generalized Definition and 197 Threads

Prove the Generalized Associative Law for Groups (i.e. a finite sum of elements can be bracketed in any way). The proof is outlined in D & F. I just want to know whether or not one part of my proof is correct. Show that for any group G under the operation °, and elements a1,...,an, any...

I've seen the standard derivation of the expression for liquid pressure P = dgh where, d = density of the liquid; g = acceleration due to gravity; h = height of liquid column in many textbooks has been done by using a specific example of a cylindrical vessel. In such a case, the geometry of the...

My question is kinda simple but it has been causing me some trouble for a while. In the problem of the pendulum rotating about an axis, why isn't the angle of rotation about the axis a generalized coordinate? The doubt appears when i try to write the hamiltonian for the system and i don't know...

Homework Statement Let A \in M_{22} (\mathbb{R}) with one single eigenvalue λ and one single eigenvector v. We denote w the generalized vector such that (A - λI)w = v. Prove that v and w are linearly independent. Homework Equations I know that if A has only one eigenvalue λ and one...

Homework Statement Using the generalized triangle inequality, prove |d(x,y) - d(z,w)| ≤ d(x,z) + d(y,w) Homework Equations d(x,y) is a metric triangle inequality: d(x,y) ≤ d(x,z) + d(z,y) The Attempt at a Solution I know that this needs to be proved with cases: a) d(x,y) - d(z,w)...

For calculating the force on a continuous charge distribution due to another continuous charge distribution, if F=kdqdq'/r^2 would you simply integrate first over dq' and then dq?

I am trying to understand what generalized coordinates are but I'm having some trouble. After reading up on them a bit my best understanding of the idea of generalized coordinates is the following: Because choice of coordinate system is arbitrary when solving physical systems (or anything for...

Thank you to soroban for proposing this problem! \left| (1+2i)^n \right|^2 for n=1,2,3... can be generalized in a very simple form that doesn't include any notation related to complex numbers. 1) Find a way to generalize the nth term. 2) Prove your generalization is valid Hint 1: Start with...

Dear Forum, I am familiar with the formulas between inertial frames of reference that move at a constant speed between each other. The observed object move at a constant speed or at a constant acceleration. It can be shown that while the positions and velocities are different in the two...

Homework Statement Show that |x_1 + x_2 + · · · + x_n | ≤ |x_1 | + |x_2 | + · · · + |x_n | for any numbers x_1 , x_2 , . . . , x_n Homework Equations |x_1 + x_2| ≤ |x_1| + |x_2| (Triangle inequality)The Attempt at a Solution I tried using the principle of induction here, but to no avail...

Can you help me prove this theorem regarding Fibonacci and Lucas numbers? Theorem. Let m,r ϵ Z and n be non-zero integer. Then U2mn+r ≡ (-1)mn Ur (mod Um) and V2mn+r ≡ (-1)mn Vr (mod Um).[FONT=arial]Im not that good at proving. This type of congruence is much harder than what I read in our...

Homework Statement Let ##A## be open in ##\mathbb{R}^n##; let ##\omega## be a k-1 form in ##A##. Given ##v_1,...,v_k \in \mathbb{R}^n##, define ##h(x) = d\omega(x)((x;v_1),...,(x;v_k)),## ##g_j(x) = \omega (x)((x;v_1),...,\widehat{(x;v_j)},...,(x;v_k)),## where ##\hat{a}## means that the...

Hey! I was reading Goldestein's book on classical mechanics and I came across this (Page 20 3rd Edition): "Note that in a system of Cartesian coordinates the partial derivative of T with respect to qj vanishes. Thus, speaking in the language of differential geometry, this term arises...

Hi there. I need help to work this out. A particle with mass m is studied over a rotating reference frame, which rotates along the OZ axis with angular velocity \dot\phi=\omega, directed along OZ. It is possible to prove that the potential (due to inertial forces) can be written as: V=\omega...

In some texts about Lagrangian mechanics,its written that the generalized coordinates need not be length and angles(as is usual in coordinate systems)but they also can be quantities with other dimensions,say,energy,length^2 or even dimensionless. I want to know how will be the Lagrange's...

Consider a generalized Eigenvalue problem Av = \lambda Bv where $A$ and $B$ are square matrices of the same dimension. It is known that $A$ is positive semidefinite, and that $B$ is diagonal with positive entries. It is clear that the generalized eigenvalues will be nonnegative. What else can...

I've been having some trouble with conceptually understanding the idea of a generalized eigenvector. If we have a linear operator A and want to diagonalize we get it's eigenvalues and eigenvectors but if the algebraic multiplicity of one of the eigenvalues is greater than the geometric...

Hi all, I need to find the λ and the ai that solves the Generalized eigenvalue problem [A]{a}=-λ2 [B]{a} with [A]= -1289.57,1204.12,92.5424,-7.09489,-25037.4,32022.5,-10004.3,3019.17 1157.46,-1077.94,-0.580522,-78.9482,32022.5,-57353.5,36280.6,-10949.6...

What exactly is generalized linear model? I understand you have to use the link function. Wikipedia says: "The link function provides the relationship between the linear predictor and the mean of the distribution function." So, what is this RELATIONSHIP? Maybe someone can provide an...

I just sent some time dicking around with the MacLaurin expansion of exp(-z2) to derive a series expression for √π, by integrating term-by-term along the real line. I'm not really concerned with wether this is a useful or well-studied expression, I just thought it would be a fun exercise...

Homework Statement Find a distribution g_n which satisfies g'_n(x) = \delta(x - n) - \delta(x + n) and use it to prove \lim_{n \to \infty} \frac{\sin{nx}}{\pi x} = \delta(x) Homework Equations Nothing relevant comes up at the moment. The Attempt at a Solution Well the first...

I've calculated the mean difference of my (normally distributed) data set. The mean difference is defined as: Now, I'm trying to calculate the "mean difference deviation" in order to generate a confidence interval for this quantity ( "95% of the differences in the set are greater than...

Hi, I have formulated what I believe to be a generalized(to some degree) optimization under uncertainty problem. The write up is included in the attached file. I would appreciate any and all input, help or guidance as to how this problem could be solved. If you have any questions please feel...

Hi everyone, I'm currently trying to solve this equation : x²[A+B.exp(x)]=1 for A and B real numbers, and x a complex (this comes from physics, so in my case, Re(x)>0) I know that x.exp(x)=a has a solution using Lambert function : x=W(a) I know that x².exp(x)=a may be recast to use the...

Mathematica has this command "Eigensystem[{m,a}]", which (to quote their documentation) "gives the generalized eigenvalues and eigenvectors of m with respect to a." I have never encountered this concept before, ever - that there can be eigenvectors of matrices with respect to other matrices. All...

Please note: Below, I keep trying to put [ capital B ] but it gets turned into [b]! In Dennery and Krzywicki, they give an example of how to put a matrix in Jordan canonical form (pp. 167-170). They start with a 4x4 matrix [A] that looks kind of messy and transform it to a quasi-diagonal form...

Hello, this is rather complicated to explain so bear with me. I was wondering about the coefficients of polynomials which are factorable in the integers, meaning polynomials which can be written as (x+a)(x+b) where a and b are integers. I had a curious idea about letting the x-axis...

Homework Statement Let f be a distribution on R and suppose that its kth derivative is 0. Prove that f is a polynomial. 2. The attempt at a solution I honestly haven't a clue how to start. If I could treat f like a "regular" function, this would so easy.

Hi! Im trying to do some rather easy QM-calculations in Fortran. To do that i need a routine that calculates the generalized Laguerre polynomials. I just did the simplest implementation of the equation: L^l_n(x)=\sum_{k=0}^n\frac{(n+l)!(-x^2)^k}{(n-k)!k!} I implemented this in the...

I'm looking at McMahon's Quantum Mechanics Demystified and in the angular momentum chapter he introduces "generalized" angular momentum J, which is the sum of a particle's orbital angular momentum and its spin. It seems strange to me that these two things can be simply added together...

http://arxiv.org/abs/1106.2121 Abstract: Gravitational and electroweak interactions can be unified in analogy with the unification in the Weinberg-Salam theory. The Yang-Mills framework is generalized to include space-time translational group T(4), whose generators $T_{\mu}(=\p/\p x^{\mu})$...

Hi, Does the equation AA^\dagger=I force A^\dagger to be the generalized inverse of A? That is: AA^\dagger=I\Rightarrow A^\dagger\text{ is the generalized inverse of } A? A is any rectangular matrix over the field of complex numbers. It is very easy to verify the first three properties, but...

I would like to work out the following commutation relations (assuming I have the operators right...:-p) (1) \left[\hat{p}^{\alpha},\hat{p}_{\beta}\right] (2) \left[\hat{p}_{\alpha},\hat{L}^{\beta\gamma}\right] (3) \left[\hat{L}^{\alpha\beta},\hat{L}_{\gamma\delta}\right] where...

The title pretty much says it. According to my book, Classical Dynamics by Thornton and Marion, generalized coordinates can be quantities other than position such as energy or length squared, but what about time?

Hi, In flat space-time, the Poincare group, is the symmetry group responsible for translations, rotations, and boosts for relativistic quantum mechanics. For an arbitrary Einstein metric (not Minkowski space), what Lie group is responsible for coordinate transformations in relativistic...

Which of the generalized means (like http://en.wikipedia.org/wiki/Generalized_mean and more general) do you think is most suitable to approximate the median?

I see that a generalized eigenvector can be represented as such: (A - λI)xk+1 = xk, where A is a square matrix, x is an eigenvector, λ is the eigenvalue I is the identity matrix. This might be used, for example, if we have duplicate eigenvalues, and can only derive one eigenvector from...

The picture for the double pendulum I am referring to is pretty standard, wikipedia for example uses it and so does any other textbook. I do not completely understand why one uses the second angle measured from the vertical y-axis for the second generalized coordinate. The second angle is not...

Hey guys, need some quick help before an exam I have a differential eqn. x' = | 0 1 | *x , and initial conditions x(0) = |2| | -25 10 | |3| I find that there are two eigenvalues 5, and 5 The corresponding eigenvector to 5 is [1 5]...

Hello, I wasn't quite sure where to make this topic, so I hope I didn't do wrong by putting it here. The question I'm having is somewhat difficult to describe and I guess it's more of a mathematical question really, but since I'm learning mechanics now and came up with it, I thought it...

I remember reading a theorem that said that for an n x n matrix A, there exists a basis of Cn consisting of generalized eigenvectors of A. For A = [1 1 1; 0 1 0; 0 0 1] (the semicolons indicate a new row so that A should be 3 x 3 with a first row consisting of all 1's and a diagonal of 1's)...

Homework Statement In order to solve the problem I am working on, I have to prove the following generalized problem, S(x)=\sum_{n=0}^{\infty} n x^n =\frac{x}{(x-1)^2} for |x|< 1 I evaluated this sum using Wolfram Alpha. Clearly it looks related to the geometric series solution, but I am...

I have a data set {X(t) = (x(t), y(t))}_t=1,...,N and I'm interested in modelling the changes from t to t+1, using some metric d(X(t),X(t+1)) The issue is that x(t) has some dependence on y(t), and I'd like to account for this: if there is a large change in y(t) we expect there to be a...

Homework Statement Let B be a non-empty set, and supose that {Sa : a\inB} is an B- indexed family of subsets of a set S. Then we have, (\cup a\in B Sa)c = \bigcapa\in B Sac. Homework Equations The Attempt at a Solution I tried to show that the two were both subsets of each...

For All p in Natural Number, Is \exists n , n > 1, \sum^{n}_{k=1} k^p = C^2 where C is arbitary natural number (not constant) ??

We know the the generalized momentum is P=mu + qA Can someone explain to me, what's the physical meaning of the quantity 'qA'? The particle's momentum that we measure is just 'mu', right?

Homework Statement For each linear operator T, find a basis for each generalized eigenspace of T consisting of a union of disjoint cycles of generalized eigenvectors. The find a Jordan canonical form J of T. a) T is the linear operator on P2(R) defined by T(f(x)) = 2f(x) - f '(x)Homework...

WCFSGS' Version: Generalized Second Law of Thermodynamics We have known that there has been some generalization to the second law of thermodynamics. We like to present here the Version of WCFSGS about this generalization. At this moment, we are not quite sure if our version is different from...

So I'm working on the proof of the generalized uncertainty principle and there is a step that I'm not fully understanding. There is a line were it says that for any complex number we can write the inequality as [Re(z)]^2 + [Im(z)]^2 >/ [Im(z)]^2. why are we able to get rid of the real part on...

Homework Statement Let have the problem to find the complex generalized cirlce of radius r Homework Equations |z-c|^2 = r^2 The Attempt at a Solution hvor r is the radius and c the center.. by expanding the above z\overline{z} - z\overline{c} - \overline{z}c +...