# operator

1. ### Fuzzy logic question - operator

Hello, can someone explain intuitively why is fuzzy operator "OR" of sets A and B equal to maximum of of these two sets. Thanks.
2. ### annihilation /creation operator algebra

I'm a little confused about a line of the derivation of $<\hat{X^2}>$ of the coherent states of the linear harmonic oscillator. I know this is more physics but I bet one of you will know the answer. We have the annihilation operator $\hat{A}$ and it's associated eigenfunctions $\rho_\alpha$...
3. ### Bounded Linear Operator

Hello everyone I have been trying to get a grasp of linear operators, but I am stuck in a task appearing not to be too complicated, it goes as follows. Let V denote a normed vector space, and let \left \{ v_k \right \}_{k=1}^{n} denote a collection of vectors in V. Equip \mathbb{C}^{n} with...
4. ### Unitary transformation in commutator expansion

I would like a proof or an outline of one for the following identity: exp(A)*B*exp(-A) = B + [A,B] / 1! + [A,[A,B]] / 2! + ... + [A,[A,...,[A,B]...] / n! + .... A and B are linear operators in the case I am considering, but the identity is a formal one, which I believe depends only on the...
5. ### Meaning of determinant of a differential operator

This question comes from one of my QFT texts. I am looking for the meaning of something like det( \square ). I'll give a quick list of the salient definitions the book uses. For anyone that has covered this you could probably just skip to the end. Start with \int e^{-a x^2/2} = \left (...
6. ### Lie Algebra Bracket Operator

I was reading about the bracket operator, and I don't understand how it works. I have the following set: $\displaystyle \bigcup_{ k = 0 }^\infty \mathbb{N}^k$ I then mod out by an equivalence relation which is easier to describe by example than in math terms. These n-tuples represent cycles...
7. ### Proving the relations between mean operator and central difference operator and hD

Hi members, I am trying to prove the following relation but still unable to prove. $\frac{U}{\delta\mu}=\frac{U}{\delta}\left(1+\frac{\delta^2}{4}\right)^{\frac{-1}{2}}=1-\frac{\delta^2}{6}+\frac{\delta^4}{30}-\frac{\delta^6}{140}+0(\delta^8)$ I know $\frac{U}{\delta\mu}=\frac{U}{sinh U}$ and...
8. ### Find matrix elements and expansion for operator in position representation

In 1-D let TL be an operator defined on the position eigenstates |x> such that TL|x>=|x+L>. Find the matrix elements TL(x,x')=<x'|TL|x> and construct an explicit expansion for this operator in the position representation. Show that “in the position representation” <x|T_L|\psi >= \psi (x-L)...
9. ### Spectrum of a nilpotent operator

Hi, I am trying to solve an exercise of the book "Introductory Functional Analysis with Applications" from Kreyszig, section 7.5: What is the spectrum of linear operator T on a Banach space $T:\,X\rightarrow\,X$ that is nilpotent $T^m = 0$? Using the information that $\lambda^m$ is an...
10. ### Fourier Integral Operator

Hi everybody! I'm studying the Fourier integral operators but I can't resolve a pass. I'm considering the following operator: $Au(x)=\frac{1}{{(2\pi h)}^{n'}}\int_{\mathbb{R}_y^m\times\mathbb{R}_{\theta}^{n'}} e^{i\Psi(x,y,\theta)/h}a(x,y,\theta,h)u(y)\, dy\, d\theta$ where \$Au\in C^0...
11. ### Show that an operator that computes the median of a subimage area, S, is nonlinear

For example, the median of the set of values {2,3,8,20,21,25,31} is 20. Show that an operator that computes the median of a subimage area, S, is nonlinear My attempts at a counterexample have all failed For linear operator: H[a1*f1(x,y) + a2*f2(x,y)] = a1*H[f1(x,y)] + a2*H[f2(x,y)] Let a1=2...

Am really not sure how to prove this question, does \nabla . (\frac{1}{r}r) = \frac{\partial}{\partial x} + \frac{\partial}{\partial y} + \frac{\partial}{\partial z} \dot (\frac{1}{x^{2}+y^{2}+z^{2})^{1/2} ( x+y+z) [/tex]
13. ### properites of curl operator proof

suppose F and G are vector functions of (x,y,z) and \alpha , \beta are scalar functions of (x,y,z) then prove 1) \nabla \times (\alpha F) = \nabla \alpha \times F+ \alpha \nabla \times F
14. ### Operator well-defined

Hello, I just can't see why the following operator is well-defined: Let x \in \ell^{2}(\mathbb{N}), a \in \ell^{1}(\mathbb{N}) . Define the operator A so that Ax = (\sum_{k=1}^{\infty}a_{k+n-1}x_{k})_{n \in \mathbb{N}}. Also, assume that a_n is monotonically decreasing. Can anybody show me why...
15. ### Use of # symbol as a connected sum operator

I have just completed a post in the Differential Geometry Forum and there is some problem with the Latex The post is Homology of Connected Sum of Two Projective Planes, P^2 # P^2 and I am using the usual symbol for connected sum, namely # I have tried to produce the symbol # by using \#...
16. ### Another i operator

Here's the problem, (1-i)(1-i)^2 so I worked it like this, (1-i)(1^2-i^2)= (1-i)(1-(-1))= (1-i)(2)= (2i) Does that look correct, or should I have started it by working the square in the second set of parentheses like this (1-i)(1-i)(1-i)
17. ### i operator help

I think I should make sure I have this right so I don't screw up the rest of my homework. Here's what I've got, (j3)(-j7)(-j)(j)=(j^2)(-j^2)(21)=(-1)(1)(21)=-21 So, am I right or where did I go wrong. I think it's whether or not I signed that right. Thanks for the help, Jarod.
18. ### Show this defines a unitary operator

I need help getting started with this question: Show that (U_Af)(x)=f(A^{-1}x) \mid det(A) \mid^{-1/2} defines a unitary operator U_A in L^2(R^n,d\lambda). A is a n by n invertible matrix with real entries. Any hints will be greatly appreciated.
19. ### Linear Operator Question

I'm working on a practice final exam, and I'm stuck on a problem: Let T: \mathcal{P}_2 \to \mathcal{P}_2 be a linear operator defined by T(f) = f' + f''. (\mathcal{P}_2 is the set of all polynomials of order \leq 2). (Note that T is corresponding to a matrix) (1) Find the matrix A. For this...
20. ### Compactness of the operator

Hello there, I'd like to ask for help with this exercise: Let X=L^2(\mathbb{R}) and \varphi : \mathbb{R}\rightarrow\mathbb{R} continuous function, for which there holds \lim_{x\to\pm\infty}f(x)=0 and \exists x\in \mathbb{R}\,:\, \varphi (x)\neq 0 . In addition let be A:X\rightarrow X linear...