Random stuff from the net, links, wikipedia…

1. RAND: Living Well at the End of Life (via Razib Khan). Here’s a link to one of the sources, a book which deals with some of the same questions: Approaching Death: Improving Care at the End of Life. Looks interesting, don’t have time to read it at the moment.

2. Fatal familial insomnia. “Fatal familial insomnia (FFI) is a very rare autosomal dominant inherited prion disease of the brain. It is almost always caused by a mutation to the protein PrPC, but can also develop spontaneously in patients with a non-inherited mutation variant called sporadic Fatal Insomnia (sFI). FFI is an incurable disease, involving progressively worsening insomnia, which leads to hallucinations, delirium, and confusional states like that of dementia.[1] The average survival span for patients diagnosed with FFI after the onset of symptoms is 18 months.”

Sleep’s important.

3. False consensus effect.

“In psychology, the false consensus effect is a cognitive bias whereby a person tends to overestimate how much other people agree with him or her. There is a tendency for people to assume that their own opinions, beliefs, preferences, values and habits are ‘normal’ and that others also think the same way that they do.[1] This cognitive bias tends to lead to the perception of a consensus that does not exist, a ‘false consensus’. This false consensus is significant because it increases self-esteem. The need to be “normal” and fit in with other people is underlined by a desire to conform and be liked by others in a social environment.

Within the realm of personality psychology, the false consensus effect does not have significant effects. This is because the false consensus effect relies heavily on the social environment and how a person interprets this environment. Instead of looking at situational attributions, personality psychology evaluates a person with dispositional attributions, making the false consensus effect relatively irrelevant in that domain. Therefore, a person’s personality potentially could affect the degree that the person relies on false consensus effect, but not the existence of such a trait.

The false consensus effect is not necessarily restricted to cases where people believe that their values are shared by the majority. The false consensus effect is also evidenced when people overestimate the extent of their particular belief is correlated with the belief of others. Thus, fundamentalists do not necessarily believe that the majority of people share their views, but their estimates of the number of people who share their point of view will tend to exceed the actual number.

This bias is especially prevalent in group settings where one thinks the collective opinion of their own group matches that of the larger population. Since the members of a group reach a consensus and rarely encounter those who dispute it, they tend to believe that everybody thinks the same way.

Additionally, when confronted with evidence that a consensus does not exist, people often assume that those who do not agree with them are defective in some way.[2] There is no single cause for this cognitive bias; the availability heuristic and self-serving bias have been suggested as at least partial underlying factors.

The false consensus effect can be contrasted with pluralistic ignorance, an error in which people privately disapprove but publicly support what seems to be the majority view (regarding a norm or belief), when the majority in fact shares their (private) disapproval. While the false consensus effect leads people to wrongly believe that they agree with the majority (when the majority, in fact, openly disagrees with them), the pluralistic ignorance effect leads people to wrongly believe that they disagree with the majority (when the majority, in fact, covertly agrees with them).”

4. Malthus, An Essay on the Principle of Population. Salman Khan recently made a video on the subject, here’s wikipedia.

5. Marital Rape License (warning, tvtropes link).

“Only a few decades ago, it was legal for a man to rape his wife. Sweden was the first country to explicitly criminalize it in 1965, and it has only been illegal in all fifty US states since 1993. Fifty-three countries around the world still don’t consider it a crime.

In some old patriarchal systems, a woman belonged first to her father (or closest living male relative if the father was dead) and then to her husband. Once married — and in some systems she could be married off without her consent to some old man she despised or had never met — her husband had a legal and “moral” right to her body whether she liked it or not. It gets even creepier when the bride is underage.”

We tend to take a lot of stuff for granted. Another reason why you should read Nothing To Envy.

6. Schema (psychology)

“A schema (pl. schemata or schemas), in psychology and cognitive science, describes any of several concepts including:

*An organized pattern of thought or behavior.
*A structured cluster of pre-conceived ideas.
*A mental structure that represents some aspect of the world.
*A specific knowledge structure or cognitive representation of the self.
*A mental framework centering on a specific theme, that helps us to organize social information.
*Structures that organize our knowledge and assumptions about something and are used for interpreting and processing information.

A schema for oneself is called a “self schema”. Schemata for other people are called “person schemata”. Schemata for roles or occupations are called “role schemata”, and schemata for events or situations are called “event schemata” (or scripts).

Schemata influence our attention, as we are more likely to notice things that fit into our schema. If something contradicts our schema, it may be encoded or interpreted as an exception or as unique. Thus, schemata are prone to distortion. They influence what we look for in a situation. They have a tendency to remain unchanged, even in the face of contradictory information. We are inclined to place people who do not fit our schema in a “special” or “different” category, rather than to consider the possibility that our schema may be faulty. As a result of schemata, we might act in such a way that actually causes our expectations to come true.”

7. Koch Snowflake Fractal (a structure with infinite perimeter but a finite area). Couldn’t remember if I’ve already blogged this at one point, but no harm done in case I have:

The Gauss Christmath Special

It’s that time of the year again:

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Update: Here are a few other videos. If you’re sick of christmas songs, you can have those playing in the background while you’re busy making a christmas cake (or whatever) instead:

Part 4, The Foundation of Rome:

(His youtube channel has a lot more)

Pi is wrong?

This is cute:

More here and here. She’s a bit like Salman Khan – she makes learning more fun and she’s very pleasant to listen to (in my opinion). They work well together too.

Here’s another video from her youtube channel:

Euler’s formula and Euler’s identity

The video is more than half-way through the calculus coursework, so if you’re unfamiliar with this stuff there’ll probably be some things you don’t understand even if he keeps it simple and don’t go through a more formal proof. The Maclaurin series he’s talking about is just a Taylor series evaluated at x=0, at uni we always call them Taylor series or Taylor expansions but apparently naming conventions differ.

The three videos before that one builds up to this, but if you’re familiar with maths and can remember how to do Taylor expansions and how to deal with trigonometric functions, you should be able to follow this quite easily without watching those as well; I could, as he doesn’t deal with anything here that I haven’t had exams in at a previous point in time. It probably didn’t do any harm that I read 100 pages in Discrete Mathematical Structures this weekend, parts of which contained a brush-up on permutations and factorials (the “!”-thingies in the formulas).

Videos like these were the kind of stuff I had to cut down on a lot during the last month leading up to the exam, those and non-study books. I’m behind on the blogging of the books I’m reading but I’ll get to it.

What do economists learn when taking their education? Most people would probably guess that they/we learn a lot of stuff about markets, industries/firms and some political economy (‘how the economy works’) and such. Maybe something about ‘how to calculate the numbers’. This is another side of the coin. Even though we wouldn’t be asked to go through that proof at an exam, we are (at least some of us) probably expected to know enough math to be able to understand something like this (it depends on the courses). There’s a lot of math and statistics in (some areas of) economics. There’s actually enough to make a guy who voluntarily decides to watch a video like the one above in his spare time think it’s a little too much. Though of course part of the reason why I feel that way is the fact that I suck at math, which is also why I try to get better at it – at least I’m not a math atheist. Now that we are dealing with comics, there’s also this.

Discrete Mathematical Structures

Here’s the link, below a few quotes to illustrate what kind of book this is:

1) “Theorem 3: Let A, B, and C be finite sets. Then |A∪B∪C| = |A|+|B|+|C|-|A∩B|-|B∩C|-|A∩C|+|A∩B∩C|.”

2) “Existential quantification may be applied to several variables in a predicate and the order in which the quantifications are considered does not affect the truth value. For a predicate with several variables we may apply both universal and existential quantification. In this case the order does matter.”

3) “Theorem 2 The Extended Pigeonholde Principle: If n pigeons are assigned to m pigeonholes, then one of the pigeonholes must contain at least ⌊(n-1)/m⌋ + 1 pigeons.

Proof ( by contradiction ) If each pigeonhole contains no more than ⌊(n-1)/m⌋ pigeons, then there are at most m * (n-1)/m = n-1 pigeons in all. This contradicts our hypothesis, so one of the pigeonholes must contain at least ⌊(n-1)/m⌋ + 1 pigeons.”

The above perhaps points to part of the reason why I haven’t quoted from the book before. Given that the exams are getting closer every day, it’s unlikely that I’ll do much more reading in this book (or perhaps any non-directly study-related book) in the next month’s time. The book contained a few remarks on ideas as to how to construct proofs in chapter 2, which though most of the ideas were familiar to me are not completely exam-irrelevant. Pretty sure most of the other stuff is. Though I’ll perhaps not get a lot of non-exam relevant reading done I’ll try to keep blogging over the coming weeks, I’ve almost returned to ‘one post/day’ and I like that very much though it’s uncertain if I can keep up that kind of activity level in the longer run.

(More) random stuff

Can’t let the blog die so I sort of have to at least post something from time to time. So here goes…

1. Global sex ratios:

At birth: 1.07 male(s)/female
Under 15 years: 1.07 male(s)/female
15-64 years: 1.02 male(s)/female
65 years and over: 0.79 male(s)/female
Total population: 1.01 male(s)/female (2011 est.)

Link. Here’s an image of child sex ratios in India (via brownpundits:

Here’s one for the whole population, image credit: Wikipedia (much larger version at the link):

I’ve from time to time read about the Chinese gender ratio problem, I didn’t know there were much going on on that score in India. The clustering of gender ratio frequencies seems in my opinion sufficiently non-random to merit some explanation or other, especially when it comes to the northern provinces (Punjab, Haryana & Kashmir). Here’s a pic dealing with more countries:

Link.

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2. Gambler’s ruin. I remember having read about this before, but you forget that kind of stuff over time so worth rehashing. I think the version of the idea I’ve seen before is the first of the four in the article; ‘a gambler who raises his bet to a fixed fraction of bankroll when he wins, but does not reduce it when he loses, will eventually go broke, even if he has a positive expected value on each bet.’ I assume all readers of this blog already know about the Gambler’s fallacy but in case one or two of you don’t already do click the link (and go here afterwards, lots of good stuff at that link and I shall quote from it below as well) – that one is likely far more important in terms of ‘useful stuff to know’ because we’re so prone to committing this error; basically the important thing to note there is that random and independent events are actually random and independent.

A couple of statistics quotes from the tvtropes link:

“The Science Of Discworld books have an arguably accurate but somewhat twisted take on statistics: the chances of anything at all happening are so remote that it doesn’t make sense to be surprised at specific unlikely things.”

“There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.” (Mark Twain. Maybe it’s more of a science quote really – or perhaps a ‘science’ quote?)

“People (especially TV or movie characters who are against the idea of marriage) often like to cite the “50 percent of marriages end in divorce” statistic as the reason they won’t risk getting hitched. That is actually a misleading statistic as it seems to imply that half of all people who get married will wind up divorced. What it doesn’t take into account is the fact that a single person could be married and divorced more than once in a single lifetime. Thus the number of marriages will exceed the number of people and skew the statistics. The likelihood that any one person chosen at random will be divorced during their lifetime is closer to 35 percent (the rate fluctuates wildly for males, females, educated and uneducated populations). It’s still a huge chunk of people, but not as high a failure rate for marriage for an individual as the oft-cited “50 percent of all marriages” statistic would leave you to believe.” (comment after this: “How can you give that setup and not deliver the punchline. “But the other half end in death!”")

A mathematics quote:

“Black Mage: 2 + 2 = 4
Fighter: You can’t transform numbers into other numbers like that. It’d just go on forever. That’s like Witchcraft! “

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3. Messier 87. Interesting stuff, ‘good article’, lots of links.

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4. Substitution cipher. I’d guess most people think of codes and codebreaking within this context:

“In cryptography, a substitution cipher is a method of encryption by which units of plaintext are replaced with ciphertext according to a regular system; the “units” may be single letters (the most common), pairs of letters, triplets of letters, mixtures of the above, and so forth. The receiver deciphers the text by performing an inverse substitution.

Substitution ciphers can be compared with transposition ciphers. In a transposition cipher, the units of the plaintext are rearranged in a different and usually quite complex order, but the units themselves are left unchanged. By contrast, in a substitution cipher, the units of the plaintext are retained in the same sequence in the ciphertext, but the units themselves are altered.

There are a number of different types of substitution cipher. If the cipher operates on single letters, it is termed a simple substitution cipher; a cipher that operates on larger groups of letters is termed polygraphic. A monoalphabetic cipher uses fixed substitution over the entire message, whereas a polyalphabetic cipher uses a number of substitutions at different times in the message, where a unit from the plaintext is mapped to one of several possibilities in the ciphertext and vice-versa.”

The one-time-pad stuff related is quite fascinating; that encryption mechanism is literally proven unbreakable if applied correctly (it has other shortcomings though..).

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5. Evolution may explain why baby comes early.

“there’s only so much a human female pelvis can increase in terms of width before serious functional problems in locomotion make change in that direction unfeasible. [...] If the pelvis was prevented from getting any wider due to biomechanics, and a large adult brain was a necessary condition of high fitness value for humans, then one had to accelerate the timing of childbirth so that the neonate exited while the cranium was manageable in circumference.”

Interesting stuff.

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6. Random walk. The article actually has some stuff related to the previous remarks on gambler’s ruin.

Fermat’s Last Theorem

I read the first 150 pages of it yesterday, I’ll complete it today – it’s a real pageturner. Even though it’s a book about mathematics (and mathematicians) I’ve yet to stumble upon anything which is harder to deal with than the stuff I encounter in my textbooks on a daily basis – it’s a very accessible book so far. Some quotes:

1. “Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.” An introductionary quote by G. H. Hardy. I thought I’d post it also because of the recency of this discussion on this blog.

2. “It is true that I had hit the date of the displacement of the judges of Castres to Toulouse, where he [Fermat] is the Supreme Judge to the Sovereign Court of Parliament; and since then he has been occupied with capital cases of great importance, in which he has finished by imposing a sentence that has made a great stir; it concerned the condemnation of a priest, who had abused his functions, to be burned at the stake. This affair has just finished and the execution has followed.”

From a letter by the mathematician Sir Kenelm Digby to his collegue John Wallis. On the next page Singh drily remarks: “he devoted all his spare energy to mathematics and, when not sentencing priests to be burned at the stake, Fermat dedicated himself to his hobby.”

Another quote touching upon the same theme: “One story claims that a young student by the name of Hippasus was idly toying with the number √2, attempting to find the equivalent fraction. Eventually he came to realise that no such fraction existed, i.e. that √2 is an irrational number. [...] Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. [...] To his eternal shame he sentenced Hippasus to death by drowning.”

The above quotes are also relevant to the previous discussion – the point being of course that it’s not just ‘the dietary habits of Cleopatra’ that is forgotten over time. Aristotle at one point argued that the number zero should be outlawed. And these are only the things that we haven’t forgotten yet.

3. “The fame of Fermat’s Last Theorem comes solely from the sheer difficulty of proving it.”

4. The book explains a lot of mathematical concepts and deals with many interesting examples of stuff mathematicians have been working on during the ongoing development of (in particular) number theory. Some stuff I didn’t know, or didn’t know that I knew, or had forgotten:

“Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number.” [...] “During the twentieth century mathematicians have extended the idea further and have searched for so-called ‘sociable’ numbers, three or more numbers which form a closed loop. For example, in the loop of 5 numbers (12,496; 14,288; 15,472; 14,536;14264) the divisors of the first number add up to the second, the divisors of the second add to the third, the divisors of the third add up to the fourth, the divisors of the fourth add up to the fifth, and the divisors of the fifth add up to the first.”

“Prime numbers are the numerical building blocks because all other numbers can be created by multiplying combinations of the prime numbers.” – I knew this at one point, but I’d forgotten. Basically, there are primes and then there’s everything else. At least as long as you only deal with natural numbers. It was somewhat important when it came to the proof because: “If one can prove Fermat’s Last Theorem for just the prime values of n, then the theorem is proved for all values of n.”

“All prime numbers (except 2) can be put into two categories; those which equal 4 n + 1 and those which equal 4 n – 1″

“26 is indeed the only number between a square and a cube.”

“The most significant and rarest of numbers are those whose divisors add up exactly to the number itself and these are the perfect numbers. The number 6 has the divisors 1, 2 and 3, and consequently it is a perfect number because 1 + 2 + 3 = 6. The next perfect number is 28″ [...] “Euclid discovered that perfect numbers are always the multiple of two numbers, one of which is a power of 2 and the other being the next power of 2 minus 1.”

“If you take any number and multiply it by 2, then the new number must be even. This is virtually the definition of an even number.” – you kinda knew this, and then again maybe you didn’t really. Also, “If you know that the square of a number is even, then the number itself must also be even.”

“The network formula shows an eternal relationsship between the three properties which describe any network:

V + R – L = 1,

where

V = the number of vertices (intersections) in the network,
L = the number of lines in the network,
R = the number of regions (enclosed areas) in the network.” (the book also contains the proof of this formula, created by Leonhard Euler.)

5. “Dear …………………,

Thank you for your manuscript on the proof of Fermat’s Last Theorem.
The first mistake is on :
Page ……… Line ………..
This invalidates the proof.

Professor E. M. Landau”

Landau was head of the mathematics department in Göttingen from 1909 and 1934, and it was his job to examine the entries submitted for the Wolfskehl Prize, a big prize offered by the estate of Paul Wolfskehl to anyone who proved Fermat’s Last Theorem. He printed out cards like the one above and gave them to his students to let them fill in the blanks. Because of the substantial amount of money on the line, there were a lot of faulty attemps submitted.

Maybe I’ll write another post on this book, maybe I won’t, but I have no problem recommending it. You can order the book here.

Random wikipedia links of interest

1. Corrosion.

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2. Demographics of the People’s Republic of China. A few quotes from the article:

a) “Census data obtained in 2000 revealed that 119 boys were born for every 100 girls, and among China’s “floating population” the ratio was as high as 128:100. These situations led the government in July 2004 to ban selective abortions of female fetuses. It is estimated that this imbalance will rise until 2025–2030 to reach 20% then slowly decrease.[2]“

b) “Average household size (2005) 3.1; rural households 3.3; urban households 3.0.
Average annual per capita disposable income of household (2005): rural households Y 3,255 (U.S.$397), urban households Y 10,493 (U.S.$1,281).”

c) A map of the population density (darker squares have higher density):

The ‘average population density’ of 137/km2 is not an all that interesting variable. The Gobi desert is not a nice place for humans to live: The temperature variation in the area is extreme, ranging from –40°C in the winter to +50°C in the summer.

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3. Cost overrun. An excerpt:

“Cost overrun is common in infrastructure, building, and technology projects. One of the most comprehensive studies [1] of cost overrun that exists found that 9 out of 10 projects had overrun, overruns of 50 to 100 percent were common, overrun was found in each of 20 nations and five continents covered by the study, and overrun had been constant for the 70 years for which data were available. For IT projects, an industry study by the Standish Group (2004) found that average cost overrun was 43 percent, 71 percent of projects were over budget, over time, and under scope, and total waste was estimated at US$55 billion per year in the US alone.”

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4. Tensor. This is difficult stuff.

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5. Eye.

Random wikipedia links of interest

(/Well, I found them interesting…)

1) Banach–Tarski paradox

2) Origin of the domestic dog

3) War of the Polish Succession

4) Lotka–Volterra equation

5) Empire of Brazil

Random articles about mathematics

I have known about Wikipedia’s Search:Random function for some time. It’s a link that lands you on a random wikipedia article, and it’s a great timewaster. However, if you – like me – think that there’s just a little too much randomness about that search function (ie. too many articles about music bands you’ve never heard about and couldn’t care less about, star trek episodes, unimportant clergymen who died more than 600 years ago, articles about pseudo-scientific mumbo-jumbo ect., it is actually possible to target your search somewhat while still maintaining the element of randomness; at least when it comes to the subject of mathematics. I recently learned that Wikipedia has a Search:Random function that deals only with articles about mathematics – here’s the link.

Both the standard Random function and the math one is, with P->1, likely to turn up an article containing knowledge I didn’t know I didn’t know. But the math one is in my case much more likely to contain knowledge I didn’t know I didn’t know and actually do want to know, than the unspecified random function is (as said, the unspecified random function is unfortunately very likely to direct me towards articles containing knowledge I didn’t know I didn’t know, and didn’t want to know. Sometimes there are even articles containing knowledge I didn’t know I didn’t want to know I didn’t know… ), and if one is completely lost, then backtracking is almost always as option when you move around wikipedia.

This article was my first hit using that function, this one was the second. I guess I have just found myself a great new startpage…

Mathworld

Blogging is about a lot of things, but one of the most important one of them surely is information sharing.

I assume my readers know about the existence of youtube, google, myspace, wikipedia ect. Of course you do. Well, I have a treat for you: Here’s one online ressource you probably didn’t know about: Mathworld.

It really is a wonderful site, and unless you are like an expert on cryptography working as a codebreaker for the NSA or something like that, there are very few questions you would be able to ask about math that this site is unable to answer. Wikipedia is no longer my first place to go for mathematics anymore, as this site is superior in this area.