matt grime

You didn't ask for a definition. And as i said, arbitrary is an adjective it is not a noun, it doesn't make sense to ask what "an arbitrary" is. Seriously, find a dictionary and look it up; it's not maths, and i guess that English isn't your first language so rather than you have to mistranslate my words just look them up in the library, or use babelfish. If you want to do it by example: Let P be a proposition, then P(x) is true for arbitrary x if it is true for all x for which it makes sense to ask for P(x). As you can see it has nothing to do with parameters, that is why your question is impossibly difficult to answer sensibly. It is like asking someone to define the difference between 'fish' and 'fast'.

Look in a dictionary, for they are used in maths as their common english usage. oone is a noun the the other an adjective. that is sufficient difference between them

IF you read the replies here it is evident that people have interpreted it in several different ways. Which one is it? Is it that unreasonable to ask, Bigmoosie, exactly which meaning the question ought to have? If s is a shuffle in S_52, then the number of times it needs to be done to obtain the initial ordering again is (any multiple of) ord(s). Often s is presumed to be the distinguished shuffle of a perfect riffle shuffle, which is eminently practically attainable as any card sharp can tell you. I don't know its order. If we want a number that works for all shuffles then it is lcm(ord(s)) for s in S_52. If we mean: shuffle at random, then another at random, the initial formation occurs once every 52! times on average, this is the probabilistic version. The person who mentioned chaos has introduced a spurious concept, and it should be ignored.

That certainly works if you mean 'repeat an arbitrary shuffle' until you get back to the beginning, but that is a weak bound, and in general the largest order of an element in S_n (shiffles of n objects) is not n! but considerably smaller. S_4 for instance has order 24=4! but the largest order of an element in S_4 is 4. But then there is the interpretation that you mean 52! shuffles will reorder the pack for all shuffles, which is certainly true (not sure if its optimal with this property since in S_4 the universal number ie the lcm of the orders of elements is 12) Would the OP be so kind as to write his question unambiguously? Ie, what do you mean by shuffle, precisely? A perfect riffle shuffle (interleaving), an arbitrary shuffle repeated, or just a series of random shuffles (when there is no deterministic answer, but merely a probablistic one)

Who says you can't do a perrfect shuffle?

There is an element of logic (as a subject in mathematics) in what you want to know. However mathematics simply ignores such problems by declaring them to be disallowed, thus one cannot have the set of sets, or the set of sets that do not contain themselves, which is equivalent to the liar paradox (if somethingn is true then it's false, if it's false then it's true). Thus these problems are not resolvable in mathematics since they are not a valid part of mathematics. Or if you prefer the resolution is to not study situations where they may arise. That may displease you, but that's they way it is. If you want to debate them and argue that you shouldn't be allowed to ignore these statements then you're into philosophy. If you like mathematics is the art of the possible to paraphrase someone. What you want is a system that is impossible, if your situation were to arise. We ignore lots of things in mathematics to make life easier. Sometimes we consider some problems (eg usually we do not square root -1 in front of the beginner student since they;ve been told that you can't do this by teachers who have no idea about mathematics; we can allow square rooting of -1 and the expense of having to consider large sets than just real numbers; you're kind of problem is far more fundamental than that and cannot be allowed since it creates an inconsistent system). A metestatement is just my loose term for a statement that is about truth/falseness. It refers to 'first order' things like true/false and is thus at a higher level in some sense. Not of course that i really see what your point is: so what, you can write down some statements that are mutually inconsistent. who cares? so can anyone.

what paradox? just because you say something is possible does not make it realizable. note that the completely acceptable statement: this statement is false, is not allowed in logic (it is one of the assumptions, or axioms). Self references are disallowed, and it is any rate a meta statement or a second order statement. cf Russell's paradox. no, we don't. you might but that is not our problem. <meaningless stuff snipped> so it has nothing to do with real life then. you make certain hypthoseses, that are contradictory, and make some deductions based upon some even more dodgy assumptions, most of which are not expressed. it isn't maths. it might be philosophy. it probably isn't. even assuming your assumptions are more properly expressed, at *best* you're just producing a set of assumptions that do not make 'consistent' sense. that isn't surprising, a lot of people do that.

Seeing as the definition of pi is that the circumference of a circle of radius r is 2*pi*r, then obviously the answer is no. Of course in practical considerations we always use some rational approximation (such as 3.14) so the answer is yes. Or we could conclude that the question is ambiguous.

No you won't.

The roots don't repeat with period 3. They aren't cube roots of unity. They are cube roots of -1. h^4=h^3h=-h. They satisfy x^3+1=0, not x^3-1=0.

Because it is: i presume you're subbing h^2=h-1 in at some point. Why not do it at the start instead of expanding a sextic unnecessarily? (you're algebra is just wrong)

Those random number generators are random because they are to all intents and purposes as far as we can see random, in the sense of any output is equally likely as any other output. It is nothing to do with inverse operations, whatever that might mean. And rest of what you say is meaningless. What do you even care to define by random? It is only a loose term, unless you define it properly there is no answer.

No, just very misused nomenclature by non-mathematicians. A probabilistic system that displayed some evidence of unpredictability wouldn't exactly be news would it?

Chaotic systems are (normally) governed by entirely deterministic systems; there is no probabilisitc input. The 'chaos' appears because of certain properties of these deterministic systems, and not because of any inherent 'randomness'. There is a variant called quantum chaos but I know nothing about that.

That isn't a description of chaos theory. Chaotic systems are in some obvious sense highly deterministic systems. The map from [0,1] to [0,1] given by x goes to kx(1-x) [the logistic map, 0<k<=4] is the canonical example of a chaotic system and as you can see there is only one variable, and we can tell its evolution in time explicitly, but the system displays topological transitivity and sensitieve dependence on initial conditions, ie the two properties we (usually) take to mean what the layman calls chaos. I wish I had a pound for every person who's mis-cited chaos theory as an example of something. It seems that the non-mathematical world hasn't got chaos at all in perspective.

Then you must belong to the (even smaller?) subset of nonmathematicians who read the mathematics threads and who know what max-flow min-cut is. (I imagine lots of computer scientists also know what it is.) However it is a reasonable observation that the number of mathematically minded who read this site is a lot lower than say sci.math, or other places.

Not sure what you were expecting really: the number of mathematicians reading this site is negligible, and the number of those who've done enough optimization to know what max-flow/min-cut is must be, oh, about 3. I can only see two possible networks to construct. One has as vertices all the elements and is complete with the weight of edge i to j the number of subsets i and j lie in, the other is sort of dual: the complete graph on all the subsets with the weight of each edge the cardinality of the intersection. However, it is up to you to figure out what condition 'there is a set if distinct representatives' corresponds to, if anything at all.

If you believe that then perhaps you either did not understand what I meant or maths is not the subject for you. Mathematics is all about definitions. Notice how no mathematician has trouble with infinity yet that doesn't stop people asking dumb questions on it here simply because they do not konw or accept some definitions

Well, that is the nice wooly idea, but it needs formalizing if you are to undertake mathematical analysis of these things as abstract objects. Ultimately you aren't even reasoning about things that 'occur' but just formal random variables. This becomes especially true when one passes to infinite (discrete or continuous) sample spaces. For instance we commonly refer to picking an integer at random, but actually that is a very loose terminology indeed, and needs plenty of clarification as to what you mean if you were to be pressed as to what you actually meant.

The statements are true in some larger model (and also false in another larger model), if this is the thing I'm thinking of. The notion of 'constructed' but not provable simply means that the statement that cannot be proven is made up from statements (constructed) in the system. (if it were made of statements not in the system it would be a silly result!) the statement is a little like the liar's paradox: this theorem is not provable from the axioms of the system. or at least that's what it is supposedly like. by the liar's paradox, if that is a theorem it is not a theorem, and if it is not a theorem it is a theorem...

random? not really, they are highly deterministically placed (look at the sieve or erastrothenes) and there is for instance always a prime in the interval [n,2n] (russel's postulate) and in particular the n'th prime is less than or equal to 2^n. something like that couldn't be true of a randomly chosen set of integers (where we'll take random to mean in some kind of asymptotic probabilistic sense, eg a genuinely random selection from the interval [0,M] would contain on average m/N numbers in any subset of size m) but they behave approximately randomly, indeed things that are true for 'random' numbers in some proper technical sense are true for the primes in particular. this could of course just be terminology. second, there *is* a series the nth term of which is the nth prime, but there is no simple way to calculate this in an reasonable amount of time. And for whomever asked, the digits of pi are conjectured to be *normal*, this cannot be proven. And no there is nothing a priori random about the digits of ALL irrationals, indeed most (all?) we know about are either not normal or are only conjectured to be possibly normal. The digits of the first known transcendetal number (as in the first number known to be transcendetal) are very highly specified (something like sum over n of 10^{-n!}) http://mathworld.wolfram.com/NormalNumber.html seems to indicate that the number of known normal numbers is practically zero, though there are uncountably many normal numbers, indeed the nonnormal numbers have measure zero i'm told.

it's maths, not physics, the answer is pi-1, not pi^c -1 pi is just a number.

Any set is well ordered (or can be; it is not required to agree with the ordering of < that is given on the reals) if we assume the axiom of choice. How do you add the series x_n=1/n^2 and get pi^2/6? By adding the first term and the second and so on, and I'm certainly adding a countably infinite set.... the point is if you say to me "add up these numbers" i am entitled, nay necessitated, to say "what is the first, second, third term" etc, ie ask the order of summing as my example shows. it makes no sense otherwise. or to put it another way, no matter what the two people discussing this first thought, the question as asked is completely meaningless. I am asking them to demonstrate to me that they know how to sum something that is not either a finite sum or a series, because it is not an obvious thing to be able to do.

I wouldn't know where to begin with the problems in this.... Oh, no, I do: how about: that makes no sense? Really, it doesn't. There are words there and all the words individually make sense, but not together.

Unless you specify some total ordering of the numbers there is no canonical way to add up the infinite set. Addition doesn't work like that. You even rely on picking a series that diverges which is false on two counts: firstly there is no ordering so there is no such thing as a series, and secondly 1-1/2+1/3-1/4+1/5-... converges to log 2, yet it has divergent subseries, moreover rearranging terms allows me to sum that puppy to log3/5 or something. And the situation of adding "from" -a to a demonstrates that I can pick a total ordering which has a vaguely meaningful transfinite sum, quite possibly to almost any limit I wanted. Presumably you were thinking of strictly positive numbers, and adding an infnite number of them which do not converge to zero.