His discovery was made in 1683 when he was investigating a question concerning compound interest. Remember, compound interest is when you get a certain percentage interest on the amount you have in e.g. a bank, but the interest is calculated not on the initial amount but on the amount after the previous period’s interest has been added.

Suppose we invest £10 in a bank account which pays an interest of 5% per year, and we leave it there for 3 years. Assuming the interest is added just once a year, after the first year our £10 will have earned £0.50 interest, so we will have £10.50. At the end of the second year the interest earned will be 5% of £10.50 which is £0.53 (rounding up to the nearest penny), so we will now have £11.03 at the start of year 3. The interest at the end of year 3 is going to be 5% of £11.03 which is £0.55, so the total at the end of the 3  years is £11.58.

Doing this for just 3 years manually is quite easy, but if we wanted to do it for e.g. 25 years, there would be a lot of tedious calculation. Also, often the interest is added more than once a year. So, for example, you may have an annual interest rate of 5% but it is added each quarter. You can quickly see that even doing this manually over a 3-year period would be a lot of calculating.

Thankfully, there is a simple formula for calculating the total accumulated value, which is

P(1 + i/n)^{nt}

where P is the initial amount invested (the ‘principal’), i is the rate of interest, n is how often each year the interest is added (called the ‘compounding frequency’) and t is the time for which we are making the calculation, expressed in years.

If we go back to our example above, and stick to n=1, we have P=10, \; i=0.05, \; t=3 and so

P(1 + i/n)^{nt} = 10(1+0.05)^{3} = 10(1.05)^{3} = 10(1.157625) = 11.58

exactly as we calculated manually.

But, you may be wondering, what is the similarity between the formula

P(1 + i/n)^{nt}

and the formula for e

\displaystyle e \equiv \sum_{n=0}^{\infty} \frac{1}{n!}?

What Bernoulli noticed was that, if you make n larger and larger (do the compounding daily instead of quarterly or once a year), the sequence approaches a limit. So, for example, in the above example, with n=1 we found the amount at the end was £11.58. If we made n=4 (compounding quarterly) we would get £11.61 and if we compounded the interest every day (n=365) we would get £11.62. If we compounded every hour (!!!) (n=8760) we would get £11.62, the same answer as if we compound every day, so we have reached the limit.

What Bernoulli did was consider the formula with P=1, \; i=100\% and t=1. If we do this for different values of n we get the following curves. The first one is just showing n from 0 to 20, and it is clear that the values are flattening out. The second plot goes from n=0 to 500, and I have just shown on the y-axis values from 2.5 to 2.72. This shows even more clearly that, as n gets larger, the value of 1(1+1/n)^{n} tends towards a particular value, and that value turns out to be 2.71828… (which is e).

 

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y = 1(1+1/n)^{n} for n from 0 to 20

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y=1(1+1/n)^{n} for n from 0 to 500, but note the y-axis is only displayed between 2.5 and 2.72. The curve is clearly tending towards a value, and that value is e=2.71828...

e in calculus

Even if you don’t know how to do calculus, you have probably heard of it. It was co-invented by Isaac Newton and Gottfried von Leibniz (see my blogpost “Who Invented Calculus” for the fascinating story of the 30-year feud between Newton and Leibniz). Without going into too much detail about all the various things one can do with calculus, one thing is that it gives is the gradient (slope) of a curve at a given point (something which is sometimes called the derivative).

There are an infinite number of different mathematical functions, for example y=x^{2}, or y=x^{3} - 2x + 3, and we can use differentiation to determine the gradient of these functions for any particular value of x. But, the function y=e^{x} is unique; it is the only mathematical function whose derivative is the same as the function. To put this another way, the slope of the curve y=e^{x} is e^{x} for any value of x, and there is no other mathematical function whose derivative is the same as the function, only f(x)=e^{x}.

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A plot of e^{x} as a function of x. At x=0, \; e^{x}=1. As x \rightarrow -\infty, \; x \rightarrow 0.

In addition, when we integrate something like dx/x, we get a logarithm; but the base of that logarithm is e, not base 10 (our usual base of counting). In fact, we call logarithms in base e natural logarithms. Because the derivation of the variation of pressure with altitude involves integrating dP/P, we find that the vertical distribution of pressure is logarithmic, but in base e, P=P_{0}e^{-z/H}, where H is the scale height and P_{0} is the pressure at sea level. It is because of the pressure’s exponential dependence on altitude that H is usually expressed as the value for the pressure to drop by a factor of 1/e.

The normal (or gaussian) distribution

As one last example of where e pops up in mathematics, it arises in the equation which describes the normal or gaussian distribution. I blogged about that distribution in this blogpost here “What does a 1-sigma, 3-sigma or 5-sigma detection mean?”. The function which describes the normal distribution has the form

f(x) = \frac{1}{\sqrt{2\pi}} e^{-x^{2}/2}

where e is our friend, Euler’s number.

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The normal, or Gaussian, distribution y = \frac{1}{\sqrt{2\pi}}e^{-x^{2}/2}

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Later this morning (Monday 4 July) I will be on BBC radio talking about NASA’s Juno mission to the planet Jupiter. This is the latest space probe to be sent to study the largest planet in the Solar System, and follows on the highly successful Galileo spacecraft which studied Jupiter in the 1990s.

Juno left Earth in August 2011 and is due to arrive at Jupiter today. But, in order to go into orbit about the planet a rocket needs to be fired to slow the spacecraft down and put it into orbit. This is due to happen tomorrow (Tuesday 5 July). The rocket engine which will do this was built in England. If the ‘burn’ fails, the mission will fail, as the space probe will just hurtle past Jupiter and continue on into the outer Solar System.

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NASA’s Juno satellite was launched in August 2011 and arrives at Jupiter this week. It will be put into a polar orbit about the planet, but with a highly elliptical orbit which will take it out beyond Callisto’s orbit. Each orbit will take 14 days.

What are Juno’s scientific objectives?

In addition to studying Jupiter, the Galileo spacecraft spent a great deal of time studying her four large moons; Io, Europa, Ganymede and Callisto. Galileo was in an equatorial orbit. Juno, on the other hand, will be put into a polar orbit. Its main objective is to study Jupiter, rather than its moons.

Jupiter is what is known as a gas giant. It is mainly hydrogen, and contains more mass than all the other planets in the Solar System put together. In fact, it is a failed star; if it were some 10 times more massive it would have had enough mass to ignite hydrogen fusion in its core. Even though it is not burning hydrogen, it is still leaking heat left over form its collapse into a planet 4.5 billion years ago.

In the last 20 years we have discovered many Jupiter-like planets orbiting other stars. Most of these are much closer to their parent star than Jupiter is to the Sun, and this has raised questions about how gas giants can be so close to their parent star, and how is Jupiter where it is in our Solar System? Jupiter is about five times further away from the Sun than the Earth is, and much further away than the Jupiter-like planets we have found around other stars. Did Jupiter start off closer to the Sun and get kicked further out, or did it migrate inwards from further away? We don’t know.

Some of the things Jupiter hopes to determine are

  • the ratio of oxygen to hydrogen in Jupiter’s atmosphere. By determining this ratio it will effectively be measuring the amount of water, which will help distinguish between competing theories of how Jupiter formed.
  • the mass of the solid core believed to lie at the planet’s centre, deep below the very thick and extensive atmosphere. This also has implications for its origin.
  • the internal structure of Jupiter – it will do this by precisely mapping the distribution of Jupiter’s gravitational field.
  • its magnetic field to better understand its origin and how deep inside Jupiter the magnetic field is created.
  • the variation of atmospheric composition and temperature at all latitudes to pressures greater than 100 bars (100 times the atmospheric pressure at sea level on the Earth).

Juno has a funded operational lifetime of about 18 months. In order to better study the interior of Jupiter, the spacecraft will plunge into the planet’s atmosphere in February of 2018, making measurements as it does so.

++UPDATE++

Juno’ rocket successfully fired at about 3:20 UT today (Tuesday 5 May) and is now in orbit about Jupiter. It will complete two large 53-day orbits before being inserted into its 14-day orbit for science operations. This 14-day orbit is highly elliptical, and at its closest the probe will come to within 4,300 km of the cloud tops. 

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Yesterday (Saturday 2 July) saw reigning Wimbledon champion Novak Djokovic crash out of Wimbledon in the third round to American Sam Querrey. Not only was Djokovic defending his title,  but he is currently the holder of all four Major titles, a feat not done since Rod Laver in 1969, as I mentioned in this blog here.

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Novak Djokovic suffered a shock defeat in the 3rd round of Wimbledon, brining to a halt his dream of doing the Grand Slam (holding all four Major titles in the same calendar year).

Djokovic’s defeat of course opens up the draw to other contenders. As Djokovic was seeded number 1, and Andy Murray number 2, it will make no difference to Andy Murray unless he reaches the final as they are in opposite parts of the draw. However, it may make a difference to my own favourite, Roger Federer, as he is seeded number 3. Federer would have met Djokovic in the semi-finals if they had both progressed.

I will keep my fingers crossed that this time next week I will be looking forward to a men’s singles final featuring Federer. I so want to see him surpass the record he currently jointly holds with Pete Sampras and win his 8th Wimbledon title. We shall see.

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