strangestchild

You're simulating a full game of poker. Once a player has been given card 'i', you have to ensure that card 'i' isn't drawn again during the game. You could maintain a set containing all cards that have already been drawn, and re-select your random number if you draw a duplicate, but that's going to get awfully laggy once large numbers of cards have been drawn.

Homeopathic obesity – what an intriguing idea...

There are recruitment agencies which only take individuals with autism on their books, and try to find them roles which suit their particular skills profile. A quick Google should turn up some useful results, but be careful to not be taken for a ride.

Tech is a good industry for those on the autistic spectrum, as it values protracted periods of focus and analysis; but as with any job, communication is really important – perhaps especially so in tech. It will be important to find an employer who is prepared to take him on as an individual and work with him to minimise his difficulties and build on his strengths.

Horses for courses. Whilst I'm skeptical that we're at the stage where mechanical proof-checking is viable, such a technique would be immensely valuable.

On the other hand, it is of course true that if the proof itself is mechanistic, most mathematicians would feel that a lot of the important essence of the result had been lost.

Voevodsky's work is apparently in the former, and the blog author conflates this with the latter in order to (wrongly) criticise it.

What a ridiculous article.

The statement "70% [are] on prescription drugs" is devoid of moral content. If the title had been "70% are on prescription drugs unnecessarily" or indeed "70% misuse prescription drugs", we would have had a story. As it is, the implicit title "70% are currently being treated for medical conditions, using drugs that have passed several rounds of peer-reviewed trial" is a testament to our healthcare system.

I'm not at all surprised by the statement that prescription opioid usage 'outstrips' heroin usage by a factor of 14. I'm glad these people are getting pain relief. Indeed, if the number of heroin users was anywhere near the number of people with badly broken bones or chronic pain, I would be alarmed.

This article very much prompts the question "What alternative would you suggest?"

Scientists definitely do have a test for the so-called IQ, and it's called the IQ test – but 'Intelligence Quotient' and intelligence are not synonymous. One need only read Mensa magazine to realise that IQ does not constitute intelligence in a meaningful sense.

It's clear that brains vary, of course, but 'intelligence' is too broad a term to say, quantitatively, to what degree a person possesses it. The term encompasses:

- The ability to acquire new knowledge quickly, or to learn difficult things at all.

- The ability to, given time, make nontrivial deductions from given data.

- The ability to make rapid deductions in short periods of time about given data.

- The ability to generate multiple unrelated solutions to a problem.

- A high level of verbal proficiency.

We mean all these things to different degrees at different times when we use the term; and although these are not orthogonal concepts, they clearly do not exist in a one-dimensional space.

Certainly if IQ is taken to mean 'the raw, unchanging potential of a mind', the Flynn effect is inexplicable given that the timescales involved are far too small for our neurobiology to have adapted evolutionarily. Given that it means very little other than 'the ability to pass IQ tests', the Flynn effect is perhaps not surprising at all.

What I mean is that since Goodstein's theorem is provably true for the naturals, but is not a consequence of the Peano axioms, then the definition of the naturals used to demonstrate Goodstein's must be strictly stronger than the Peano axioms themselves. I was wondering what this definition might be.

I'm familiar with the distinction between formalism and Platonism, although I still haven't made my mind up yet :)

How then are the naturals defined? By the Von Neumann construction? (0, {0}, {0, {0}}, ...)

Agree - especially given that this was funded by the Campaign for Real Beauty, and it wouldn't exactly have supported Dove's ad campaign if the pictures looked the same or the sketch was reversed.

I know, I was being facetious. I just like the thought of the campaign finishing and runn1ng posting each of his sponsors a $0.99 protractor.

I use a similar workflow with SVN to great effect. I have never had a sync issue - whether this is luck or that the repository structure is more robust in this setting than Git's, I don't know. I would strongly recommend against using this as a collaboration tool, though!

The problem has already been solved: http://goo.gl/NLdou

Looks good - I've signed up :)

I would say though that I think your articles and lectures could do with a bit more organisation. In both cases, the user would benefit from posts having tags denoting subject content, and from a search facility so that I can find things that specifically interest me. If you have five hundred videos about mathematics, finding one on Analytic Topology, say, by scrolling through them would be a bit painful. I also think it would be good if the lectures had some way of tying subsequent videos together - it seems a tiny bit messy to have 'Cosmology - Lecture 1', and 'Cosmology - Lecture 2' as separate entries. I'm also not certain that you want to limit lectures to being videos. A well-written explanation of a broad subject is sometimes better than a video that I can't search or skim through - and I think it serves a different purpose from the articles. It's also easier for your community to add written content than videos.

Hope that was useful to you - but obviously take it all with a pinch of salt. You know your strategy better than some guy off the internet who's only spent ten minutes on your site :)

Perhaps your experience has been different from mine, maybe because I'm based in the UK - I know that US education tends to be more generalised. Among my friends with or pursuing postgraduate degrees in pure mathematical disciplines, none have any particular knowledge of stats above the undergraduate level. As a master's student, I wouldn't imagine I count for much - but what I know about statistics could be written on the back of an envelope. It's something I've been meaning to remedy for a while now. It's possible that tenured professors have a wider breadth of knowledge than the average PhD - and I admit that I wouldn't know if that were the case.

As for the depth of statistics as a field, and its reliance on other disciplines - I agree entirely. I think pure mathematicians are far more likely to be ignorant of statistics than statistical mathematicians are of, for example, analysis.

That seems a pretty strong generalisation. A doctorate in Model Theory doesn't give you the authority to talk stats.

I agree with you that the ... notation is ill-defined. It's common in mathematics to use conventions that are a little woolly, but only where everyone understands how to express the idea more correctly.

Here's one way of presenting this formally. Define a 'decimal' to be an ordered sequence of integers (called 'digits') a_1, a_2, a_3, and so on. (By 'and so on', I formally mean that for each positive integer k we have a digit a_k at the kth position in the sequence). Let's say each a_k has to be between 0 and 9 inclusive.

For each positive integer k, define the 'kth partial sum' of the sequence to be the sum from j=1 to j=k of (1/10^j) x a_j.

I'll skip over what it means if we say that the partial sums converge as k->Infinity, because it sounds like you understand what limits are and how they work. If not, I'd be happy to explain.

Now, if the partial sums converge to some value 'd', we say that the decimal has value equal to d. It can be shown that any decimal has at most one such d (which is good, because a decimal shouldn't have two values).

Now, I think you'd be happy to say that when we write 0.9999999..., what we mean is the decimal where a_k=9 for all k. Given this definition, it follows that the value of the decimal is precisely 1, using properties of geometric sequences.

It is up to you how you define '...', but all mathematicians would agree that 0.99999... should be interpreted as above if it is to have any meaning at all.

If you really want a more precise eplanation of '...' at the end of a truncated decimal, I would provide the following: "Writing 0.abcdef... asserts that the digits abcdef of the truncation provided have a pattern which should be obvious to the reader. Assign the first few digits a_1, a_2 etc as per the part of the decimal that is explicitly given; and then assign all subsequent digits values according to said pattern." - it's not a formal notation, as I say, but rather a convenient shorthand that is understood by working mathematicians. It is always possible to be more precise if one has to be.

All good fun problems - and some less well-known as well as the old favourites. That said, calling these the "12 most controversial facts in mathematics" is like calling the truth of the moon landings the most controversial fact in astronautics. But I suppose "The Most Unintuitive Mathematical Results That Laymen Can Be Made To Understand" is not quite so catchy.

I think the point is that storing even encrypted passwords is not as safe as storing (salted) hashes, because if the database was compromised, the encryption key would likely be compromised as well. It's safer if even the site themselves do not know your password.

Technically, you are right to say that there's no evidence passwords are being stored in plaintext, but encrypted stores really aren't any better.

I had to read this three times before I realised you were being sarcastic. My blood pressure does not thank you.

I think this is very dependent on the area of maths. In applied mathematics, obviously simulation is of huge importance; and in very fundamental pure maths the language is close enough to formal logic to make it easier to apply computer-aided proof.

But in very pure disciplines which rely on several layers of supporting definitions and theorems, there is little to be gained from numerical computation - but huge amounts of bootstrapping are still required before the computer can prove results of its own using logical manipulation.

To take a simple example, writing a computer program capable of proving that there are infinitely many primes - without embedding so much domain knowledge in it as to render it useless - seems a pretty nontrivial task.

Big fan of this app - definitely something I'll make use of.

One idea though: something that as a singer I would find useful is the ability to select a single base note and practice against that but without the base note being played each time - if that makes sense.

This would be handy because a big challenge for singers (or players of instruments without clear note separation) is placing the notes you hear relative to the key you're in.

I realise it's a lightweight app and making it too feature-rich wouldn't necessarily be a good thing, but I know a lot of people would find it helpful.

Absolutely. For example, the ingroup bias allows a group of related individuals to benefit at the expense of those with whom they are less likely to share genetic material; and the observation bias allows our brain to focus on information that is more likely to be of interest - if I tell you tigers have stripes, you're likely to notice more stripes and maybe spot more tigers.

In general, though, fallacies like this arise because the brain prefers rules that are simple and quick to apply - they may not be optimal in terms of the solution obtained, but they are effective heuristics once cost and time are factored in. It's better to spot a hidden tiger quickly but occasionally get it wrong, than to be the world's greatest tiger spotter given half an hour to think about it. Most probabilistic and decision-making fallacies fall into this bracket - the middle-choice fallacy is actually a pretty good heuristic (as another poster pointed out)- but there are edge cases where it can trick us if we don't think over our decisions rationally when we have the time and freedom to.

It's a shame to see the driving/flying statistic trotted out again. My absolute chance of dying in a car crash as opposed to a plane crash is irrelevant when deciding which to take - as an individual I'm far more likely to die of drowning than of acid burns, but I'd still rather go swimming in plain old H2O. Neglecting the base rate is itself a common probabilistic fallacy.

In fact the relative risk of flying versus driving depends significantly on the way you choose to assess risk: chance of death per journey, per mile, or per hour. Further, a straightforward mortality assessment does not factor in the risk of nonlethal but debilitating injury. Like all good questions, it's not as simple as it first appears.

This is really great - love the idea. Just in case you hadn't spotted, I think there's a punctuation bug whereby characters are sometimes omitted, although it doesn't happen all the time. Not a big deal though.

Example:

Correctly punctuated original: http://tinyurl.com/b6sbozh

Bookbookgoose ignores the apostrophes: http://imgur.com/4U9Ow

[EDIT: Dropbox link was broken - using imgur instead]

In principle, not every number can be directly expressed as the result of a nontrivial Ackermann function or Knuth operator (just as not every number is a square or a cube) - although there is a section in the Knuth page you linked on how to represent any nonnegative integer as a series of Knuth-type operations.

I would imagine, though, that to represent an arbitrary integer between 0 and n still needs ceil(log_2(n+1)) bits - since you have to distinguish between n+1 choices, whichever notation you use.

Presumably it rather depends on what you're storing such large numbers for. If you want to store a single number n (for the sake of argument, let's say a positive integer) for later reference, and have no knowledge in advance of any special properties of the number, you can't do better than ceil(log_2(n)) bits of information, whatever scheme you use for storing it.

On the other hand, if you know your number has particular properties (for example, say you knew it was an even number between 10,000 and 1,000,000 digits long) you would be able to use that information to store it more efficiently. See http://en.wikipedia.org/wiki/Information_theory for the mathematical background for treating this sort of problem.

However, many space-efficient ways of storing data are not time-efficient when it comes to retrieving it or performing operations on it. I can refer to huge numbers using very dense mathematical notation - for example, I could mention A(100,100) (see Petrushka's comment on the Ackermann function) - which would allow me to refer to an astronomically huge number in ten characters - but this is useless for most practical purposes as it would take an inordinate amount of time to calculate. Similarly, I could 'store' the quadrillionth prime in memory by simply writing "the quadrillionth prime" - but this is probably not what you wanted.

The data structure you use will depend enormously on what you want to use it for. For phone numbers, it is helpful to be able to auto-complete numbers a use is typing, which would be difficult to achieve if you simply concatenated them with separators and compressed the data. For storing words in a dictionary, you might use a trie, but that becomes a poor choice of structure if what you actually want is to be able to easily identify words that rhyme with each other.

In summary, it is important to identify what sort of numbers you are storing, what operations you intend to be able to support on your dataset, and how you intend to access the elements. In principle, for arbitrary positive integers, you can't do better than the bound I provided - and if you want more than to just store the data without using it, you may end up using rather more than that.