Functional Definition and 380 Threads

Homework Statement find the functional groups in the following compound: Homework Equations C8H16O4The Attempt at a Solution I know there is an ether but there is also something else. What is it? I have tried to find a group that is in the compound, but I have had no luck. It seems to be only...

If $f$ is a Real valued function on the set of real no. such that for any real $a$ and $b$ and $f(af(b)) = ab$. Then $f(2012) = $

I just finished my undergrad in Chem Eng and am very interested in energy field ( I was thinking of doing something with battery and storage systems as it is closest to my field). I was recently offered a masters in MSE for possibly working on battery materials using Density Functional Theory...

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Homework Statement A function from R-->R is differentiable and follows f( (x+y)/3 ) =( 2 + f(x) + f(y) ) / 3 Derivative of f(x) at x=2 is 2 Find the range of f ( |x| ) Homework Equations The Attempt at a Solution Well the questions asks me the range of f( |x| ). But i don't...

Hey all, I have been reading up on Green Functions and I stumbled upon the term "linear functional". I know the properties of the linear operator, but i can't really grasp what a functional does. In my notes it says that it indicates a linear function whose domain is a function space, and...

Hi, I have a question about a functional derivative. When determining the condition that the functional derivative have a stationary value of 0, do I use a partial derivative or a regular derivative? I would really appreciate the help. Thank you! David

Homework Statement i am doing project on gold nanoparticles i was reading a per regarding synthesis of gold nanoparticles. then i came across a statement "GNPs bearing functional moieties, which are anchored with thiol-linkers, in their monolayers" i didn't understand the word functional...

I've been watching Sidney Coleman's QFT lectures (http://www.physics.harvard.edu/about/Phys253.html). I've gotten up to his discussion of functional integration, and I have some questions. He starts out by discussing a finite-dimensional integral of a Gaussian function: \int{\frac{d^n...

I have a question regarding functional differentiablility. I understand that Frechet differntiability of a functional T with respect to a norm \rho_1 implies Hadamard differentiability of the functional T with respect to the same norm. However, it is no surprise that there would be cases...

Homework Statement I'm reading Kittel&kroemer's Thermal physics. How can I know Entropy's functional dependency? Author assume entropy's functional dependecy without explanations and derive some equaltities. So I can't follow it. N the number of particles. U Energy of the system. V. volume of...

As a basic exercise in C++ functional programming, I wrote the following code: #include <iostream> #include <string> using namespace std; template <class T> void Wib (T& a,T& b) { T temp = a; a = b; b = temp; }; int main() { string A = "World!"; string B = "Hello, "; Wib <string>...

To be specific, with total derivative I mean the linear map that best approximates a given function f at a given point. For f:ℝ\toℝ we have D(f,x_0):ℝ\toℝ, i.e. D(f,x_0)(h) \in ℝ. Often it is also denoted as just \delta f. Now in physics, in particular in the area of the Lagrangian, I find...

I know function is just a subset of functional but physical example helps to understand this difference.."Any physical situation" thanks in advance

Suppose $f(x)$ is continuous for all $x$ and $f(a+b)=f(a)+f(b)$ for all $a$ and $b$. Prove that $f(x)=Cx$, where $C=f(1)$. I have shown that $f(x)=Cx$ for all rational numbers. How do I use the continuity of $f$ to show it is true for all $x$?

I'm looking for a rigorous introduction to functional analysis in the style of Apostol. I've looked at Introductory Functional Analysis with Applications by Kreyszig, but I find it slightly too conversational. I know that Rudin has a Functional Analysis book, but it seems to be out of print...

Homework Statement S[y] = \int21dx ln(1 + xny'), y(1) = 1, y(2) = 21-n where n > 1 is a constant integer, and y is a continuously differentiable function for 1 ≤ x ≤ 2. Let h be a continuously differentiable function for 1 ≤ x ≤ 2 and ε a constant. Let ∆ = S[y + εh] − S[y]. Show that...

Homework Statement Let E be a dense linear subspace of a normed vector space X, and let Y be a Banach space. Suppose T0 \in £(E, Y) is a bounded linear operator from E to Y. Show that T0 can be extended to T\in £(E, Y) (by continuity) without increasing its norm. The Attempt at a Solution...

Is an open normed subspace Y (subset of X) primarily defined as a set {y in X : Norm(y) < r}? Where r is some real (positive) number. I know the open ball definitions and such... but it seems like this definition is saying, an open normed space, is essentially an open ball which satisfies...

Alright, so I feel kind of dumb...but: I have been working on some QFT material, specifically derivation of Feynman rules for some more simple models ( \phi^{4} for example), and I have been seriously failing with functional derivatives. Every time I try to use the definition I mess up...

Hello! I was wondering how I could find the following derivatives from the given function using Jacobian determinants. f(u,v) = 0 u = lx + my + nz v = x^{2} + y^{2} + z^{2} \frac{∂z}{∂x} = ? (I believe y is constant, but the problem does not specify) \frac{∂z}{∂y} = ? (I...

Folks Are there any introductory functional analysis books which show calculus examples to illustrate the different axioms? thanks

Folks, Is there a difference between l subscript infinity and l superscript infinity. I believe the latter is the space if bounded sequences? thanks

Hi,all Is there someone's research related to DFT? I'm an undergraduate trying to get into it. I hope I could get some help hear if I have any question about that! Thanks a lot,Euphemia

"Applied Functional Analysis" by Zeidler In my book, "Applied Functional Analysis" by Zeidler, there's a question in the first chapter which, unless I got my concept of density wrong, I can't seem to see true : Let X=C[a,b] be the space of continuous functions on [a,b] with maximum norm. Then...

Homework Statement J(f)=\int 2xf−f′2+3f2f′dx f(0)=0,f(1)=−1. Homework Equations Ff-\frac{d}{dx}Ff'=0 The Attempt at a Solution Ff=2x+6f f'' Ff'=-2f' + 6f2 Plugging in, I get: 2x+6f f''- [itex]\frac{d}{dx} (-2f' + 6f2) 2x+6f f''-12f f'-2f''=0 Which doesn't look...

Homework Statement Let V be a finite dimensional vector field over F. Let T:V→V Let c be a scalar and suppose there is v in V such that T(v)=cv, then show there exists a non-zero linear functional f on V such that Tt(f)=cf. Tt denotes T transpose. Homework Equations Tt(f)=f°T...

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 =...

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 [FONT="Times New Roman"]N is an (n-1)-dimensional subspace of an n-dimensional vector space V, show that [FONT="Times New Roman"]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...

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...