Functional Definition and 416 Threads

Folks, I am starting a module in functional analysis undergrad level. I have been suggested introductory functional analysis by Kreyszig, but in instead of buying another expensive book is there a good online source like a pdf on in this topic that I could avail of? Any help will be...

I was reading on wikipedia about "functional powers", but I can't seem to find anything on it outside of this one section. I was wondering if there's any way to show anything for f^n(x). This is more of a general plea for more information on the topic than a specific question. Oh and here's...

Homework Statement the sulphur atoms in the self-assembled monolayers are ~ 4.99Å apart, and that they form a hexagonal close-pack structure, estimate the number of functional molecules/cm2 of the substrate Homework Equations 1 angstrom = 1.0 × 10^{-10} metres Area of hexagon =...

Homework Statement Hi, I'm working on research and I hit a roadblock with something that should be very simple but I can't solve it because it gets so messy. If anyone can let me know how to do this, it would be greatly appreciated. I have a functional T: T = \int_{\lambda_{1}}^{\lambda_{2}}...

Hello everyone :) I'm reading the book QFT - L. H. Ryder, and I don't understand clearly what are the generating functional Z[J] and vacuum-to-vacuum boundary conditions? Help me, please >"<

I have n elements. Say n = 3. Suppose I have an association matrix that gives the relationship between each element \begin{array}{cc} 0 & 0 & D3\\ D1 & 0 & 0\\ 0 & D2 & 0 \end{array} I have a function in mind now, I want to operate and the physical variables representing my three...

A functional analysis' problem I hope this is the right place to submit this post. Homework Statement Let A be a symmetric operator, A\supseteq B and \mathcal{R}_{A+\imath I}=\mathcal{R}_{B+\imath I} (where \mathcal{R} means the range of the operator). Show that A=B. 2. The attempt at a...

Hi, in their book ''Density-Functional Theory of Atoms and Molecules'' Parr and Yang state in Appendix A, Formula (A.33) If F ist a functional that depends on a parameter \lambda, that is F[f(x,\lambda)] then: \frac{\partial F}{\partial \lambda} = \int \frac{\delta F}{\delta f(x)}...

Hi everyone, I have been studying "Optimization by Vector Space Methods", written by David Luenberger and I am stuck in an obvious point at first glance. My problem is in page 105, where the norm of a linear functional is expressed in alternative ways. The definition for the norm of a linear...

Hello everybody here, I'm taking Functional Analysis this term, and the textbook is : "An Introduction to Hilbert Space, Cambridge, 1988" by N. Young. Unfortunately, we have to solve most of the book's problems. So, does anyone has some of them ? I found a list of solved problems on...

Hi All, is there anybody to give me some help on how I can calculate the Euler Lagrangian equation associated with variation of a given functional? I am new with these concepts and have no clue about the procedure. thanks a lot

Concerning Hardy-Littlewood approximate functional equation for the \zeta function \zeta(s) = \sum_{n\leq x}\frac{1}{n^s} \ + \ \chi(s) \ \sum_{n\leq y}\frac{1}{n^{1-s}} \ + \ O(x^{-\sigma}+ \ |t|^{\frac{1}{2}-\sigma}y^{\sigma - 1}) does somebody know of any similar result for the...

Please teach me this: In section 11.4 chapter 11 of QTF theory book of Peskin&Schroeder,computing Effective Action,they calculate a functional integral of product of two exponentials of ''exact'' Lagrangian and ''counterterm'' Lagrangian with the same variable of integral(value of field).I do...

Homework Statement The bilinear form are symmetric, i.e. a(u,v) = a(v,u) for all u and v. Find the bilinear form and the linear functional for the problem -\Deltau + b . \nablau + cu = f(x) in \Omega u = 0 on the boundary. Is this bilinear form for this problem symmteric? Is it coersive...

Given that N is an (n-1)-dimensional subspace of an n-dimensional vector space V, show that N is the null space of a linear functional. My thoughts: suppose \alpha_i(1\leq i \leq n-1) is the basis of N, the linear functional in question has to satisfy f(\alpha_i)=0. Am I correct? Thanks

Hi- This lab question has me stumped... Homework Statement Derive an equation describing the effect of the control voltage on the pules width for a 555 pulse width modulation circuit (monostable circuit). Homework Equations width = 1.1*R*C The Attempt at a Solution I took...

My professor tried to show the following in lecture the other day: If T is a linear operator on a Hilbert space and (Tz,z) is real for every z in H, then T is bounded and self-adjoint. Below, I use (*,*) to indicate the Hilbert space inner product. He told us to use the identity (which I've...

I am working on a functional and I need to find its minimum, the conventional procedure is to use Lagrange-Euler method and find the minimum state of the function, but if I need to impose a constraint to the function, I don't know what I need to do J=int(F(t, f(t), a, b)) minimize(f) and...

Hello, I'm reading through John Conway's A Course in Functional Analysis and I'm having trouble understanding example 1.5 on page 168 (2nd edition): Let (X, \Omega, \mu) and M_\phi : L^p(\mu) \to L^p(\mu) be as in Example III.2.2 (i.e., sigma-finite measure space and M_\phi f = \phi f is a...

Please teach me this: When calculating something with Grassman numbers without changing order of the numbers,then there are nothing different from ordinary numbers.So I think it would be contrary if we define the complex conjugation of a product of two Grassman numbers to reverse the order of...

it could be by the power of Newton's laws and energy conservation principles, one can sort out the equation of a wave classically... y=a exp(-iw(t-x/v) ) ; ----1. in the quantum domains where classical situations are ruled out, how is it apt to say or on which basis can we say that...

Can someone please explain why the following three definitions for the norm of a bounded linear functional are equivalent? \| f \| = \sup_{0 < \|x\| < 1} \frac{|f(x)|}{\| x \|}, and \| f \| = \sup_{0 < \| x \| \leq 1} \frac{|f(x)|}{\| x \|}, and \| f \| = \sup_{\| x \| = 1}...

Im trying to calculate the ground state energy of Helium using a density functional theory approach combined with the local density approximation. So far I have set up universal functionals and I mainly need help with the actual algorithm the evaluation of the Hartree energy functional.

Hey All, Here's a stupid and probably ridiculously easy question, but I want to make sure that I have it right. Let G be a Lie group with Lie algebra \mathfrak g . Assume that H \in \mathfrak g and \phi \in \mathfrak g^* the algebraic dual. Assume that X(t) is an integral curve...

Homework Statement Given http://www.mathhelpforum.com/math-help/attachments/f33/20928d1298610998-function-msp281219ebge8he857gc6900005ba9285dff0f5h79.gif , find the values of ''a'' for which the value of the function f(x) <= 25/2. The answer is a<= 1/2. Homework Equations The Attempt at a...

Homework Statement I want to understand the proof of proposition 7.1 in Conway. The theorem says that if \{P_i|i\in I\} is a family of projection operators, and P_i is orthogonal to P_j when i\neq j, then for any x in a Hilbert space H, \sum_{i\in I}P_ix=Px where P is the projection...

Hi! I am doing some numerical calculations recently. I need to calculate the functional derivative. eg. functional : n(\rho)=\int dr'r'\rho(r')f(r,r') it need to calculate: \frac{\delta n(r)}{\delta\rho(r')} I think the...

Is there a way to "remove" functional groups? I see a lot of pages online that show how you can change them, but not how to completely remove one from a molecule. Is it even possible?

What is meant by functional derivative? Thanks in well advance.

I cannot work out the following functional derivative: \frac{\delta}{\delta g_{\mu\nu}} \int d^4 x f^a_{\phantom{a}b} \nabla_a h^b Where f is a tensor density f= \sqrt{\det g} \tilde{f} ( \tilde{f} is an ordinary tensor) and should be consider as independent of g. In my opinion this is not...

Hello guys. This is my first post at physics forums, so please be gentle :) I am trying to understand functionals, so I am solving as many exercises from these lecture notes that I downloaded. Homework Statement Let f:[a,b]\rightarrow\mathbb{R}^k be a continuous function and define...

"Functional gages cannot be used to inspect features specified at LMC." What does that statement mean? I am going through "GD&T for Mechanical Design" by Cogorno (McGraw-Hill) and it makes that statement on page 24.

Homework Statement Find the curve y(x) that passes through the endpoints (0,0) and (1,1) and minimizes the functional I[y] = integral(y'2 - y2,x,0,1). Homework Equations Principally Euler's equation. The Attempt at a Solution We choose f{y,y';x} = y'2 - y2. Our partial...

Roger Penrose, in The Road to Reality, introduces the idea of what he calls a "functional derivative", "denoted by using \delta in place of \partial; "Carrying out a functional derivative in practice is essentially just applying the same rules as for ordinary calculus" (Vintage 2005, p. 487). He...

Hi, I want to start reading about Density Functional Theory and get through some of its approaches. I have a vey weak back ground of solid state physics. Please guide me what is the best resource to start reading. Regards

Let V be a finite-dimensional vector space over the field F and let T be a linear operator on V. Let c be a scalar and suppose there is a non-zero vector \alpha in V such that t \alpha = c \alpha. Prove that there is a non-zero linear functional f on V such that T^{t}f=cf, where T^{t}f=f\circ T...

How would I find the amount of energy that is stored in a particular functional group? I know things like Azide, Nitro, Alkynyl, Cyanides, etc. would all store a lot of energy.

My background is in physics, not pure mathematics, so please try to explain in ways that we lay-people could understand ;) I'm brushing up on my calculus of variations--specifically Hamilton's principle--in which it is stated that the integrand is a 'functional,' not a 'function.' I've read...

Hi all, Long time stalker, first time poster. I've finally got stumped by something not already answered (as far as I can tell) around here. I'm trying to make sense of double functional derivatives: specifically, I would like to understand expressions like \int dx \frac{\delta^2}{\delta...

In Random Operators in Fixed point theory of functional analysis, Is there any relation between the saparable space and measurable functions?,, what are the random operators?

For my current research, I need to prove the following: \int_0^1 \frac{dC(q(x) + k'(q'(x) - q(x)))}{dk'}\,dk' = \int_0^1 \int_L^U p(q(x) + k(q'(x) - q(x)))(q'(x)-q(x)) dx dk where C(q(x)) = \int_0^1 \int_L^U p(kq(x)) q(x)\,dx\,dk Here's what I've tried using the definition of functional...

Can anyone tell me the Engineering applications of Functional analysis with a real world example, If possible? Thanks in advance.

Hey, I am working with the equation y=(x+10)/(x+1), and have calculating the iterations of the sequence s_(n+1)=(s_n + 10)/(s_n + 1). I find that whatever value of s(1) is chosen (the initial value) the sequence converges to root 10. However I am now trying to prove why this happens, and...

Are there any theorems concerning this expression \frac{1}{z}f\left(\frac{1}{z}\right). I appreciate posts of any theorems you can think of.

Hi all, I've been studying the path-integral quantisation of gauge theories in Zee III.4. My understanding is roughly as follows: that one can think of the differential operator in the quadratic tems in the lagrangian as a linear operator between infinite dimensional spaces (morally...

If X is Banach space and F:X \rightarrow X is a linear and bounded map and that F^n(x)\rightarrow0 pointwise .. How can I show that it converges to zero uniformly also? Thanks

I have the following relation: W_{\varepsilon}[J] = \mathrm{exp} \left[ - \varepsilon \int \mathrm{d} x \, \left( \dfrac{\delta}{\delta J(x)} \right)^{n} \right] \mathrm{exp}(W[J]) where W is a functional of J. Now I read in a textbook that it follows W_{\varepsilon}[J] = W[J] -...

In the literature (Ryder, path-integrals) I have found the following relation for the functional derivative with respect to a real scalar field \phi(x) : i \dfrac{\delta}{\delta \phi(x)} e^{-i \int \mathrm{d}^{4} x \frac{1}{2} \phi(x) ( \square + m^2 ) \phi(x)} = ( \square + m^2 ) \phi(x)...

We know that a linear operator T:X\rightarrowY between two Banach Spaces X and Y is an open mapping if T is surjective. Here open mapping means that T sends open subsets of X to open subsets of Y. Prove that if T is an open mapping between two Banach Spaces then it is not necessarily a closed...

Does anybody know of any good resources for this? Specifically for real analysis, I'm looking for something that covers calculus on manifolds, differential forms, Lebesgue integration, etc. and for functional analysis: metric spaces, Banach spaces, Hilbert spaces, Fourier series, etc. Thanks!