Representation Definition and 722 Threads

In atiyah's book on commutative algebra page 106 it says that elements in graded modules can be written uniquely as a sum of homogeneous elements. More precisely: If A = \oplus^{\infty}_{n=0} A_n is a graded ring, and M = \oplus^{\infty}_{n=0} M_n is a graded A-module, then an element y \in...

Hi all. I need an integral representation of z^{-\nu}K_{\nu} of a particular form. For K_{1/2} it looks like this: z^{-\frac{1}{4}}K_{1/2}(\sqrt{z}) \propto \int_{0}^{\infty}dt\exp^{-zt-1/t}t^{-1/2} How do I generalize this for arbitrary \nu? A hint is enough, maybe there's a generating...

\hat{S}^+_i=\sqrt{2S}(\hat{a}_i-\frac{1}{2S}\hat{a}^+_i\hat{a}_i\hat{a}_i) \hat{S}^-_i=\sqrt{2S}\hat{a}^+_i, \quad \hat{S}^z_i=S-\hat{a}^+_i\hat{a}_i Why is in solid state physics often convenient to use this representation? It is obvious that (\hat{S}^-_i)^{\dagger}\neq \hat{S}^+_i...

Basic question, but nevertheless. In a non-Abelian gauge theory, the fermions transform in the fundamental representation, i.e. doublets for SU(2), triplets for SU(3), while the gauge fields transform in the adjoint representation, which can be taken straight from the structure constants of...

[SIZE="4"][FONT="Book Antiqua"]How can I find the matrix representation for Green's function. If somebody have any reference please write it to me.

Hi, I currently have a Gaussian distribution (Normalized Frequency on the y-axis and a value we can just call x on the x-axis). So for the sake of simplicity, let's say that I ignore any values below 0 and any values above 1 on the x-axis. Then what I will do is take 10 equal segments...

There is something that I don't quite understand in relation to the Bloch Sphere representation of qubits. I've read that any vector on the sphere is a superposition of two basic states, like spin up and spin down, denoted by |1> and |0>. So does this mean that if the vector is at z=0...

Homework Statement Obtain the Fourier series representing the function F(t)=0 if -2\pi/w<t<0 or F(t)=sin(wt) if 0<t<2\pi/w. Homework Equations We have, of course, the standard equations for the coefficients of a Fourier expansion...

Propose a parametric representation of a spiral. Hint: Use the parametric representation of a circle. This is the parametric representation of a circle we are given : x = r * Cos(Theta) y = r * Sin (Theta) 0 <= Theta <= 2 Pi Nope, we are not given anything background on spirals...

Homework Statement y^2 + 4y + z^2 = 5, x = 3 Homework Equations The Attempt at a Solution I know that the calculated coordinates must satisfy the above equation, but I don't know how to go about solving for those coordinates. The best I could do was to equate z = \sqrt{(-y + 5)(y + 1)}.

Homework Statement There are two questions, 1) straight line through (2, 0, 4) and (-3, 0, 9) 2) straight line y = 2x + 3, z = 7x Homework Equations r(t) = a + tb = [a1 + tb1, a2 + tb2, a3 + tb3] The book also explains how to calculate the line if b is a unit vector, but I don't...

This is a carryover from a previous thread: https://www.physicsforums.com/showpost.php?p=2875138&postcount=68 Sports Fans: I am familiar with the Minkowski equations and the Lorentz transformations in one or two dimensions: A) In algebraic form (1) t2 - x2 = t'2 - x'2 (2) t' =...

Hi, How can we represent covariant and contravariant vectors on curved spacetime diagrams? How can we draw these vectors on a spacetime diagram? Contravariant vectors are really vectors, therefore we can represent them on the diagram with directed line elements. Covariant vectors are...

Homework Statement Show graphically how \vec{a}\times\vec{x}=\vec{d} defines a line. \vec{a} and \vec{d} are constants. \vec{x} is a point on the line.Homework Equations \vec{a}\times\vec{x}=a\cdot x\cdot sin(\theta)\cdot \hat{n}The Attempt at a Solution Not sure if the included relevant...

The SU(2) and SO(3) groups are homomorphic groups. Can we say that the SU(2) group is representation of SO(3) and vice versa (SU(2) representation of SO(3))? Is a representation R of some group G a group too? If so, is it true that G is representation of R?

Homework Statement Show that \delta_\epsilon(x) = \frac{\epsilon}{\pi (x^2 + \epsilon^2)} is a representation of the Dirac delta function. Homework Equations I already know how the function can satisfy the first two requirements of being a dirac delta function, namely...

Hi Everyone, I just registered for PF today because this problem was driving me nuts and I was hoping to get some help. It comes from pg. 5 of Peter Miller's "Applied Asymptotic Analysis" and goes like this: The Euler gamma constant has one definition as \gamma := \int_0^\infty...

Hey, I'm hoping this thread can clear up some confusion I have with complex signals and moving back and forth from physical signals to the mathematical models. I'll probably ask some questions specifically, but if you would like to help me please treat this whole post as a question because I'll...

Homework Statement See figure. Homework Equations N/A The Attempt at a Solution I've dealt with parametric equations for lines before in my linear algebra class but I'm not sure how I'm suppose to model two curves with one. Anyone have any suggestions?

Homework Statement Lim (t->0+) [exp(-x^2/t)] / sqrt(t) The Attempt at a Solution Ive looked at series representation: 1/sqrt(t) * [exp(-x^2) - 1+t^2/2 + t^3/6 + ...] but that doesn't help. L'Hopital's rule isn't any use either because the t terms are in the denominator but I know...

The two sets of matrices: {G_1} = i\hbar \left( {\begin{array}{*{20}{c}} 0 & 0 & 0 \\ 0 & 0 & { - 1} \\ 0 & 1 & 0 \\ \end{array}} \right){\rm{ }}{G_2} = i\hbar \left( {\begin{array}{*{20}{c}} 0 & 0 & 1 \\ 0 & 0 & 0 \\ { - 1} & 0 & 0 \\ \end{array}} \right){\rm{...

Hi guys, I have a question which is very fundamental to representation theory. What I am wondering is that whether a first rank cartesian representation of so(3) is irreducible. As I understand first rank cartesian representation of so(3) can be parametrized in terms of the Euler angles. That...

Homework Statement I am trying to evaluate exp(i*f(A)), where A is a matrix whose eigenvalues are known and real. Homework Equations You can expand functions of a matrix in a power-series. I think that's the way to get started on this problem. I foresee the exponential of a power-series...

Homework Statement Find the rational number representation of the repeating decimal. 1.0.\overline{36}Homework Equations The Attempt at a Solution I know it has something to do with infinite geometric sequences but I'm not sure what. what would your ratio be for a repeating decimal, I've...

Homework Statement F(x)=∫(0 to x) tan^(-1)t dt. f(x)= infinite series ∑n=1 (-1)^(en)(an)x^(pn)? en=? an=? pn=? I know en = n-1 Homework Equations The Attempt at a Solution Start with the geometric series 1/(1 - t) = ∑(n=0 to ∞) t^n. Let t = -x^2: 1/(1 + x^2) = ∑(n=0 to ∞)...

Hi, When I solve the diffusion equation for a spherically symmetric geometry in spherical coordinates I obtain the following general solution (after application of the boundary conditions). T(r,t) = \sum_{n=1}^{\infty}\, \frac{A_n}{r}\sin(\lambda_nr)\exp(-\alpha\lambda_n^2t) So to...

I've never properly studied complex numbers but I will soon (in September). Basically: We get taught from a young age that: the real root of f(x)=x²-4 is where the graph of y=f(x) cuts the x axis But is there a graphical representation of a complex root? What's so special about the...

with the series representation of sin or cos as a starting point (you don't know nothing else about those functions), how to prove: a. they are periodic. b. the value of the period.

This question comes from the proof of Riesz Representation Theorem in Bartle's "The Elements of Integration and Lebesgue Measure", page 90-91, as the image below shows. http://i3.6.cn/cvbnm/ac/9a/a3/3d06837bc78f74ba103b6d242a78e3a1.png The equation (8.10) is G(f)=\int fgd\mu. The...

Having trouble drawing the wave representations for this diagram. [PLAIN]http://img156.imageshack.us/img156/94/diag.png I'm not sure what the incident and reflected waves should look like. The question is asking to draw the waves in each region. I understand how to calculate the waves just...

f(x)=1/(1-x^2)^(1/2) 1/x^(1/2)=1+ sum(( (-1)^n 1*3*5*7...(2n-1)(x-1)^n )/(2^n n! ) , n=1, infty ) thus 1/(1-x^2)^(1/2) = 1+ sum(( 1*3*5*7...(2n-1)(x^2)^n )/(2^n n! ) , n=1, infty ) is this a correct taylor series representation centered at 1

In the coordinate representation of a quantum mechanical system, is it always true that the Hamiltonian of the system is diagonal? If so, can someone explain to me why this is true?

\int \frac{x-arctanx}{x^3}dx \frac{d}{dx}( x-arctanx ) = 1-\frac{1}{1+x^2}=\frac{x^2}{x^2+1} = x^2 \sum_{n=0}^{\infty}(-1)^nx^{2n} = \sum_{n=0}^{\infty}(-1)^nx^{2n+2} \int \sum_{n=0}^{\infty}(-1)^nx^{2n+2} dx = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+3}}{2n+3}+C C=0? \int...

I'm having problem finding the representation for the Fourier series with function f of period P = 2*pi such that f (x) = cosαx, −pi ≤ x ≤ pi , and α ≠ 0,±1,±2,±3,K is a constant. Any help is appreciated...

Homework Statement The problem is: (x2 - 4) y′′ + 3xy′ + y = 0, y(0) = 4, y′(0) = 1 Homework Equations Existence of power series: y = \sum c(x-x0)^n or y = (x-x0)^r\sum c(x-x0)^n The Attempt at a Solution I know the point x=2 is an ordinary point of the differential...

I am trying to show that the laplacian: L = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} can also be represented as: L = \frac{1}{2}(\vec{E}^{2}-\vec{B}^{2}) where F^{\mu\nu} = \partial{}^{\mu}A^{\nu} - \partial{}^{\nu}A^{\mu} and F_{\mu\nu} = g_{\mu\alpha}F^{\alpha\beta}g_{\beta\nu} A is the...

how can i find a power series representation for a function like f(x)= ln(1+7x)?

how can i find a power series representaion of d/dx (1/x-9)

I have come across this expression in some notes \Gamma (z) = \frac{1}{z} \prod \frac{(1+ \frac{1}{n})^{z}}{1+ \frac{z}{n}} Do you think it's accurate? I have some doubts because I have looked for it on wokipedia, and I couldn't find it.

Homework Statement Draw an argand diagram to represent the follwing property: real(z) < abs(z) < real(z)+img(z) Homework Equations z = x+iy; real(z) = x abs(z) = sqrt(x^2 + y^2) img(z) = y The Attempt at a Solution substituting original expression with x, y, and sqrt(x^2 + y^2)...

Hi Could anyone recommend me a good book that will teach me the kind of group/representation theory I would need to understand these things when applied to QFT (Lie Algebra, Lorentz group, SU(2) etc)? Thanks

Hi guys, Not actually a mathematics question as such (sorry) but does anyone know where i can get my hands on a copy of Chen's paper "On the representation of a large even integer as the sum of a prime and the product of at most two primes". For the life of me all i can find is references to it...

explicit representation?? I got to show the explicit representation of 2cos(omega)t - 1/4 sin(omega)t = 0. what is this? is this operator notation??

Homework Statement Since I don't know how to insert equations into a message here, I've scanned both the problem and my attempt at a solution. Where I run into problems is how to find an. I'm not completely sure how to treat that integral and was hoping somebody could nudge me in the...

Homework Statement Let T be the linear operator on R3 that has the given matrix A relative to the basis A = {(1,0,0), (1,1,0), (1,1,1)}. a) Determine whether T can be represented by a diagonal matrix, and b) whenever possible, find a diagonal matrix and a basis of R3 such that T is represented...

Hi all! Does anyone know the position space representation of the Feynman propagator on the cylinder? The momentum space representation is the same as in Minkowski 2D space, but the position space representation is different because the integrals over momenta are now sums. Or could someone...

D(g) is a representaiton of a group denoted by g. The representaion is recucible if it has an invariant subspace, which means that the action of any D(g) on any vector in the subspace is still in the subspace. In terms of a projection operator P onto the subspace this condition can be written...

Does anyone know how to prove that any irreducible representation of a finite group G has degree at most |G|? Equivalently, that every representation of degree >|G| is reductible. Thx!

Hi guys, Just brushing up on my GR for a project and i have a silly question: For the spherical polar representation of the schwarzschild metric, the fact that there are no infintesimal-cross terms implies that the non-diagonal entries in the matrix representation are zero, correct? I...

base representation please help~ It is known that if asks+as-1ks-1+...+a0 is a representation of n to the base k, then 0<n<=ks+1-1. Now suppose n=asks+as-1ks-1+...+a0 and m=btkt+bt-1kt-1+...+b0 with as,bt not equal to 0, are two different representations of n and m to base k, respectively...