Functional Definition and 416 Threads

I'm going to be applying to grad schools next year (I have an undergrad degree in math and phyisics), and I have narrowed down my areas of interest to two fields: functional analysis and it's involvement in QFT; and computational/theoretical neuroscience. I find pure math more enjoyable, but I'm...

Hi PF, I am currently trying to teach myself the rudiments of differential forms, in particular their application to physics, and there's something I'd like to ask. It seems like diff forms can be used to express all kinds of physics, but the area I haven't been able to figure out is stuff...

Hello, could you explain me what's the right way to solve these equations. I've never solved it before. f(x+y)+f(x-y)=2f(x)f(y)\,\;\;\forall x,y\in\mathbb{R} f(x)+\left(x+\frac{1}{2}\right)f(1-x)=1\,\;\;\forall x\in\mathbb{R} thank you...

Homework Statement Hi all, i have to identify 5 samples (1,2,3 were solids, 4,5 were liquids) by classifying them as 1) Aliphatic or aromatic and 2) Carboxylic acid, amine (primary, secondary, tertiary) or ammonium carboxylate We did a burn test on the solids, tested solubility in water...

Hello, Is there any place I can find the equation for the Taylor expansion of a functional around a function ?? Particularly, I want something like: f[x(t)] = f[\hat{x}(t)] + (f[\hat{x}(t)] - f[x(t)] \frac{\delta f}{\delta x(t)}|_{x(t)=\hat{x}(t)} + \frac{(f[\hat{x}(t)] -...

Where can I get a very basic introduction to the current research directions in functional analysis? I have done a basic course in it. Also I am interested in knowing about applications of Ramsey theory to functional analysis. Thanks.

Homework Statement Is the solution correct Homework Equations The Attempt at a Solution all are in the file

Homework Statement What functional groups are present based on the compound's names? A. Methyl Hydroxybenzoate B. 2-Hydroxypropanoic acidHomework Equations The Attempt at a Solution We've learned about the basic Hydrocarbon derivatives in class, but only dealing with problems like...

I'm reading Quantum Field Theory Of Point Particles And Strings, by Brian Hatfield, chapter 9 called Functional Calculus. But he seems to assume some famiality with the subject. I'm intriqued by his notation. He uses notation for functional derivatives almost as if it were ordinary derivatives...

Homework Statement Maximize the functional \int_{-1}^1 x^3 g(x), where g is subject to the following conditions: \int^1_{-1} g(x)dx = \int^1_{-1} x g(x)dx = \int^1_{-1} x^2 g(x)dx = 0 and \int^1_{-1} |g(x)|^2 dx = 1. Homework Equations In the previous part of the problem, I computed...

Hi everyone, I'm working through Section 9.2 (Functional Quantization of Scalar Fields) from Peskin and Schroeder. I have trouble understanding the absence of a term in equation 9.41 which I get but the authors do not. Define \phi_i \equiv \phi(x_i), J_{x} \equiv J(x), D_{xi} \equiv...

Hi everyone, I'm reading section 9.2 of Peskin and Schroeder, and have trouble understanding the origin of a term in the transition from equation 9.26 to 9.27. Specifically, equation 9.26 is \frac{1}{V^2}\sum_{m,l}e^{-(k_m\cdot x_1 + k_l\cdot x_2)}\left(\prod_{k_{n}^{0}>0}\int d \Re...

Homework Statement Suppose we define an absolute value on the rationals to be a function f: Q -> Q satisfying: a(x) \geq 0 for all x in Q and a(x) = 0 \Leftrightarrow x = 0 a(xy) = a(x)a(y) for all x,y in Q a(x + y) \leq a(x) + a(y) for all x,y in Q Determine all such functions and prove they...

Homework Statement Use the method of dimensional analysis to show that the functional dependence in equation (1) can be derived from an observational expression: lambda = k*mu*f^m*T^n. Homework Equations lambda=k\sqrt {{\frac {T}{\mu}}}{f}^{-1} (1) lambda = k*mu*f^m*T^n \mu={\frac...

Hi, I am trying to minimize: \int_0^\infty{\exp(-t)(t\,f'(t)-f(t))^2\,dt} by choice of f, subject to f(0)=1 and f'(x)>0 for all x. The (real) solution to the Euler-Lagrange differential equation is: f(t)={C_1}t rather unsurprisingly. However, this violates f(0)=1. If...

hi, i'm studying the functional equation of riemann zeta function for Re(s)>1; my book(complex analysis by T. Gamelin) use contour integral in the proof, where the contour is taken on the usual 3 curves (real axis and a small circle C\epsilon around the origin). I'm not able to figure why...

In Rudin's Functional Analysis (in theorem 3.4), he says: "every nonconstant linear functional on X is an open mapping". X is topological vector space. This seems like a strengthening of the open mapping theorem, which requires X to be an F-Space, and that the linear functional to be...

Could any of you recommend a functional analysis textbook? I have looked at "Methods of modern mathematical physics" by Reed&Simon, but they assume a pure-maths BSc background, thus this book is not ideal for me. About my background: I have an Applied Physics BSc and starting a Theoretical...

Homework Statement Show that the range \mathcal{R}(T) of a bounded linear operator T: X \rightarrow Y is not necessarily closed. Hint: Use the linear bounded operator T: l^{\infty} \rightarrow l^{\infty} defined by (\eta_{j}) = T x, \eta_{j} = \xi_{j}/j, x = (\xi_{j}). Homework Equations...

This is from Rudin, Functional Analysis 2.1. Not homework. If X is an infinite-dimensional topological vector space which is the union of countably many finite-dimensional subspaces, prove X is first category in itself. What about this example? Take R^n (standard n-dimensional space of...

Hello all, I have been trying to fill in the gaps in the example of a functional given in chapter 3 of Hartle's book "Gravity" and I am not having much luck. I exhausted wikipedia for help to no avail. Does anyone know of or can provide a good simple example of a functional or just the...

Hi All, I am asked to produce a function such that, literally, increasing the indipendent variable by lambda will produce an increase in the function of a*lambda. I thought about setting up an equation as follows y(lambda*x)=a*lambda*y(x) In general a simple solution of the...

Homework Statement If f'(x)>0 for all real positive x, where f:R+ ---> R and f(x)+(1/x)=f-1(1/(f(x))), f-1(1/(f(x)))>0 for all x>0. Find all the possible values of (i) f(2),(ii) f'(2) and (iii) Limit (x f(x)) as x ----->0 . The Attempt at a Solution Guessing from the last...

I am reading a book (Di Francesco's "CFT", pg 337) in which it is given that if we take the operator that translates the system along some direction (which is a combination of time and space) as 'A', then the partition function is just trace(A). How do we get this?

Homework Statement Suppose a function satisfies the conditions 1. f(x+y) = (f(x)+f(y))/(1+f(x)f(y)) for all real x & y 2. f '(0)=1. 3. -1<f(x)<1 for all real x Show that the function is increasing throughout its domain. Then find the value: Limitx -> Infinity f(x)xThe Attempt at a Solution I...

Homework Statement I am to illustrate a particular theorem by considering a functional f on R^2 defined by f(x)=\alpha_1 \xi_1 + \alpha_2 \xi_2, x=(\xi_1,\xi_2), its linear extensions \bar{f} to R^3 and the corresponding norms. I'm having a couple problems with this problem. For one, I...

is there a functional equation for \sum_{n=0}^{N}n^{k}=Z(N,k) where k and N are real numbers, in case N tends to infinite we could consider the functional equation of Riemann zeta but what happens in the case of N finite ??

See the attachment.

I want to show that: Ai(x)+jAi(jx)+j^2Ai(j^2x)=0, where: Ai(x)=\int_{-i\infty}^{i\infty}e^{xz-z^3/3}dz and j=e^{2i\pi/3}, so far I got that I need to show that: e^{zx}+je^{jxz}+j^2e^{xzj^2}=0 but didn't succeed in doing so. Any hints?

Homework Statement Hi, I am in a variational calculus class and am working on a homework and need a bit of help with two of the problems. the first one is to prove that the functional: J(u)= \int (A u'^2 + B u u' + C u^2 + D u' + E u) dx where A,B,C,D,E are constants and A \neq 0 has no...

I had a quick question on a part of a proof in chapter 1 of Functional Analysis, by Professor Rudin. Theorem 1.10 states "Suppose K and C are subsets of a topological vector space X. K is compact, and C is closed, and the intersection of K and C is the empty set. Then 0 has a...

Hi, if I want to calculate the generating functional for the free Dirac Field, I have to evaluate a general Gaussian Grassmann integral. The Matrix in the argument of the exponential function is (according to a book) given by: I don't understand the comment with the minus-sign and the...

Hello, I want to derive the connected two point function for the interacting boson-fermion theory. I know that the generating functional is Z(J, \overline{\eta}, \eta) = N \; exp \left( \int d^4 z \; L_{int} \left(-i \frac{\delta}{\delta J(z)} \right) \left(-i \frac{\delta}{\delta...

I'm thinking about getting this book. I'm a physics major, and I think the only analysis course I'm required to take later as a prerequisite for graduate courses is Introduction to Complex Analysis. So far, I've taken Cal I-III and Linear Algebra. Differential Equations will probably be in the...

My second PHP question this week...:smile: I'm writing a PHP app which includes things such as form validation and database interaction. So far, almost everything has been written procedurally. However, I started playing around with PHP's OO stuff, and it's cool. Problem is, this app is being...

In free-field theory, the functional integral \int \mathcal{D}\varphi \exp\left(i \frac{1}{2} \int d^4 x (\partial_\mu \varphi \partial^\mu \varphi - m^2 \varphi^2)\right) can be done exactly (see e.g., Peskin and Schroeder p. 285). I'm tyring to understand the step in their derivation...

Can you recommend me some low-cost universities in Canada(or US) that specialise in functional programming? I am looking for Bachelor Degrees. I have noticed that many interesting stuff, that I would call functional proramming, are in degrees such as electrical engineering and computer science...

Homework Statement I have two scalar functions u(x,y,z) and v(x,y,z) which are differentiable..Now it is required to prove that a necessary and sufficient condition for these two to be functionally related by equation F(u,v)=0 is [\nablau] \times [\nablav]=0 The Attempt at a Solution...

Hello, I am currently looking for a book on functional analysis. However most books I have seen assume knowledge real and complex analysis. But I am looking for a more superficial introduction covering the important results, some examples of applications (mainly to computational problems)...

Could someone please give me an example of an ether and an ester. In words, not drawing them. Thank you!

I'm looking for a entry-level book discussing the application of functional analysis to differential equations- mostly the Navier-Stokes equation, but PDEs in general. The books I have or have seen are either math books, full of proofs and definitions without application, or physics papers...

Homework Statement http://img357.imageshack.us/img357/8695/38808719uw6.png Homework Equations \lim_n a_n := \lim_{n \rightarrow \infty} a_n The Attempt at a Solution I'm stuck at exercise (c). Since if n heads to infinity the m doesn't play the role the limit must be one. So...

Does anyone know of where I should look to find lots of good functional analysis problems? I am currently reading Kreyszig which has great commentary, but the majority of the exercises are simple.

Something seems a little weird to me: What are the dimensions of a generating functional, Z[j] -- say for real scalar field theory? Z[j]=\int\mathcal{D}\phi\,\exp\, i\!\int d^4x\left(\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2+j\phi\right) Also, what about mass dimensions...

I've had a Meade DS 114 for a while. My kids dismantled it and apparently a piece is missing. The focuser tube is 2"; my eye pieces are 1.25" Meade says there was a piece which adapted the 1.25" eye pieces for use in the 2" hole (which also allowed for use of 2" eye pieces I guess). My...

Given that F = \int{f[h(s),s]ds} does \frac{\partial}{\partial h}ln(F)=\frac{1}{F}\frac{\delta F}{\delta h}=\frac{1}{F}\frac{\partial f}{\partial h} ?

While working out a problem I got a result which gave rise to this doubt regarding value of action functional. Suppose I start from an action, obtain the equation of motion and when I try to check if that solution gives a finite value of action, I get, surprisingly, vanishing value. The actual...

I try to learn DFT by myself(Kohn-Sham Equations), but the concept is still not so clear for me. So far, if I start with assuming any density, and then I would be able to find V(KS) Then I use this hamiltonian and solve for a wave function. And I use this wave function to find another...

Is it possible to find the extrema of an integral equation if the integral depends on a variable and an integral of that variable, i.e. the integrand is f(x) * g(integral(x)). I'm not sure if this is a "nonlocal" functional, or not a functional at all, but I can't find any references that...

what is the result of Fourier transformation of functional theta [Heaviside] function?