Representation Definition and 722 Threads

Consider continuous function x(t), which has zero time average: \lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2} x(t)\,dt = 0 and exponentially decaying autocorrelation function: \lim_{T\rightarrow\infty}\frac{1}{T}\int_{-T/2}^{T/2} x(t)x(t-\tau)\,dt = C_0e^{-\gamma |\tau|}...

How do I go from the entropy of a system, S(U,V,N), to its internal energy, U(S,V,N)? For instance, for an ideal classical gas, we have S=(3/2)N*R*ln(U/N) + N*R(V/N) + N*R*c where R is the Boltzmann constant, N is the particle number, V is the volume and "c" is a constant. How can I...

Homework Statement I have my quantum mechanics final creeping up on me and I just have a question about something that doesn't appear to be covered in the text. Let's say you have a wave function of the following form for a linear harmonic oscillator: \Psi = c_1 | E_1 \rangle + c_2 | E_2...

Homework Statement Represent 5x / (10 + x) as a power series, find c0, c1, c2, c3 and c4, find the radius of convergence The Attempt at a Solution I think I got the representation fine: Ʃ from 0 to ∞ = 1/2 [ (-1^n)(x^n+1) / 10^n ] Radius of Conv. = 10 But what the hell are those...

Is this interesting? I have a way to represent all of the real numbers with a subset of the open interval (0,1). I write as a binary decimal x = .a_{0}a_{1}a_{2}a_{3}... where x = \sum -1^{a_{0}+1}/p_{i} where p_{i} is the i^{}th prime. Since the reciprocals of the primes diverges, in this...

Hello, Typically the area of a parallelogram if give by A=b*h The det(M) =ad-bc, where m=2x2 matrix. How they are related? -- Shounak

Hello all, Recently I've found something very interesting concerning Taylor series. It's a graphical representation of a second order error bound of the series. Here is the link: http://www.karlscalculus.org/l8_4-1.html My question is: is it possible to represent higher order error bounds...

Homework Statement I am asked to find a power series for the function f(x) = 2/(1 - x^2), centered at 0. Homework Equations The Attempt at a Solution The only part I can't determine is the interval of convergence. I get stuck on the step |x2| < 1. What am I to do next?

Hello Forum, When we represent a vector X using an orthonormal basis, we express X as a linear combination of the basis vectors: x= a1 v1 + a2 v2 + a3 v3+ ... Each coefficient a_i is the dot product between x and each basis vector v_i. If the vector x is not a row (or column vector)...

Homework Statement Write an iterative program to find the largest integer n such that the equation 6a + 9b + 20c = n where a, b, c are nonnegative integers, has no solution.Homework Equations We are given that the set of n in {x, x+1, x+2, ..., x+5} has nonnegative solutions a, b, c to the...

Homework Statement I need to verify an integral representation of the Euler constant: \int^{1}_{0}\frac{1-e^{-t}}{t}dt-\int^{\infty}_{1}[\frac{e^{-t}}{t}dt=\gammaHomework Equations The Attempt at a Solution OK, I'm supposed to use this fact (which I have already proved)...

Homework Statement O hai, my minions. So...we made the xantoprotheic reaction in the lab, but I have to attach the representation of the reaction. I haven't found it anywhere. So I came to you. Homework Equations Protein + NaNO3→(180s@373K)→(room temp) + NaOH→ Weird yellow precipitate...

Hello, everyone. I understand the Standard Model and all of its quarks, leptons and bosons, but could someone show me the equation and simply explain it? I haven't been able to find it or an explanation of how it relates to the graphical representation of the Standard Model. Thanks in advance!

I've used many different power series representations of functions and seem to always take it for granted that functions which are "nice" and continuous have such a representation. But what is the criteria for a function to have a power series representation? I know of some that don't, but...

The solution to a linear differential equation is, y=exp(ax). If a is complex ,say a=b+ic, then the period is T=2pi/c. My question is, if a is in polar form, a=r*exp(iθ), how is the period then T=2pi/θ. Any help would be great, Thank, Will

Homework Statement The quantum numbers +1/2 and -1/2 for the electron spin represents? Homework Equations a) rotation of electron in clockwise and anticlockwise directions respectively b) tow quantum mechanical spin states which have no classical analogue c) Magnetic moment of electron...

I've been trying to do the following question but I can't get the right answer. Also, I put in the wrong thread title but I don't know how to change it (sorry). An ambulance is at rest in a traffic jam in front of an intersection. At that moment, Jim is driving his car at 88.0 km/h through the...

Homework Statement I'm having trouble understanding a concept in representation theory. I've been reading several texts on the application of rep. theory to quantum mechanics ("Group Theory and Quantum Mechanics" by Tinkham and "Group Theory and Its Application to Physical Problems" by...

I know how to use determinants to solve a system of linear equations, I know I can use them to find the rank of a matrix and find out if a system is linear dependant/independant. However, I still don't really "get" determinants. To me they're some sort of magic box that I can use to calculate...

I'm self-studying representation theory for finite groups using "Group Theory and Quantum Mechanics" by Michael Tinkham. Most of it makes sense to me, but I'm having difficulty understanding what is meant by saying a function "belongs to a particular irreducible representation", or "has the...

It seems that there’s a loose way of discussing group representations that’s fine for the initiated, but confusing for the neophyte: I understand that the objects that represent group operations are usually (or always?) square matrices, which naturally can be described as “matrix...

Hi, I have the following problem, I have an electric field (which no charge) which satisfies the usual Laplace equation: \frac{\partial^{2}V}{\partial x^{2}}+\frac{\partial^{2}V}{\partial y^{2}}+\frac{\partial^{2}V}{\partial z^{2}}=0 in the region \mathbb{R}^{2}\times [\eta ,\infty ]. So...

Could you advise a DSP book which explains state-space representation with initial conditions. Initial conditions part is important for me. Thanks.

The position operator in coordinate representation is: Xab=aδ(a-b) this is diagonal as expected The momentum operator turns out to look like Pab=-ih∂aδ(a-b) Now, this is not supposed to be diagonal because it does not commute with X. However it looks pretty diagonal to me. What am I...

I'm thinking of expanding the numerical value of a real number in different bases. I want you guys to ensure some things for me so I know I'm not wrong. If X is the representation of a in the base B and Y is the representation of b in the same base(radix), can we say that X+Y = a+b? In...

Please forgive such a simple question, but how does a particle 'remember' what direction to travel in from instant to instant? I mean, say we take 2 particles way out in space, we kick one say along a +x axis, and the other with the same force along the -x axis... How is the Direction...

hi, is it misleading to write omega = 1.0000... for the ideally flat space?

I am trying to prove a specific representation of Euler's constant, but I am not really getting anywhere. I hoped you could help me with this one, because I looked it up on the Internet and even though the relation itself is found in many webpages, its proof is in none. The relation is...

Do Lorentz Transformations or their products having irreducible representation, and is superposition allowed or special consideration are needed?

The general evolution of a ket |\psi\rangle is according to -i\hbar\frac{d}{dt}|\psi\rangle=H|\psi\rangle without specifying a representation. From this equation, how can you simply get a equation in a certain representation F as below: -i\hbar\frac{\partial}{\partial t}\langle...

\stackrel{\rightarrow}{PQ} bisects \angle{ABC}. If m\angle{ABD} can be represented by 3a + 10 and m\angle{DBC} can be represented by 5a - 6, what is m\angle{ABC}? What I did was add the two representations together and multiplied by two, which would give me 16a + 8. Does that sound correct?

Hello there I am sorry if this sounds silly or out dated i want to ask anyone, how can i model the mathematical representation from music signal(time domain). lets say i recorded the signal from guitar through my music production software and print screen that signal as shown here...

Homework Statement for a.) f(x) =1/ ( (1+x)^2 ) what is the radius of convergence? b.) Use part a.) to find a power series for f(x)=1/ ( (1+x)^3) c.) Use part b.) to find a power series for f(x) =x^2 /( (1+x)^3) Homework Equations I want to check my work. I used properties of functions...

Hey, I have been churning the idea of Dirac notation around in my head and I am thinking about the position and momentum basis representation of a wavefunction in a Hilbert space. Wikipedia mentions the following in the article 'Bra-ket notation' under the heading 'Position-space wave...

RThe "canonical representation phi" (measure theory) (Royden) Homework Statement I need some help understanding the canonical representation of phi as described on p. 77 of Rodyen's 3rd edition. I've transcribed it below for those of you who don't own the book. Homework Equations The...

When you have an ordinary (read: spin zero) particle with a quantum state |ψ>, the corresponding position-space wave function <x|ψ> is an ordinary scalar function of position. How do things work when the particle has spin? In that case its quantum state will live in the tensor product of two...

Hi all! I have a discrete 2D vector field with a particular characteristic: At every point, instead of having a single vector, I have two vectors which are in the opposite direction. For example, at point p(x,y)=p(0,0) I have two vectors: v1(1,1) and v2(-1,-1). And so on for all points. I...

Homework Statement http://gyazo.com/7b2a903b6b3165595b8766d3540f43d9 What is this really saying? I can see that a functino is the inverse Fourier transform of the Fourier transform... and it doesn't matter which way round you integrate. Is that all it's saying. What's the difference...

Homework Statement Is the fundtion defined f:P(\mathbb{N}) \to [0,1] by f(X) = 0.a_{1} a_{2} \dots in binary representation where a_{k}=1 if k\in X and otherwise 0 one-to-one? (*note: N does not have 0) If not, can you change bit so that the changed funtion becomes one-to-one...

Hi all, do you know where i can find the 3x3 matrix representation of SU(2)? Which means basically rotation matrices for particles of spin 1. Thanks!

Homework Statement The problem is taken from Callen's book (page 36). Find the three equations of state in the entropy representation for a system with the fundamental equation u =A\frac{s^{5/2}}{v^{1/2}}. Show by a diagram (drawn to abitrary scale) the dependence of temperature on volume...

I have been banned, maybe my nickname was not so kind. I let the topic continue here. I report my last comment: "Ok, I got the point. thanks for replying! It's just a change of basis that under boundary condition diagonalize the Hamiltonian. But then a subtle point: In order for...

Hi all, I have a question. For sure the momentum representation used in solid state physics works for infinite lattices or periodic ones. But when it comes to finite lattice, i.e. 100 sites, can the momentum representation be used? What are the errors? Where does this fail? Thanks for...

I am reading Dummit and Foote on Representation Theory CH 18 I am struggling with the following text on page 843 - see attachment and need some help. The text I am referring to reads as follows - see attachment page 843 for details \phi ( g ) ( \alpha v + \beta w ) = g \cdot ( \alpha v +...

Hello My question is about the ground state of vibrations for a solid. I'm working with graphite and have found out that for k=0 (The Gamma symmetry point), the vibrational modes can be decomposed into irreducible represenations in the following way Vibration = 2 * E1u + 2 * E2g + 2 * A2u...

Often a time varying E-field is represented in complex format. I have a simple E-field (uniform in space) given by \vec{E}(t)=E_o\cos(\omega t)\cdot\hat{k} or equivalently, the real part of \vec{E}(t)=E_o e^{\omega t}\cdot\hat{k}. We know the potential is the negative gradient of the...

Hi,all, I m an undergrades and I am suffering on understanding the different representation transformations, namely from schrodinger picture to interaction picture tupically, my lecturer didn't state which representation he was using and I m so confused, any helps would be great. Shall I bring...

Confused.. please help! Often when an electric field varies sinusoidally with time, it is represented as a complex number. Say, \vec{E}(t)=A\cos(t) \cdot \hat{k} We know at any time, the magnitude of E is A\cos(t). Alternatively the same vector E is understood to be the real part of the...

If you consider the permutation representation of Sn in ℂ^n, i.e the transformation which takes a permutation into the operator which uses it to permute the coordinates of a vector, then of course the subspace such that every coordinate of the vector is the same is invariant under the...

I don't understand what electrical engineers mean by the frequency of a signal... Frequency is the inverse of the period, but they speak of the frequency of non-periodic signals. I know that I can derive the function using the Fourier transform, I just don't understand what frequency means in...