Baseball oh baseball.
Another wonderful season has just ended. A late congratulation to Lamigo monkeys who gave a blessed display in G5-G7, the no hitter has been stunning and it was unimaginable that he had just played as a relief pitcher in the last game. Whilst on the opposite side of the globe, Cubs fans may have to wait another hundred years...
Baseball is the last sport that I've been paying attention to its professional leagues, other than football. That's a nice complement out of the football season after all, but that isn't the end of the story. They are completely different in nature.
Football, I am not sure if all of you would agree, is a chaotic process in nature. From the extremely wide varieties of tactics that leads to rigidity in substituting players, to a full dynamic process during the game, has made the whole season full of dependence and has random events leading all the way through. That doesn't seem to be obvious because every particular professional games has their strategy cautiously set up and ruthlessly executed, and being a full dynamic game the advantage accumulates --- and become decisive in the game. However looking the game in a more macroscopic way tells a different story. I learned that after following football for 10 years, and I bet it also took Abramovich also 8 years to realize that --- in 2012 --- the best team is not always going to win the champions league --- the phantom goal in 2005, the penalty showdown in 2006 and 2008, the Stamford Bridge massacre in 2009...and the against-all-the-odds display in 2012. It's the accumulation from the whole season that makes the slightest edge in the later stages of the Champions League. [Just a famous quote from zonal marking that I do agree quite a lot: semi-final of the champions League is always the most exciting, unpredictable, technically rich game in the whole tournament. And for me, being the semi-finalist is all about team strength and going any further requires a steel of mind with brilliantly delivered tactics, and loads of luck.]
Baseball however, has it quantity overwhelmed so that the game boils down onto data that really matters. Batter faces all sorts of pitchers and pitcher faces all kinds of batters, on a daily basis so the data representing their efficiency shows high correlation with their real performance. AVG [well some may think that being not very useful, especially for Moneyball fans], OBP, OPS --- accurately measures how efficient they score; ERA, H/9, WHIP --- predicts how likely pitches are going to allow runs. Whether it is a 100 mph 4-seam blitz, or a nasty curveball --- it does not matter --- in long run. Teams with proper farms may substitute a player by another without much difficulty, perhaps a drop of .05 in OBP, no one is really untouchable statistically. Drafting and exchange market, FA system, money and players lurking around --- definitely not the so called 'traditional' way in sports industry huh? But that's not the reason I love baseball. Taking a closer look into the game every single duel between the batter and the pitcher must be the star of the game. Look at the mind battle between them, their logical reasoning and physical reaction [baseball is 90% physical, and the other half is mental.], how can't you love that?
Two sports from two different culture, starring two different aspects in nature. I don't expect many to follow both as I do, but that will be on my recommended list seriously.
Oh, of course professional leagues aren't all of the sports available. We always have something called a union --- once again big, big congrats to the All Blacks, who again gave an astonishing display throughout. Hopefully I will have the honor to watch it live in Japan, 2019.
Hopefully.
Wednesday, 4 November 2015
Thursday, 25 June 2015
Math Girls; Galois theory
It is never easy to teach someone else, and it is even harder to teach others an advanced topic. In university the ideal case would be giving the motivation during the lecture and students do the rest by themselves during self-study or supervision. But, if we really ought to teach someone from the very beginning, what would be the best choice? This is of course a very complicated question and for sure I cannot give the answer here.
My way to do it is to follow the historical treatment - doing what those mathematicians did hundreds of years ago. What they did, why they did so --- this is exactly what helps students to understand the mechanism behind a topic. Landau styled definition-theorem-proof aren't bad, they are just a bit too hard for those understanding ability not being the best.
Based on the above I have found a series interesting, the Math Girls by H.Yuki. There are 5 vol. available in Chinese and Japanese, and the first 3 vol. are published in English. Such series perfectly illustrates the above. Topics delivered via conversation and story-telling, and by merging yourself into the discussion the history behind will simply push you all the way through to the end of the journey.
It is mostly easy for senior secondary school students --- mostly refers to the most part of the book. It starts basically from stretch and the difficulty gradually increase, but it is usually tolerable till the last chapter. Nonetheless one will be able to appreciate the theory without precisely understanding the technical procedures.
A few months back I've just finishing reading his 5th book of the series on Galois theory. One of the clear motivation behind the theory is the irresolvability of quintic equations in radicals. This is of course a very hard question, that had puzzled mathematicians for three hundred years, until a genius called Galois came up with his genius idea (that no one can realize till some years later) a day before his duel, where he lost and died. It was a sad story (he is one of the many French mathematicians that died in weird ways), but this is not the main point --- where was his idea came from?
It is the permutation of roots.
Classical algebra books will illustrate this via quadratic equations, cubic equations and quadric equations. However this is not very clear under numerical examples beyond the quadratic case. In particular, Cardano's solution for cubic equation is fairly unpleasant. There are so many radicals added --- which one gives an extension? Which one does not? What is the degree of extension?... When I first tried to read through graduate texts on Galois theory I kept asking questions like this to myself and got myself puzzled.
In the book they put some effort analyzing those equations in an elemental way. No groups, no fields --- this is also what mathematicians did in the past. I personally found that extremely useful by the end of the day, and I loved that book.
It might be a bit more approachable using modern language than Galois' first thesis, but it does not change the fact that his thesis is showing the core idea of the whole theory, so following the historical treatment it would be the best for one to follow the thesis to explore the elemental part of his theory. That was not obvious until the very last bit [Kummer extensions, main theorem for quintic equations], so I will try to fill that gap here, if I have time, probably in the coming entry.
And yeah, for those who are interested in mathematics [in particular if you know how to read Chinese or Japanese], I would sincerely recommend that to you.
*
Theorem. $x^5-16x + 2 = 0$ has no radical solution over $\mathbb{Q}$.
*
My way to do it is to follow the historical treatment - doing what those mathematicians did hundreds of years ago. What they did, why they did so --- this is exactly what helps students to understand the mechanism behind a topic. Landau styled definition-theorem-proof aren't bad, they are just a bit too hard for those understanding ability not being the best.
Based on the above I have found a series interesting, the Math Girls by H.Yuki. There are 5 vol. available in Chinese and Japanese, and the first 3 vol. are published in English. Such series perfectly illustrates the above. Topics delivered via conversation and story-telling, and by merging yourself into the discussion the history behind will simply push you all the way through to the end of the journey.
It is mostly easy for senior secondary school students --- mostly refers to the most part of the book. It starts basically from stretch and the difficulty gradually increase, but it is usually tolerable till the last chapter. Nonetheless one will be able to appreciate the theory without precisely understanding the technical procedures.
A few months back I've just finishing reading his 5th book of the series on Galois theory. One of the clear motivation behind the theory is the irresolvability of quintic equations in radicals. This is of course a very hard question, that had puzzled mathematicians for three hundred years, until a genius called Galois came up with his genius idea (that no one can realize till some years later) a day before his duel, where he lost and died. It was a sad story (he is one of the many French mathematicians that died in weird ways), but this is not the main point --- where was his idea came from?
It is the permutation of roots.
Classical algebra books will illustrate this via quadratic equations, cubic equations and quadric equations. However this is not very clear under numerical examples beyond the quadratic case. In particular, Cardano's solution for cubic equation is fairly unpleasant. There are so many radicals added --- which one gives an extension? Which one does not? What is the degree of extension?... When I first tried to read through graduate texts on Galois theory I kept asking questions like this to myself and got myself puzzled.
In the book they put some effort analyzing those equations in an elemental way. No groups, no fields --- this is also what mathematicians did in the past. I personally found that extremely useful by the end of the day, and I loved that book.
It might be a bit more approachable using modern language than Galois' first thesis, but it does not change the fact that his thesis is showing the core idea of the whole theory, so following the historical treatment it would be the best for one to follow the thesis to explore the elemental part of his theory. That was not obvious until the very last bit [Kummer extensions, main theorem for quintic equations], so I will try to fill that gap here, if I have time, probably in the coming entry.
And yeah, for those who are interested in mathematics [in particular if you know how to read Chinese or Japanese], I would sincerely recommend that to you.
*
Theorem. $x^5-16x + 2 = 0$ has no radical solution over $\mathbb{Q}$.
*
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