Functional Definition and 380 Threads

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...

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...

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.

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

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!

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...

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?

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...