matt grime

it is. it;s a well known case where knowing conditional probablity increases your chances of winning.

There are two possibilities here, that affect the choices. 1. If the host knew where the goats were, then the conditionals tell you to swap. 2. If the host picked a door at random, then don't change, or change, the odds are the same.

0.9 recurring equals one by definition of the terms involved. There is no debate. If you doubt the result then you aren't doing maths, you're doing you're own particular brand of maths that isnt' the same as the established definitions.

But you are going outside the realms of the reals and into the hyperreals which means that you have all kinds of set theoretic problems (coming up to the set of all sets) to deal with. One need not even pass to the notion of infintesimals to have a system where "1/infinity = 0". The extended complex plane will suffice, with a little topology, and "infinity" is properly the point at infinity. I'm not sure that even in the hyper-reals your statement is actually true, ed84c, since it presumes that 1/H, where H is infinte, is real, and infinitesimal, which I don'y believe is actually true - infintesimal yes, but real, no.

Hyperbolic geometry is the most prevalent form of geometry in applications (eg GR, SR, whatever the physics is). Euclidean geometry, whilst apparently the most useful, turns out to be the least interesting. You may wish to learn about Riemann surfaces and quotients by actions of groups to see why this apparently odd statement is actually very natural.

It isn't that infinity cannot be defined, but that it has no place in a discussion about arithmetic in the real numbers. Use of the word should be avoided altogether except when it is an explicitly and carefully defined object in the mathematical sense. The question, as stated, is just unmathemaitcal. There are "points at infinity" but this is a careful mathematical construction. We say 1/x tends to infinity as x tends to zero as a shorthand that means EXACTLY that the modulus of 1/x grows without bound. Some people, including wolfram I believe, abuse notation to say that 1/0 =infinity, but that isn't good mathematical practice. Every statement involving limits and infinity can, and should, be stated in terms of finite things until such time as people accept that there is no sich thing as "infinity" as a point in the real number system.

lambda, not lamda. all greek letters are given by slash(full name), capitalized for capital letters. there is also varphi and varrho if you like that kind of thing

Not necessarily, since you've not said if there is a countable basis. There is no such thing as "the" infinite dimensional vector space. You are asking is [math]\coprod_{a \in \Alpha} F_a[/math] countable if each F_a is countable. That depends on the indexing set \Alpha

apologies, then. can i suggest "wolfram locus points" and google?

1 is famous enough to be in any good analysis book: Let I(x) be the integral you require, then [math]I(x)I(y) = \int \int exp(-x^2-y^2)dxdy[/math] make a substitution and do it in spherical polars [math] \int\int e^{-r^2}rdrd\theta[/math] which is some multiple of pi the second: |x| is defined in two parts, so it must be integrated in two parts, unless n is even in which case the module sign can be dropped.

Vector State-Space and,, is backwards. Eigenvalues are not the study of stablity, stability is studied using eigenvalues. It's a shame people don't see more linear algebra, really. It's the introduction to so much interesting mathematics, both pure and applied (crystallography, game theory. spectral analysis..)

catenary means shaped like a cosh curve (hyperbolic trig function) It is the shape adopted by a freely hanging chain in uniform gravitational field.

I used prime numbers since I thought there was a chance you had heard of them. I can give you a list of 1001 open problems without any stretching of the imagination, however, I doubt that more than2 of them would mean anything to you right now (There is a derived equivalence between the princpal blocks of a group and the normalized of its sylow-p subgroup if the sylow group is abelian? there is a Krull-shcmidt type theorem for relatively stable categories? the constant coefficient of some L functions is non-zero iff there are infinitely many rational points....) the factorize thing states that: given any integer, there is a guaranteed way to find its factors, sadly this takes more than a reasonable number of operations (universe would end before you'd factored any big numbers such as those in RSA encrpytion). So, can you find some tricks and short cuts that'd do it in a reasonable length of time.

Mathematics is never ending, don't worry. Research Maths, for want of a better phrase, has approximately two extremes of style. 1. To answer a specific question. For instance, the following are open questions that don't look like being solved anytime soon: Are there an infinite number of twin primes (that is prime numbers differing by 2, such as 3,5, or 11,13. 2. Can any even number be written as the sum of two primes? 3. Given n objects to be arranged in m ways satisfying p constraints is there any reasonably short way of finding the optimal arrangement? 4. Can you factorize numbers quickly? 5. What is the distribution of the primes? These are reasonably easy to understand, since they do not involve anything very complicated. And that is precisely what makes them hard to solve. You know that Wiles proved Fermat's last theorem: there is no solution in integers to x^n+y^n=z^n for integer n greater than or equal to 3. Well, the proof has nothing to do with anything that you can think of. This leads us into style 2. You take some set of rules, see what you can deduce, what happens if you add in more rules. Here you tend not to have any particular result in mind, you just play around. And there are a lot of unexplored things to play around with, and sometimes by doing this you get some odd results. Most of us operate somewhere between the two extremes. That is we have some ideas of explicit problems in the background that motivate us to define abstract new systems to study to hope to get an idea how to solve the problems. Wiles classified some objects called modular forms, these are functions of a complex variable satisfying some rules. He also showed that the data that classifies these things can be used to classify semi-stable elliptic curves, and this classification implies that there are no solutions to Fermat's equation in a very odd way. He was building on 50 years of collective work: that's how fast pure maths develops. The problem is that in order to explain to you the pure parts I'd have to introduce so much more material. Applied is a little easier. Essentially the equations that govern physical systems are too complicated to solve. So we make approximations, eg linearization. But we are continually trying to improve these approximations and find better ways of allowing more vairiables to be taken into consideration.

they are defined as x^y = exp{ylogx} this works for all real positive x, and can be extended with some care to all (non-zero) x.

You recognize that that is an ellipse in R^3 right? So there are only a finite number of things to check.

Because U is specified as open in the question....

The image of any constant map is closed.

Up to transpose that is correct. Though I can never remember which way round it is. The OP should do the mental maths to figure out which way round it is.

yes. write u_i in terms of the e_i, presumably one is given in terms of the other, even in a general form.

You can summarize why it is nonsense quite simply: friendliness is defined in terms of another undefined unmathematical term, being ethical. The first proposition then states that minds are utility functions. A quick check on utility function will show you that is an idiotic pronouncement, mathematically. Nor does the author even propose to show what MAY be reasonable, that decision making processess may be modelled with a utility function. It also continues in that proposition with a series of unrelated deductions without any explanation or evidence. It also contains a very odd statement that any mathematical "function" that assigns "beauty" is uncomputable. This is clearly a dubious statement. The universe contains only a finite number of objects, so assigning a functional value seems eminently computable. It strikes me the author is confusing the philosophical difficulties inherent in defining such a subjective attribute with a mathematical statement that has no such problem.

Anyone heard of Sokal? Just because you can phrase it in pseudomathematical language doesn't make it maths. In fact it appears to be nonsense since there are no references to explain any of the undefined terms.

Like I say, these aren't questions I've considered for a long time but for this bit, G/N is n't a subgroup of G so you can't talk about it's intersection with N, really. Your other points are good, but I can't figure out a good answer right now.