THIS IS CAKETIME
This is one of these funny yet slightly bizzare internet videos you get every now and then. Irritatingly it appears not be available on you tube, which will soon host no videos of any interest....
22
Yesterday, I'm not sure that I feel old- actually I think I feel about 17 still, I'm not entirely sure how I'm going to remain at that mental age... preferably a long time. Am off for yet another interview tommorrow, after a catastrophic one at GCHQ, where I was somewhat baffled by the questions- my knowledge failed me quite badly, but never mind. I scribbled down some nonsense answers based on what I could remember from a course in first year oh so long ago.
Oh well, at least I got to go into a high security building, even if was only a visitor centre, I'm going to count myself as a secret agent from now on. I kind of liked the idea of telling people that I can't tell them what I do at work. Top secret, yeah? I never liked martini anyway....
New name, and a confession of Harry Potter obsession
Eventually I'll settle on a name for this blog, and it shall be a wonderful day... Possibly.
I have a pathological need to have a double seat to myself while on the train or bus. Sadly this requirement is rarely fufilled, leading me to develop a deep resentment of the poor person next to me. And god forbid they have the teremity to actually say something to me.. for such offenders no punishment is bad enough for them. I'm not declaring this to get any sympathy here, just to point out that I am probably officially insane. This hatred does not extend to friends, of course, but it does extend to people I only know a little bit... they're the worst kind because you are expected to talk to them, despite having very little in common. Apparently it's considered rude just to read your book, no matter how good it is.
Speaking of good books, it is only 4 months now until the final Harry Potter comes out. If Rowling kills off Harry I shall be somewhat displeased, but I would be far more upset if she saw fit to take down Ron or Hermione. Wild predictions-
- Voldemort will die. Well. This one isn't wild
- Harry will not- I just don't think it's that kind of book. If he does die, it will obviously be in the final battle
- Snape will die, relatively soon after revealing that A-he was in love will Lily and B-he was on the side of good. He will sacrifice himself to help fight Voldemort
- A main good character will die. Possibly a Weasly, possibly Hagrid. If it's Lupin I will be depressed.
- RAB=Regulus Black. I have a lot to back up this theory, primarily that it required two wizards to get the locket, and only one wizard can get on. Solution? A house elf! Kreacher would be bound to help his young master, especially if he was tricked. Additionally drinking that potion could have been what drove Kreacher mad. Rowling has mentioned that Kreacher is very important- I figure he either knows the location of the locket, or at least the back story.
- Dumbledore is dead, but will have certainly have left some kind of message- possibly the penseive will give clues.... The look of triumph has yet to be explained, and shall doubtless be the key to Voldemort's defeat. We still have yet to see exactly how love will defeat Voldemort
- I honestly can't guess where the horcruxes are. But I predict that one will be located somewhere in Hogwarts, leading to a big show down- either the final battle or penultimate.
- Malfoy will have to redeem himself- possibly by uniting slytherin to help fight against Voldemort.
- Pettigrew has to pay off his blood debt somehow, mentioned in Azkaban.
- It'd be nice if Harry stopped using bloody unforgivable curses already! He's earned about twenty sentences in Azkaban.
- The door they could not open in the ministry of magic will probably feature.
So theres some predictions. I'm confident about most of these, I have made them bold 'cause I'm crazy like that- no hedging about on some of these! My record is not excellent, I will add... I was convinced Harry was going to get together with Luna....
Maths-statistics time!
Ok, enough with infinity, lets move onto something conceptually much easier. Statistics. You will, of course, remember the phrase "lies, damn lies, and statistics", which I'm pretty sure has been attributed to more than one person in it's time... Statistics don't really lie, but they can certainly be misinterpreted for someones gain. Statistics can sound impressive out of context, and without any kind of background, a statistic is fairly meaningless.
Generally, when someone says something like "57% of people hate you", they mean in a survey 57% hated you. It is very rare to talk about census data in statistics, although it of course occurs, and thats when you get your best results, because, well, you are talking about the whole population. The lies tend to occur on the lower level, but manipulation is everywhere.
The problem about statistics is that the results are not always obvious- they are sometimes counter intuitive. A nice example is tests. Lets suppose I have a great medical test that is correct 99% of the time. Awesome, you might think, lets use it!
That might not be such a good idea....
Suppose only 1 in 1000 people suffered from the illness. Then when I test the population if Britain (say) then there will be 60,000 with it- the test will catch 59,400. However there will be 59,940,000 people without it, and the test will think 599,400 of those have the disease! This is ridiculous, we have 10 times as many people who do not have the disease as do. This is why many tests that seem to be excellent are not used on a wide level, simply because you would end up scaring many many more people than you would save.
This is also worth baring in mind when you think about how strict the law should be. I suspect giving the police 99% accuracy is somewhat generous, but from this you can see that they would end up being wrong most of the time- this is partially why our legal system has to be so exacting, simply because we are not populated by criminals.
Labels: maths
SU elections
I'm not entirely sure voting for someone in the SU is entirely worthwhile. They hardly have much policy power- at least my impression of the SU has always been one of a body that bows down to whatever the university administrative staff choose. So why vote? You wouldn't vote for administrators, and thats essentially what sabbs seem to be- it's like voting for civil servants. Every single candidates poster involves some kind of bizzare pun about their name, or the position they are applying for, and very few have much to say on policy..... Finally of course I'm leaving next year, so it's hardly as if my the new set of sabbs will effect me.
Despite all these excellent arguments I am nevertheless currently voting. I honestly don't know why....
Maths part 6 the nature of infinity
So i'm here. Finally. Remember all our components so far? We have our types of numbers, each set bigger than the other, remembering that a set is just a collection of objects, in this case numbers, and functions, which given a value from one set, give you a value from the other set.
And we now know functions can be "bijective". The actual meaning is not important now- although it's nice if you understand it. All you need to know that if there exists an injection from A to B then there are at least as many objects in B as there are in A- so the number of objects in B is greater or equal to that in A, and with a surjection, the opposite is true- the number of objects in A is greater or equal to that in B. So if we have both, the number in both is equal.
The great thing about the way we have set up our definition is that we don't need to go through assigning a number to each element in a set- we just have to show it's possible. So it's actually possible to apply these definitions to infinite sets.
A simple example, the natural numbers (1,2,3,4....) to the even numbers (2,4,6,8,....) Now f(x)=2x will easily give us a bijection here- we can assign a natural number to every even number. So while both sets have an infinite amount of objects, we can see that they (sort of), have the same amount. Because amounts don't necessarily mean much when we are talking about infinity, mathematicians say "cardinality" instead. But it's reasonable for you to say they have the same amount of objects.
Now this an interesting result, because one would expect the number of natural numbers to be double that of the even numbers- but is not true, at least not according to the version of counting we are using.
To give us our idea of infinity, we tend to split infinite sets into broad categories, countable and uncountable. Countable sets are ones which have a bijection to the natural numbers, that is we can assign a natural number to each member of the set. If you imagine an infinite cars driving by on the road, they will never stop, but you can certainly count them as they go.
Uncountable sets are bigger sets than the natural numbers- that is there is an injection from the natural numbers to the bigger set, but no surjection. So we cannot assign a natural number to each object. Imagine yourself plunked in the middle of an infinite desert and trying to count all the sand... there is no real order you can count them, no sensible pattern which will allow you to count them. Uncountable sets are huge, ridiculously huge in fact, which we shall talk about a little later.
Now the integers (all the whole numbers, including the negative ones) are countable- assign 1 to 0, 2 to 1, 3 to -1, 4 to 2, 5 to -2.... and so on, and you can describe each one.
Whats more, it is possible to show that even the rational numbers (all the fractions) are countable- after all they are just integers on top of natural numbers.
So where do we get uncountable numbers from? Well the real numbers are uncountable- there are a multitude of proofs for this, but all of them will involve mathematics a little too hard for here, and also I can't remember the easier ones (easier conceptually anyway. The mathematics involved is quite hard...). Bizzarely, or probably not, only the transcendental numbers (remember them- they are the extremely weird numers like pi and e) are uncountable.
Now whats odd about this is it's possible that compartively, the countable numbers take up a very tiny space in infinity, so if you were to pick out a number at random you'd always get an uncountable one- which is why, incidentally, all those physical constants are so inconviniently long..... The weird thing is that as a fourth year mathematician I know of very few examples of transcendental numbers... but they make up almost all of the numbers... crazy no?
Finally we have a notion of size in infinity, with uncountable sets sitting at the top of the roost. I have some more things to say about mathematics, and a little bit more about infinity, but for now... fare thee well.
Labels: maths
Maths office
So I have received an interview offter from Leicester, along with a request for my references. I said, truthfully, that while I had the references, I did not have the course transcripts, which the office have nearly taken a week to do. One of them is ill, and apparently work does not get done if this person is ill.... sigh. Oh well, thats two interviews I have managed to notch up, so go me! My other interview is for GCHQ, who are so tight that they don't provide travel expenses or even provide bloody lunch. Which is nice. Oh well, it's gonna be a pretty tense time come the end of march....
Spelling
So I got back a set of references today, which is good as I need to send the things off soon. Now, there were three forms, each of which I filled in the details of myself and the course, leaving my lecturer to write the reference (well...obviously). So how in gods name did she spell my name incorrectly on the envelope containing them? Hopefully she didn't refer to me by name much... but... for crying out loud! This has happened to me more than once, and it is somewhat frustrating when the only guide the person has to spelling my name is the spelling I give them.... sigh.
Math student humour
I found
this slightly amusing.....
ghost rider
Is a mind boggingly awful mess. See, it had got three stars in the usually fairly reliable Empire magazine, so while I was hardly expecting a masterpiece, I was ready to have some mindless fun.
The only problem is, when a film gets as ridiculous as Ghost Rider it is somewhat difficult to keep your brain switched off. The soundtrack is very poor, ridiculously over dramatic at dramatic points, and generally disconcerting enough to throw you out of the action. I'd like to talk about how awful the acting is, but, to be fair, the lines are bad enough that it would take Ian McKellen to deliver them (Ian McKellen can deliver pretty much any line in the universe. Genuine fact.)
Nicholas Cage is probably the best thing in this movie, a setence I honestly never thought I'd write. Many have written about his over-acting in scenes where he changes to the ghost rider, but to be honest thats not unusual, and at least theres some bloody energy there. Sadly once he becomes the ghost rider we pretty much lose Cage and get an (admittedly cool looking) cgi creation. The fights are probably this films greatest let down- generally when one sees a super hero movie one expects some impressive fight scenes. Instead we discover that the ghost rider is just ridiculously over powered- he disposes most of the bad guys within 5 seconds, with very, very, little fighting, although I suppose one could argue that a skeleton could not engage in a large range of martial arts......
The plot goes from ludicrous to insane as the story progresses, and the acting just gets ridiculous until you become completely detached from the story and pray for the damn thing to end (it clocks in at just under 2 hours for some reason.....) Do not see this film....
Maths part 5- sets and counting
So, let's recap. We have our numbers- the naturals, the rationals, the reals, and the complex numbers, and we have our functions. To reiterate functions, they just tell you what to do to a number. Give me a number x (which can be 1, 4, pi... whatever you like), and I'll look at my function, do something to it, and give it back to you.
So next we need to look at sets. A set is simply a collection of objects. For example, the collection of all star signs is a set, as is the collection of people you'd rather jump off a bridge than sleep with. When I am talking about sets though, I will generally be talking about a collection of numbers. This set can be infinite- the natural numbers compose of a set... the set of all natural numbers. Note here that the notion of a set is just a conceptually useful concept, theres nothing particularly special about it, although it is worth noting that sets don't have repitition- it won't have 1 twice, for example.
Generally sets are written with these curly brackets {N}, for example, or {1,2,4,7,9}. They can also be empty too- we call such a thing the empty set, it's a useful concept to have.
See, sets aren't too scary (although I have skipped various notions for clarity).
Now I get to talk about counting. Counting is vital to an understanding of infinity. That is, an understanding of counting anyway. Suppose I have a deck of cards, and want to see if it has the requisite 52 cards in it. How do I count it? Well I can do it the usual way, but I am a fallible creature after all, and might miss count at some point. Another way to do it would be to get a deck we know has 52 cards (perhaps we went through and numbered each card). Now all we have to do is make sure there is a card to match with each of the cards from our original deck. If they all match up, we know our original deck has 52 cards. This is actually what you are doing when you are counting- you are attempting to assign all 52 numbers to an individual card.
We can translate to the concept of a "bijection". A bijection between two sets exists if we can find a function between those two sets that is both injective and surjective.
Those are some scary words, but conceptually all we are doing is what I described above. An injection says that for every element in our first set (call it A), we can assign it to a unique element in our second set (B). So in our pack of card examples, if we only had 51 cards, we would have one number left over- it would have nowhere to go to, so we would have to assign it to the same card as another number. Notice that if we had 53 cards we would be able to assign each number to it's own card... there would be one card left over, but it wouldn't matter- we would have created an injection.
So an injection by itself is not a strong enough concept to describe counting. We have a surjection, which says that for every single element of B, there exists an element of A assigned to it. In our card example, if we have 53 cards, then we only have a number available for the first 52 cards... one card is left over, so we do not have a surjection. However, if we have 51 cards, there will certainly be a number for each card, so we will have a surjection.
So you see that if we have an injection, we know we have at least as many cards as we have numbers, and if we have a surjection, we have at least as many numbers as we have cards. This traps us to saying we have
exactly as many numbers as cards.
Now the mathematical versions of an injection and a surjection are a bit more... well mathsy, for example we have an injection if f(x)=f(y) implies x=y, but it describes exactly what we were saying.
Well, now we have a tool to see if two sets have the same amount of elements. Thats kind of useful, but in finite sets it's not terribly exciting. The beauty of the way we have set up our definitions is that there is nothing to stop us using them on infinite sets. So finally we can talk about infinity! Next time anyway......
