Functional Definition and 416 Threads

In the space of Riemannian metrics Riem(M), over a compact 3-manifold without boundary M, we have a pointwise (which means here "for each metric g") inner product, defined, for metric velocities k^1_{ab},k^2_{cd} (which are just symmetric two-covariant tensors over M)...

In the space of Riemannian metrics Riem(M), over a compact 3-manifold without boundary M, we have a pointwise (which means here "for each metric g") inner product, defined, for metric velocities k^1_{ab},k^2_{cd} (which are just symmetric two-covariant tensors over M)...

This is supposedly the chain rule with functional derivative: \frac{\delta F}{\delta\psi(x)} = \int dy\; \frac{\delta F}{\delta\phi(y)}\frac{\delta\phi(y)}{\delta\psi(x)} I have difficulty understanding what everything in this identity means. The functional derivative is usually a derivative...

Show that the extermals of any functional of the form integ (a->b) F(x,y') dx have no conjugate points. Not sure how to start this question, any help would be appreciated

Hello I need help with an analysis proof and I was hoping someone might help me with it. The question is: Let (X,d) be a metric space and say A is a subset of X. If x is an accumulation point of A, prove that every r-neighbourhood of x actually contains an infinite number of distinct...

Hello Would anyone out there be able to help me with a problem I'm having? I have to prove that a function is open and that another is closed. The question is: Consider C [0,1] with the sup metric. Let f:[0,1]→R be the function given by f(x)=x²+2 Let A={g Є C[0,1]: d(g,f) > 3}. Prove...

Hello Would anyone out there be able to help me with a problem I'm having? I have to prove that a function is open and that another is closed. The question is: Consider C [0,1] with the sup metric. Let f:[0,1]→R be the function given by f(x)=x²+2 Let A={g Є C[0,1]: d(g,f) > 3}. Prove...

Homework Statement I have to show that the functional C_n on the space of polynomials on the interval [0,1], that takes the n'th coefficient ie C_n\left( \sum_{j=0}^m a_j t^j \right) = a_n is discontinuous with respect to the supremum norm \|p\|_{\infty} = \sup_{t\in[0,1]}|p(t)| ...

Hi! I was thinking about taking an introductory course in Functional analysis the commming spring, and was wondering if you more experienced guys can tell me if this is a good complement to understand theoretical quantum physics better? Cheers

first of all how do you write in latex

given the functional integral with 'g' small coupling constant \int \mathcal D [\phi]exp(iS_{0}[\phi]+\int d^{4}x \phi ^{k}) so k >2 then could we use a similar 'Functional determinant approach' to this Feynman integral ?? in the sense that the integral above will be equal to...

Is it correct to think, that with a scalar complex Klein-Gordon field the wave function \Psi:\mathbb{R}^3\to\mathbb{C} of one particle QM is replaced with an analogous wave functional \Psi:\mathbb{C}^{\mathbb{R}^3}\to\mathbb{C}? Most of the introduction to the QFT don't explain anything like...

HOw can you compute a Gaussian functional integral? i mean integral of the type e^{-iS_{0}[\phi]+i(J,\phi)} if J=0 then i believe that we can describe the Functional integral as \frac{c}{(Det(a\partial +b)} a,b,c constant so Det(a\partial +b)}= exp^{-\zeta '(0)} \zeta (s) =...

I say what is a functional determinant ?? for example Det( \partial ^{2} + m) is this some kind of Functional determinant? then i also believe (althouhg it diverges ) that Det( \partial ^{2} + m)= \lambda_{1}\lambda_{2}\lambda_{3}\lambda_{4}... (the determinant of a Matrix is the...

I was studying my brother's old notes to prepare me for my upcoming AP Biology class. I read over six main functional groups (hydroxyl, carbonyl, carboxyl, amino, sulfhydrl, phosphate) and subsequently, there were follow-up questions. One of them asked to draw a molecule with all of the six...

I've been looking for some decent info on functional space but could not find anything. Googling gives lots of defenitions, but no explanations as such. Basically I'd like to understand why a function can be decomposed into other function, e.g. understand a meaning of inner product with...

Let be a set of LInear functionals U_{n}[f] n=1,2,3,4,... so for every n U_{n}[\lambda f+ \mu g]=\lambda U_{n}[f]+\mu U[g] (linearity) the question is if we can define the product of 2 linear functionals so U_{i}U_{j}[f] makes sense.

I'm intrigued by the fact that apparently no general theory of functional integration has been developed - something along the lines of Riemann, Cauchy, and Lebesgue. Feynman developed an approach to evaluating functional integrals for paths in spacetime - but I'm wondering whether it is...

I have a problem with this simple (?) derivation u(f(t),t) = \frac{\partial }{\partial f(t)} \int_0^T \ g(f(t),f(t'),t,t') \ dt'

Hmm, I've been working with functional derivatives lately, and some things aren't particularly clear. I took the definition Wikipedia gives, but since I know little of distribution theory I don't fully get it all (I just read the bracket thing as a function inner product :)). Anyway, I tried...

Homework Statement Consider an iteration function of the form F(x) = x + f(x)g(x), where f(r) = 0 and f'(r) != 0. Find the precise conditions on the function g so that the method of functional iteration will converge cubically to r if started near r. Homework Equations I really don't know...

I need to study all of these names for my orgo II exam next week. Can someone help me to find the general trend or functional group name for the following attached compounds? I have tried some of them let me know if I made any mistake on these two sheets. ie. #1 gem-diol (hydrate)...

If to calculate the propagator K(x,x') (vaccuum)for a theory so: (i\hbar \frac{\partial}{\partial t}\Psi - H\Psi )K(x,x')=\delta (x-x') (1) we use the functional integral approach: K(x,x')=<0|e^{iS[x]/\hbar }|0> my question is, let's suppose we use the semiclassical WKB approach...

I am writing a solution for the following problem, I hope someone can correct it, because I am not sure what I am missing. Q. V is a finite dim. vsp over K, and W is a subspace of V. Let f be a linear functional on W. Show that there exists a linear functional g on V s. t. g(w)=f(w). Ans...

Suppose that we have a compact manifold \mathcal{M} with a positive definite metric g_{ij}. The volume of the manifold is then V = \int_\mathcal{M} d^3x \sqrt{g(x)}, where x^i are coordinates on \mathcal{M} and \sqrt{g(x)} is the square root of the determinant of the metric. Suppose now that...

Does there exist a chain rule for functional derivatives? For example, in ordinary univariate calculus, if we have some function y=y(x) then the chain rule tells us (loosely) that \frac{d}{dy} = \frac{dx}{dy}\frac{d}{dx}. Now suppose that we have a functional F[f;x) of some function...

suppose Fi (a1, a2, ... an) , 0 < i <= k. a1, ..., an are reals Then the Frechet derivative DF is a k x n matrix. If rank(DF) = k , does it still suggest functional dependence amonst Fi 's ? Also, when rank(DF) < n (number of independent variables) , what does it signify ? The theorem I had...

So far all I know is that a functional is a function that has a set of functions as its domain. So what does that mean? I have a functional that looks like dy/dx = a bunch of constants. What I'd like to know is how to take that and plot it. Can this be done?

im given: u=arcsin(x)+arccos(y) v=xsqrt(1-y^2)+ysqrt(1-x^2) and i need to find the functional connection between u and v. i know that: v'_x/u'_x=dv/du and i have got: dv/du=sqrt(1-x^2)sqrt(1-y^2)-xy now i need to show the rhs as a function of v or u, obviously of v should be much...

Homework Statement For f\in C([0,1]), let N(f)=\int_0^1(f(t))^2dt. Show that N\in C^{\infty} and calculate the first derivative.Homework Equations Can I use Leibniz's integral rule for this?The Attempt at a Solution If I just blindly plug in the formula, I get dN/df=\int_0^12f(t)dt

I am thinking about taking a class on functional analysis. I am eventually planning on doing derivatives trading as a career. Is this class worth taking or should I try to find something more applied. I guess I am saying that I don't see how applied functional analysis is.

I'd like to be as familiar with functional methods as with calculus. Otherwise I always fell not to grasp QFT comprehensively. Any help is much appreciated!

Hi, i am looking for papers, books, etc, related with the Density functional theory, and Kohn Sham equations, i appreciate any help. Thanks. :rolleyes:

Strangely a good introductory internet resource on this subject doesn't seem to be readily available. Where can I look for information on this subject? --I am mainly interested in the concepts behind functional multi-threading rather than the syntax for any specific language.

Hi. Can anyone tell me how to solve the path integral \int D F \exp \left\{ - \frac{1}{2} \int_{t'}^{t} d \tau \int_{t'}^{\tau} ds F(\tau) A^{-1}(\tau - s) F(s) + i \int_{t'}^{t} d\tau F(\tau) \xi(\tau) \right\} In case my Latex doesn't work the integral is over all possible forces F over...

Although this problem is meant to be easy I can't quite work it out. Let U(A) denote the set of unitary elements of a C*-algebra A. I've already shown that if u is unitary in A then the spectrum of u: \sigma(u) \subset \mathbb{T} = \{z\in\mathbb{C}\,:\,|z|=1\} which was easy. Now...

A linear functional is a function g:V to F where V is a vector space over a field F such that if u and v are elements of V and a is an element of F, then g(u+v) = g(u) + g(v) and g(au) = ag(u) Let G be the space of all linear functionals on V. Then if \oplus_{1} and \otimes_{1} are...

I have a commutative Banach algebra A with identity 1. If A contains an element e such that e^2 = e and e is neither 0 nor 1 (I think this also means to say that it contains a non-trivial idempotent), then the maximal ideal space of A is disconnected. Currently I am trying to show this but I...

I found this functional equation for the Riemann zeta function in Table of Higher Functions, 6th ed. by Jahnke, Emde, & Losch on pg. 40: z(z+1)\frac{\zeta (z+2)\zeta (1-z)}{\zeta (z)\zeta (-1-z)}=-4\pi ^2 any suggestions as to how one might consider such an equation, much less derive it...

Let,s suppose we have a functional J and we want to obtain its extremum to obtain certain Physical or Math properties: \delta{J[f(x)]}=0 Yes you will say to me " You can apply Euler-Lagrange Equation to it and generate a Diferential equation to obtain f"..of course is easier saying than...

The (main) functional group in esters are: R_1OOR_2 Where R_1 and R_2 are carbon chain. When I though about it, I've never heard the systematic name of this functional group. What is this functional group called? I've seen esterbonding etc. but that doesn't sound systematic as ex...

Hello everybody I'm given a continuous function f (from the real numbers to the real numbers) which I know obeys the following functional equation: f(x)=f(x^{2}) How can I proof that this function is constant? I started out like this: Looking at a number x in [0,1[ I said to myself that...

I put together two questions : a) suppose there is a point mass with mass M..if it is moving, then from a certain oberver, the total energy is higher, via E=Mc^2...hence, following the generaly relativity qualitatively, where the energy density defines the curvature, the gravitation should be...

let be the next functional differential equation: \delta{F[\phi]}=G[\phi,\partial{\phi}] then its solution would be: F[\phi]=\int{D[\phi]G[\phi,\partial{\phi}] would it be correct?..thanks..

Regarding the Barber of Seville paradox, I am looking for something equivalent that is expressed in functional notation. For example, this is my attempt at a piecewise definition of such a function: For a function f : \mathbb{N} \mapsto \{0,1\} f(n)=\left\{\begin{array}{cc}0,&\mbox{ if...

we know that the functional integrals are important in quantum field theory,but we have the problem that except for the semiclassical approach,they can not be solved anyway..but if we used the formula:. \int{d[\phi]F[\phi]=\sum_{n=1}^{\infty}(-1)^{n}\phi^n{D^{n}F[\phi]} where D is the...

Hi, I am reading chapter 5 of Ryder regarding path integrals and vacuum to vacuum transition amplitudes in presence of source. I follow the math but don't have a clear physical picture. The formula is: Z[J]=\int Dq \: exp ( \frac{i}{h}\int dt(L+hJq+\frac{1}{2}i\epsilon q^2) ) Can...

Im trying to prove the following proposition Let (X,\|\cdot\|_X) and (Y,\|\cdot\|_Y) be normed vector spaces and let T:X \rightarrow Y be a surjective linear map. Then T is an isomorphism if and only if there exist m,M > 0 such that m\|x\|_X \leq \|Tx\|_Y \leq M\|x\|_X \quad \forall \, x...

Why do we treat a scalar field phi and its derivatives as being independent when trying to find a stationary solution for the action? Doesn't that give too general solutions? Where does the restriction that (d_mu phi) is dependent on phi come back in?

what are the requirements of a functional J[y] to exist in the form that its minimum will yield to a differential equation?..i mean let be the functional with condition: J[y]=\int_{a}^{b}dx(p(x)(y`)^{2}+V(x)y^{2}) \int_{a}^{b}y^{2}dx=C with c a constant... then what conditions should...