Recent content by Ted123

Oh yeah of course. You don't have to write 1-cycles in a permutation so e=(1)(2)(3)(4)(5)=(1)=(2)=(3)=(4)=(5)=(1)(2)=(1)(3) etc.

Say we have the symmetric group S_5. The permutations of \{2,5\} are the identity e and the transposition (25). But what are all the permutations of \{3\}? Is it e and the 1-cycle (3)?

Is it true that if A \equiv B \mod{\varphi(N)} where \varphi (N) is Euler's totient function then a^A \equiv a^B \mod{N}? I'm not after a proof or anything but I didn't do a number theory course and it seems that this fact is used in many questions I'm currently doing.

So, assuming t\neq 0 (for if t=0 the equation is trivially true), I can conclude that the equation will be true for \alpha =1 for all c\in\mathbb{C}, a=0, b=0 and when \alpha \neq 1 it will only be true for a,b,c=0? In other words, whatever the value of \alpha, the equation will be true for...

If \alpha =1 then a=0 and b=0 If \alpha \neq 1 then the first equation implies a(1-\alpha) = 0 so a=0 since \alpha \neq 1. Subbing a=0 into the second equation gives b=\alpha b so b(1-\alpha)=0 so b=0 since \alpha \neq 1. Subbing a=0, b=0 in the third equation gives c=0

x is a variable and t\in\mathbb{R} and \alpha is a fixed constant. We want the equation to be true for all t. If you equate coefficients you get: a=\alpha a 2ta+b = \alpha b at^2 + bt + c = \alpha c For what values of a, b and c are these true?

Homework Statement For which values of a,b,c\in\mathbb{C} is the following equation true? a(x+t)^2 + b(x+t) + c = \alpha(ax^2 + bx + c) where \alpha is some scalar. The Attempt at a Solution How do I go about this?

If V is a 3-dimensional Lie algebra with basis vectors E,F,G with Lie bracket relations [E,F]=G, [E,G]=0, [F,G]=0 and V' is the Lie algebra consisting of all 3x3 strictly upper triangular matrices with complex entries then would you say the following 2 mappings (isomorphisms) are different? I...

Is the polynomial f(x) = x^7 + (3-i)x^2 + (3+4i)x + (4+2i) \in \mathbb{Z}[i][x] irreducible? 2+i is a Gauassian prime isn't it? And 2+i does not divide 1, 2+i | 3-i , 2+i | 3+4i , 2+i | 4+2i and (2+i)^2 = 3+4i which does not divide 4+2i.

My statement of Eisenstein's criterion is the following: Let R be an integral domain, P be prime ideal of R and f(x) = a_0 + a_1x + ... + a_n x^n \in R[x]. Suppose (1) a_0 , a_1 , ... , a_{n-1} \in P (2) a_0 \in P but a_0 \not\in P^2 (3) a_n \not\in P Then f has no divisors of...

Homework Statement Homework Equations Monotone Convergence Theorem: http://img696.imageshack.us/img696/5469/mct.png The Attempt at a Solution I know this almost follows from the theorem. But I first need to write \displaystyle \int_{I_n} f = \int_S f_n for some f_n in such a...

Bingo! If only I'd drawn a graph first...

But if x<0 then e^{-x^2} > xe^{-x^2}

Oh yes, it's the other way around for divergence isn't it: if \int f(x) dx diverges then so does \int g(x) dx. OK, so I need an upper bound whose integral converges then. Any suggestions?

Well, OK. Let's start from scratch. I'm trying to prove f(x) = e^{-x^2} \in L^1(\mathbb{R}) ; that is, that f is Lebesgue integrable over \mathbb{R}. Let f(x) = e^{-x^2} \chi_{(-\infty , \infty)}(x) and f_n = f \chi_{[-n,n]}. Since f\geqslant 0, (f_n) is an increasing sequence of...