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.
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.
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 :)
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 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!
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.
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 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.