Monthly Archives: May 2012

Sample variance

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In statistics, Bessel’s correction, named after Friedrich Bessel, is the use of n – 1 instead of n in the formula for the sample variance and sample standard deviation, where n is the number of observations in a sample: it corrects the bias in the estimation of the population variance, and some (but not all) of the bias in the estimation of the population standard deviation.

That is, when estimating the population variance and standard deviation from a sample when the population mean is unknown, the sample variance is a biased estimator of the population variance, and systematically underestimates it. Multiplying the standard sample variance by n/(n – 1) (equivalently, using 1/(n – 1) instead of 1/n in the estimator’s formula) corrects for this, and gives an unbiased estimator of the population variance.

— Wikipedia on Bessel’s correction

The two estimators only differ slightly as can be seen, and for larger values of the sample size n the difference is negligible. While the first one may be seen as the variance of the sample considered as a population, the second one is the unbiased estimator of the population variance, meaning that its expected value E[s^2] is equal to the true variance of the sampled random variable; the use of the term n – 1 is called Bessel’s correction.

— Wikipedia on Sample variance

2012.05.16 Wednesday ACHK

Batman Begins

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After a series of unsuccessful projects to resurrect Batman on screen following the 1997 critical failure of Batman & Robin, Nolan and David S. Goyer began work on the film in early 2003 and aimed for a darker and more realistic tone, with humanity and realism being the basis of the film.

Nolan stated his intention to reinvent the film franchise of Batman by “doing the origins story of the character, which is a story that’s never been told before”. Nolan said that humanity and realism would be the basis of the origin film, and that “the world of Batman is that of grounded reality. [It] will be a recognizable, contemporary reality against which an extraordinary heroic figure arises.” Goyer said that the goal of the film was to get the audience to care for both Batman and Bruce Wayne.

— Wikipedia on Batman Begins

2012.05.16 Wednesday ACHK

古怪題目 1.2

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這段改編自 2010 年 6 月 2 日的對話。

凡是公開試的題目，通常也不會太深，因為公開試的對象，是平民百姓，而不是一堆天才。如果有一個公開試，題目深到只有 牛頓、愛因斯坦 級數頭腦才能做到的話，那個就再不算是「公開」試了。

公開試的真正目的，並不是給予天才表演智力的機會，而是方便社會，尤其是大學，分辨學生的智力高低。換句話說，公開試的精神，是將莘莘學子，根據智力來分流。因為這個精神，每份試卷，都會分成幾個部分，由淺至深。如果是分成兩部分，甲部會是短題目，乙部則會是長題目。而每條長題目本身，亦會再分成幾部分，由淺至深。

背後的意思是，假設一題長題目分成 a, b, c, d, e 五部分，做到 a 部分的話，你就起碼可以有 E 級成績；再做到 b 部分的話，你就起碼可以有 D 級成績；如此類推。一題長題目需要完成多少，豐儉由人，視乎你想拿到什麼成績，或者你需要拿到什麼成績。真正考驗天才的，就只有 e 部分。而「e 部分」，就是你我剛才所講的「古怪題目」。連 e 部分都可以完成的話，你就可以拿到 A 級成績。

如果某一科的目標，並不是 A 級成績的話，你其實毋須太過介意，做不做到 e 部分。　

— Me@2012.05.16

2012.05.16 Wednesday (c) All rights reserved by ACHK

Forensic Accounting

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Benford’s law, also called the first-digit law, states that in lists of numbers from many (but not all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way. According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less than 5% of the time.

Mathematical statement

… this is the distribution expected if the logarithms of the numbers (but not the numbers themselves) are uniformly and randomly distributed. For example, a one-digit number x starts with the digit 1 if 1 <= x < 2, and starts with the digit 9 if 9 <= x < 10. Therefore, x starts with the digit 1 if log 1 <= log x < log 2, or starts with 9 if log 9 <= log x < log 10. The interval [log 1, log 2] is much wider than the interval [log 9, log 10] (0.30 and 0.05 respectively); therefore if log x is uniformly and randomly distributed, it is much more likely to fall into the wider interval than the narrower interval, i.e. more likely to start with 1 than with 9.

Explanations

Outcomes of exponential growth processes

The precise form of Benford’s law can be explained if one assumes that the logarithms of the numbers are uniformly distributed; for instance that a number is just as likely to be between 100 and 1000 (logarithm between 2 and 3) as it is between 10,000 and 100,000 (logarithm between 4 and 5). For many sets of numbers, especially sets that grow exponentially such as incomes and stock prices, this is a reasonable assumption.

Applications

In 1972, Hal Varian suggested that the law could be used to detect possible fraud in lists of socio-economic data submitted in support of public planning decisions. Based on the plausible assumption that people who make up figures tend to distribute their digits fairly uniformly, a simple comparison of first-digit frequency distribution from the data with the expected distribution according to Benford’s law ought to show up any anomalous results.

Following this idea, Mark Nigrini showed that Benford’s law could be used in forensic accounting and auditing as an indicator of accounting and expenses fraud. In the United States, evidence based on Benford’s law is legally admissible in criminal cases at the federal, state, and local levels.

Limitations

Benford’s law can only be applied to data that are distributed across multiple orders of magnitude.

— Wikipedia on Benford’s law

2012.05.15 Tuesday ACHK

Batman: Year One

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Aronofsky was hired to reboot the franchise, before Christopher Nolan was brought into the project instead. Aronofsky planned a radical franchise change, something unlike any previous Batman film and one that was based on the idea of “gritty realism” that was in fact too much realism for the studio, which decided it didn’t want to make a Batman film THAT gritty or THAT “realistic.”

— Darren Aronofsky (director): How would a Darren Aronofsky Batman film differ from the current Nolan series?

— Mark Hughes, Screenwriter, Forbes Blogger

Around the same time, Warner Bros. hired Darren Aronofsky to write and direct Year One, despite interest from Joel Schumacher. Aronofsky, who collaborated with Frank Miller on an unproduced script for Ronin, brought Miller to co-write Year One with him. They intended to reboot the Batman franchise, “it’s somewhat based on the comic book,” Aronofsky said. “Toss out everything you can imagine about Batman! Everything! We’re starting completely anew.” Regular Aronofsky collaborator, Matthew Libatique, was set as cinematographer, and Aronofsky had also approached Christian Bale for the role of Batman. Coincidentally, Bale would be cast in the role for Batman Begins.

— Wikipedia on Batman in film

2012.05.14 Monday ACHK

古怪題目 1.1

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這段改編自 2010 年 6 月 2 日的對話。

你以為很需要創意的東西，可能根本在 past paper（歷屆試題）中出現過。只要你前先背誦好，那些現成可用的技巧，你就不會再覺得那類題目「很需要創意」。

（CYW：平日做功課或測驗時，可能會遇到有些題目類型，我沒有在歷屆試題中見過。那樣，我想背誦那些技巧也背不到。）

凡是有些題目十分奇怪，奇怪到一個程度是，它們的類型，根本在眾多年的歷屆試題中，從來未出現過，你就會知道，那類題目不是課程的重點，在未來試題出現的機會很微。換句話說，它們對你分數的影響很少。那你又何必去理會它們？

（CYW：假設我考試想拿到 A 級成績。如果有些「十分古怪題目」，剛巧在我那一屆的試題出現，我就拿不到 A。）

無錯。所以，對於「A 級成績人士」來說，應該間中做一些「十分古怪題目」，以備不時之需。

但是，你要留意，做事應該分輕重緩急。溫習的時間相當有限。你應該先溫習「必考」的題目，然後才溫習「多考」的題目。行有餘力，就可以研究「少考」的題目。之後，竟然還有時間剩餘的話，你才可以鑽牛角尖，研究那些「十分古怪」，幾乎沒有機會考的題目。

— Me@2012.05.14

2012.05.14 Monday (c) All rights reserved by ACHK

A Few Tricks

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Every Mathematician Has Only a Few Tricks

A long time ago an older and well-known number theorist made some disparaging remarks about Paul Erdos’s work. You admire Erdos’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdos’s work could be “reduced” to a few tricks which Erdos repeatedly relied on in his proofs.

What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over.

Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert’s beautiful results have been completely forgotten. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!

— Ten Lessons I Wish I Had Been Taught

— Gian-Carlo Rota

2012.05.13 Sunday ACHK

流言終結者 3

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這段改編自 2010 年 3 月 20 日的對話。

香港其實也有貌似《MythBusters》（流言終結者），這類探究傳說流言的節目。可惜，香港的版本，是偏重於「傳說流言」本身，而不是「傳說流言的真偽」。節目的主持人和製作人，很少會花大量的時間，去做 科學實驗 或者 歷史考究，以驗證傳說流言的虛實。描述了一個傳說流言以後，結論也往往只是「那仍然是一個謎」。它們至多只可以視作是，比較高級的娛樂節目，而算不上是什麼科學資訊節目。

「那仍然是一個謎」的思考態度，與《流言終結者》「求證求真」的科學精神，相去甚遠。但是，「那仍然是一個謎」的境界，已經遠高於一般的地球人，因為至起碼，它「不知為不知」：把不知道的東西，標籤為「未知」。

一般人的水平，差到不只是「先入為主」，甚至「先入為對」。他們的（不）思考系統，會自動將第一次聽到東西，視為真實當然。往後，如果有新的講法，與自己原本的背景「知識」有衝突，他們會堅持「己」見。這個現象，我戲稱為「第一次為真」定律。

— Me@2012.05.13 

2012.05.13 Sunday (c) All rights reserved by ACHK

Batman: Arkham Asylum

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Batman: Arkham Asylum is an award-winning 2009 action-adventure stealth video game based on DC Comics’ superhero Batman.

Arkham Asylum, written by veteran Batman writer Paul Dini, is based on the long-running comic book mythos, as opposed to most other Batman games, which are adaptations of the character in other media besides the source material.

— Wikipedia on Batman: Arkham Asylum

2012.05.12 Saturday ACHK

The Incredible Hulk

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Marvel’s The Avengers (2012 movie): What did the Hulk mean when he said his secret was that he was always angry?

It seemed that early in the movie, he was not able to control himself when he turned into the Hulk. Yet, he seemed to be able to transform at will and also listen to orders or operate as part of a team in the later part of the movie.
 

Mark Hughes, Screenwriter, Forbes Blogger

If Banner had to just avoid getting angry in order to not be the Hulk, he’d never succeed, because of course it’s probably almost impossible for the vast majority of people to really just stop themselves from getting angry. But what do we do instead? We can’t avoid being angry, but we learn to control our TEMPER and we can control how we RESPOND to anger.

In the previous solo Hulk film, it ends with Banner learning that the key is less about the anger itself, than the literal physical reactions caused by the anger. If he can keep his heart rate down and his metabolism from responding to the anger and pain, then he can prevent changing into the Hulk. He maintains a constant state of anger at his situation, and at the state of the world, and at the people responsible for it, as a way to keep the emotional status constant and thus allow him to ignore controlling his emotions so he can instead focus his efforts merely on controlling his physiological responses to the anger.

With anger as a constant state, his body adapts, it returns to a moderated physical state most of the time, so anger itself won’t be a trigger. Only when he allows the anger to go unchecked and gives in to his body’s “desire”/urge to change, does he become the Hulk — unless he is too overwhelmed by physical stress to hold it back, like when he was severely injured early in the film on the Helicarrier. Otherwise, he can change into the Hulk at will because he’s already always angry, and need only let the change take place.

http://www.quora.com/Marvels-The-Avengers-2012-movie/What-did-the-Hulk-mean-when-he-said-his-secret-was-that-he-was-always-angry/answer/Mark-Hughes-1

2012.05.12 Saturday ACHK

溫習內疚 1.1

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這段改編自 2010 年 6 月 2 日的對話。

我現在教你們的讀書方法，我自己年輕時，是不知道的。還有，有時回想起年輕時候的自己，會十分驚嘆，為何一個人可以那麼愚蠢。例如：

年青人的指定劇情是，會因為沒有溫習而內疚。然後又會因為內疚，而沒有心機溫習。接著再會因為繼續沒有溫習，而加倍內疚，如此類推。

…

你們懂得笑，即是有同感。那就證明了「在不適當的時候內疚」，並不只是個別人士的問題，而且是很多人都會有的問題。

— Me@2012.05.12

2012.05.12 Saturday (c) All rights reserved by ACHK

Your Expository Work

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You Are More Likely to Be Remembered by Your Expository Work

Let us look at two examples, beginning with Hilbert. When we think of Hilbert, we think of a few of his great theorems, like his basis theorem. But Hilbert’s name is more often remembered for his work in number theory, his Zahlbericht, his book Foundations of Geometry, and for his text on integral equations. The term “Hilbert space” was introduced by Stone and von Neumann in recognition of Hilbert’s textbook on integral equations, in which the word “spectrum” was first defined at least twenty years before the discovery of quantum mechanics. Hilbert’s textbook on integral equations is in large part expository, leaning on the work of Hellinger and several other mathematicians whose names are now forgotten.

Similarly, Hilbert’s Foundations of Geometry, the book that made Hilbert’s name a household word among mathematicians, contains little original work and reaps the harvest of the work of several geometers, such as Kohn, Schur (not the Schur you have heard of), Wiener (another Wiener), Pasch, Pieri, and several other Italians.

Again, Hilbert’s Zahlbericht, a fundamental contribution that revolutionized the field of number theory, was originally a survey that Hilbert was commissioned to write for publication in the Bulletin of the German Mathematical Society.

— Ten Lessons I Wish I Had Been Taught

— Gian-Carlo Rota

2012.05.11 Friday ACHK

上班 10

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（CTC：你現在是不是失業？）

不算是失業，因為我現在有收入。

（CTC：但是，你沒有穩定的工作職位。）

那要視乎「不穩定工作」的收入多，還是「穩定工作」的收入多。如果「不穩定工作」的平均收入，遠多於「穩定工作」，那就不算是「失業」。

— Me@2011.09.23

2012.05.11 Friday (c) All rights reserved by ACHK

流言終結者 2

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這段改編自 2010 年 3 月 20 日的對話。

有一個科學娛樂節目，叫做《MythBusters》（流言終結者），內容是在每集做科學實驗，驗證各式各樣 大小傳說、市井流言 的真偽。它收集的流言千奇百怪，例如：「真正的忍者，是否和電影中的版本一樣，可以徒手接住飛來的箭呢？」又例如：「伏特加酒，是否可以驅除腳部的異味？」

有時，你會覺得節目中的「流言」有點無聊，不驗也罷。但是，科學漫畫《xkcd》的作者 Randall Munroe 卻認為，《流言終結者》對人類的進化，有驚人的貢獻。它的主要貢獻，不在於驗證個別流言本身。它的主要貢獻，在於幫助大眾驅除迷信，改為習慣使用科學頭腦。所謂的「科學態度」，其實就是指 忠於事實，多過忠於自己的喜好。在有確鑿的實驗證據之前，不要妄下判斷。

— Me@2012.05.11 

2012.05.11 Friday (c) All rights reserved by ACHK

iCloud

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The classic Buddhist image of this hack is that thoughts are like clouds passing through a spacious blue sky. All your life, you’ve been convinced that this succession of clouds comprises a stable, enduring identity — a “self.” But Buddhists believe this self this is an illusion that causes unnecessary suffering as you inevitably face change, loss, disease, old age, and death. One aim of practice is to reveal the gaps or discontinuities — the glimpses of blue sky — between the thoughts, so you’re not so taken in by the illusion, but instead learn to identify with the panoramic awareness in which the clouds arise and disappear.

— What Kind of Buddhist was Steve Jobs, Really?

— Steve Silberman

2012.05.10 Thursday ACHK

Elementary particles

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There exist heavier particle species which are relevant for shorter distance scales. Most of the matter around us is composed of electrons, protons, and neutrons, or – using the more elementary description – electrons, up-quarks, and down-quarks (which are attracted by forces mediated by photons and gluons). However, there exist many other particle species similar to electrons – the so-called leptons – and many other quarks. Many of those particles are unstable, and therefore unimportant in the composition of stable materials.

But even if heavier particles are stable, they are less important than the light ones because it is hard to create them and because their potential existence only affects the phenomena at ever shorter distances. Elementary particles heavier than the Planck mass or so – \(10^{-8}\) kilograms or so – also exist and there are many of them. However, they may be interpreted as black hole microstates and their description in terms of Einstein’s general theory of relativity becomes more natural than their description in terms of quantum field theory.

String/M-theory provides us with many detailed interpolations between the regular light particle species and the black holes – e.g. Kaluza-Klein modes i.e. particles moving in extra dimensions; excited string states and branes, and others.

— Ten new things that science has learned about matter

— Lubos Motl

2012.05.09 Tuesday ACHK