generalized Definition and 197 Threads

"Given (rn), rn E (0,1), define a generalized Cantor set E by removing the middle r1 fraction of an interval, then remove the middle r2 fraction of the remaining 2 intervals, etc. Start with [0,1]. Take rn=1/5n. Then the material removed at the n-th stage has length < 1/5n, so the total...

"Generalized" Laplace transform Hello, I'm having trouble proving injectivity of what might be called a "generalized" Laplace transform (not the one by Varma). Let f be a rational function and C be a fixed closed contour in the complex plane, (such that C contains not pole of f): The operator...

given the function Z(s)= \prod _{k=0}^{\infty}\zeta (s+k) with \zeta (s) being the Riemann Zeta function the idea is if ALL the roots have real part (i mean Riemann Hypothesis) is correct, then what would happen with the roots of Z(s) ?? what would be the Functional equation relating...

I have a system that ideally creates a real symmetric negative definite matrix. However, due to the implementation of the algorithm and/or finite-precision of floating point, the matrix comes out indefinite. For example, in a 2700 square matrix, four eigenvalues are positive, the rest are...

Homework Statement If \omega is and nth root of unity, define Z[\omega], the set of generalized Gaussian integers to be the set of all complex numbers of the form m_{0}+m_{1}\omega+m_{2}\omega^{2}+...+m_{n-1}\omega^{n-1} where n and m_{i} are integers. Prove that the products of generalized...

http://arxiv.org/pdf/0909.0939 MTd2 spotted this paper back when it came out on 6 September and posted a reminder on another thread suggesting that we should discuss it. We should. It looks like a paper that is both important in the development of LQG and also exceptionally clear and...

Homework Statement The following theorem in geometry suggests a vector identity involving three vectors A, B, and C. Guess the identity and prove that it holds for vectors in Vn. This provides a proof of the theorem by vector methods. "The sum of the squares of the sides of any...

could GR generalized to non-integer dimension?? let us suppose that the dimension of space time is NOT an integer then , could we generalize GR to obtain an expressions of Tensor, Covariant derivatives... in arbitrary dimensions ?? let us say 4.567898.. or similar, i mean GR in non integer...

Homework Statement Suppose there is a square plate, of side a and mass M, whose corners are supported by massless springs, with spring constants K, K, K, and k <= K (the faulty one). The springs are confined so that they stretch and compress vertically, with unperturbed length L. The...

If one has a short exact sequence 0-->A-->B-->C-->0 of finitely generated abelian groups, how does one show that rank(B)=rank(A)+rank(C) ? We have that A embeds in B and C is isomorphic to B/A. The natural thing to try to use I think is the uniqueness of the decomposition of a finitely...

Are generalized coordinates, as used in Legrangian mechanics, just a different name for coordinates on a chart in a manifold? The idea of generalized coordinates never quite "clicked" with me, but after reading a paper today, it seems that they are just an implicit way of working with manifolds...

Hi Everyone, Do there exist any explicit formula for Cos(x_1+x_2+...+x_n) as a sum of products of Sin(x_i) & Cos(x_i)? Or we need to expand using Cos(A+B), Sin(A+B) again & again? If it exists then what is about Sin(x_1+x_2+...+x_n)? [It is understood that there will be 2^(n-1) number of...

If I am to try and derive a set of 3x3 matrices analogous to the Pauli matrices, how would I go about doing this? I want to find the basis for all complex 3x3 matrices (analagous to the 3 Pauli matrices and the identity matrix for all complex 2x2 matrices) to expand a complicated matrix into so...

I stuck at the second "="...i know it goes like this because the formula...i just someone explain to me why it works like that. thank you soooo much!

Determinant formula in monomials -- can it be generalized? I ran across this question in one of the Usenet groups (fr.sci.maths), and after doing a double take and realizing what was actually being asked I realized I don't know the answer, and after searching a bit I haven't turned it up, so I...

So I understand that if an nxn matrix has n distinct eigenvalues that you can diagonalize the matrix into S\LambdaS^{-1}. This is important because then this form has lots of good properties (easy to raise to powers, etc) So when there are not n distinct eigenvalues, you then solve...

Hey, Is the generalized eigenspace invariant under the operator T? Let T be finite dimensional Linear operator on C(complex numbers). My understanding of the Generalized Eigenspace for the eigenvalue y is: "All v in V such that there exists a j>=1, (T-yIdenitity)^j (v) = 0." plus 0. thanks

We all know how the metric of GR is a generalization of the flat Minkowski spacetime metric. But I wonder if the SR metric is generalized from the kinetic energy term of Newtonian physics. There the kinetic energy is (1/2)m*v^2=(1/2)m*dx*dx. If the mass/2 plays the role of the metric, then this...

Hi, I have to show that if the derivate f'(x) of a generalized function f(x) is defined by the sequence f'_n(x) where f(x) is defined f_n(x)[\tex] then \int_{-\infty}^{\infty}f'(x)F(x) dx = - \int_{-\infty}^{\infty}f(x)F'(x) dx I use the limits for generalized functions and...

can anybody help me about the attached problem about generalized Crank-Nicolson scheme. I need MATLAB code. How can i write a tri-diagonal solver for this problem?

I have a couple of questions on what is possible within quantum mechanics, and the physical justifications (if any). My question is a bit subtle and tricky to explain, but I'll try to explain as well as I can. Hopefully someone here can spread a bit of light on this. This problem first puzzled...

Okay I posted a question a few days ago about Luders Rule but didn't get any responses. I've studied this stuff in Hughes (The Structure and Interpretation of Quantum Mechanics) a bit more so I can ask a slightly different question. Hughes says you can create a "generalized probability...

Homework Statement When I use generalized coordinates how do I know that I can add the kinetic contributions from each to get the total kinetic energy? How do I know that you are not "counting the same KE twice"? e.g. if you have a double pendulum how do you know that you can just add the...

does the lagrangian depend upon upon my choice of generalized coordinates

I've been discussing some things with Samalkhaiat over in the conformal field theory tutorial. A part of that conversation (indicated by the new title) was drifting away from CFT matters, so we both thought it was better to move it into the Quantum Physics forum, to minimize pollution of the...

I have the following contour integral form of Wick's theorem (C indicating contraction): C[A(z):BC:(w)]=\frac{1}{2 \pi i} \int _w \frac{dx}{x-w} C[A(z)B(x)]C(w) + B(x)C[A(z)C(w)] Does anybody know how to evaluate contractions like C[:AB:(z)C(w)]?

Is anyone here familiar with the proof (using homology) of the generalized Jordan curve theorem, that a subspace of S^n homeomorphic to S^(n-1) divides it into two components? It can be found on page 169 of Hatcher's algebraic topology book, which can be downloaded from...

Hi all. So I'm a bit confused about finding a basis of generalized eigenvectors for an operator that is not diagonalizable. I have some "steps" in mind, but maybe someone can help me out here: 1) Find the eigenvalues of the matrix/operator 2) Find the eigenspaces corresponding to each...

This is a question that I'm asking myself for my own understanding, not a homework question. I realize that in most derivations of the Euler-Lagrange equations the coordinate system is assumed to be general. However, just to make sure, I want to apply the "brute force" method (as Shankar...

Homework Statement Standard double pendulum setup. A string with mass, connected to a string with a mass, mounted to the ceiling. Given is m1,m2,l1,l2 a) choose a suitable set of coordinates and write a lagrangian function, assuming it swings in a single vertical plane (I did this, using L...

Hi folks, I have to find the generalized solution for the following Ax=y : [1 2 3 4;0 -1 -2 2;0 0 0 1]x=[3;2;1] The rank of A is 3 so there is one nullity so the generalized solution is: X= x+alpha.n (where alpha is a constant , and n represents the nullity) I found the...

Hi. I've been wondering about the following and haven't made much progress on it. (Note that I've also posted this in the relativity section since the ultimate aim of this is to apply it to canonical relativity but since this is essentially a question about tensors I thought I'd put a copy here...

Hello guys, Do you know a good source for the definition of the generalized aerodynamic force matrix? I know that the term ij is the work done by the pressure on the mode j that works with the displacements of mode i integrated over the wing domain. This formulation is derived using the...

Does change of variables generalize to situations other than integration?

[SOLVED] Measurement Issues: POVM, Neumann and generalized What is a POVM measurement? How is it different from Nuemann measurement theory and what would be the most general measurement formulation? Please keep things discrete and not continuous. -KM

I am reading a electrical engineering book about digital signal processing, in the process, those Fourier transform and discrete time Fourier transform of constant and the exponential function lead to the delta function. I understand how to manipulate them formally, but I have serious trouble...

Hello, I have a question concerning the GMM equation specification. Say we partition each day in 7 intraday intervals. We want to estimate the 7 intraday interval moments for a variable Y observed in those 7 intervals over a period of T days meaning we have t = T*7 total observations for...

I have just started to read Goldstein's classical mechanics, and he got me a bit confused: is it correct to think of polar and spherical coordinates as of generalized coordinates? the way I got it, every coordinate system different from the standard cartesian-one is a set of generalized...

This is the problem (t for theta): X ~ Expo(t) = t * e ^ (-t * x), x>0, t >0 0 otherwise Test H0: t <= 1 vs. H1: t >1 using the generalized likelihood ratio test where you have a random sample from X {X1, X2, ... , X50} and the sum of all Xi = 35. Use alpha =...

Hi, In the following document: Generalized Pell Numbers, I've defined what I call "Generalized Pell Numbers". They provide a way for computing: 1+\sqrt[m]{m}. I'd like to know if these numbers are already known or not, and if someone knows about other properties they have or if someone is...

Can anyone here help me derive the generalized Lambert function? I'm working on a solution for an ODE from the homework group which involves this function. This is what I have so far: The W-function is defined as the inverse of the following: f(x)=xe^x=y then...

This paper by Robert A. Nelson derives an exact, explicit coordinate transformation between an inertial frame of reference and a frame of reference having an arbitrary time-dependent, nongravitational acceleration and an arbitrary time-dependent angular velocity...

Dear friends, Well, I’ve got a problem to solve but I’m not going to ask you to do it for me. Instead, what I need is an explanation of what I am doing wrong. The problem is as follows: we have a rod of mass m and length l hanging of a rail (don’t know how to call it). It moves as the...

I have a conjecture that the equation X^2 - 2Y^2 = P has solutions in odd integers if P is a prime of the form 8*N+1. I know of a paper that requires one to find Q such that Q^2 = 2 mod P inorder to solve these equations using continued fractions. To get to first base in proving my conjecture...

Can someone please give me a definition of the generalized triangle inequality?

Hello, I’ve come upon a problem in my transitional mathematics course, wherein I need to prove a generalized formula for the factorization of X^n-Y^n where the first term is (X-Y). I have deduced a formula making use of the summation of X*Y over the range of positive integers ending at n...

In some of my previous posts, I referred to GPDs, a recently developped formalism describing hadronic states. I am afraid that this formalism remains largely ignored partly because few introduction review are only recently available. See for instance : Markus Diehl, Phys.Rept. 388 (2003)...