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  1. J

    Differentiability of a complex function.

    (PROBLEM SOLVED) I am trying to think of a complex function that is nowhere differentiable except at the origin and on the circle of radius 1, centered at the origin. I have tried using the Cauchy-Riemann equations (where f(x+iy)=u(x,y)+iv(x,y)) \frac{\partial u}{\partial x}=\frac{\partial...
  2. J

    Perturbation Theory Formulas

    Does anybody happen to know where to find the perturbation theory formulas for the energies and states up to fourth order? I have to do a calculation up to this order and don't want to have to derive them if I don't have to (I know that Wikipedia has high order energies, but they only have the...
  3. J

    Trouble with use of the Residue Theorem

    I have just learned the residue theorem and am attempting to apply it to this intergral. \int_{0}^{\infty}\frac{dx}{x^3+a^3}=\frac{2\pi}{3\sqrt{3}a^2} where a is real and greater than 0. I want to take a ray going out at \theta=0 and another at \theta=\frac{2\pi}{3} and connect them with an...
  4. J

    Retake General GRE?

    I just took the General GRE and recieved V: 560 Q: 740. Would it be advisable to take it again, or would the improvement in quantitative not matter that much? Thanks for any advice.
  5. J

    Mathematica Question (Matrix Multiplication)

    I am trying to compute the following, \prod_{j=0}^{N-1}\left[\hat{I}+\hat{M(j)}\left(\frac{T}{N}\right)\right] where \hat{I}, \hat{M(j)} are matrices. My problem is that Mathematica interprets this product as element-wise with respect to the matrices, but I of course want it to use matrix...
  6. J

    Question on Galois Group

    I am trying to show that if z, z2, z3, ..., zn=1 are n distinct roots of xn-1 in some extension field of Q (the rationals), then GalQQ(z) (the galois group of Q(z) over Q) is abelian. Would I be wrong to say that since the galois group we're talking about here only involves an extension field...
  7. J

    Finite Field Question

    I am trying to prove that if c is a root of f(x) in Z_p then c^p is also a root. It seems very simple but I can't think how to approach it. Any insight on this would be greatly appreciated, and sorry for not using the latex but it seems to be acting up.
  8. J

    Simple Field Extension Question

    I'm currently trying to prove that (for a field extension K of the field F) if u\in K and u^2 is algebraic over F then u is algebraic over K. I thought of trying to prove it as contrapositive but that got me nowhere--it seems so simple but I don't know what to use for this. Any help with this...
  9. J

    Statistical Mechanics (multiplicity/accesbile microstates question)

    I am trying to show that the change in number of accessible microstates, and therefore the change in the multiplicity function g of a simple system is g=\left( \frac{\tau_F^2}{\tau_1\tau_2}\right)^{\frac{mC_V}{k_B}} where the system is two identical copper blocks at fundamental...
  10. J

    An interesting inequality-question on the proof

    An interesting inequality--question on the proof... I am working on the following proof and have gotten about half way; If a_1, a_2, \ldots , a_n are positive real numbers then \sqrt[n]{a_1 \cdots a_n} \leq \frac{a_1+a_2+\ldots +a_n}{n} By induction, I started by showing it for n=2...
  11. J

    Proof on functions of an intersection of sets

    I'm working out of Abbott's Understanding Analysis and I'm trying to show the following, For an arbitrary function g :\mathbb{R}\longrightarrow \mathbb{R} it is always true that g(A\bigcap B) \subseteq g(A) \bigcap g(B) for all sets A, B \subseteq \mathbb{R}. I'm confused on how to get...
  12. J

    Polynomials in Z modulus 9

    I'm trying to figure out how to prove that every polynomial in \mathbb{Z}_9 can be written as the product of two polynomials of positive degree (except for the constant polynomials [3] and [6]). This basically is just showing that the only possible irreducible polynomials in \mathbb{Z}_9 are the...
  13. J

    LaTeX Simple latex notation question

    Is there a symbol in latex for a divide bar that works with absolute values? The problem is for example if you want to write |a| divides |b|. It ends up looking horrible because you can't tell the difference between the abs value signs and the divide bar... |a| \vert |b| Any suggestions?
  14. J

    Simple question on disproving a group isomorphism

    I am trying to prove that the additive groups \mathbb{Z} and \mathbb{Q} are not isomorphic. I know it is not enough to show that there are maps such as, [tex]f:\mathbb{Q}\rightarrow \mathbb{Z}[/itex] where the input of the function, some f(x=\frac{a}{b}), will not be in the group of integers...
  15. J

    Proof on Order of Elements in a Group

    I'm trying to figure out how to prove the following... If a, b \in G where G is a group, then the order of bab^{-1} equals the order of a. I'm rather stumped because the group is not necessarily abelian and it seems like it would have to be in order to directly show that you can rearrange...
  16. J

    Simple prime/GCD proof question

    Hello, I'm working out of Hungerford's Abstract Algebra text and this proof has been bothering me because I think I know why it works and it's so simple but I can't figure out how you would show a rigorous proof of it... If a=p_1^{r_1}p_2^{r_2}p_3^{r_3} \cdots p_k^{r_k} and...
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