Functional Definition and 416 Threads

I'm in a problem where I have to solve the following functional equation : F(n)^2=n+F(n+1) Does anyone know some methods to solve this kind of problems ? A similar equation happens in Ramanujan example of root denesting : http://en.wikipedia.org/wiki/Nested_radical#Square_roots

Homework Statement Suppose f(z) is a possibly complex valued function of a complex valued function of a complex number z, which satisfies a functional equation of the form af(z)+bf(\omega ^2 z)=g(z) for all z in C, where a and b are some fixed complex numbers and g(z) is some function of z and...

Hello, I'm not really sure where does this question fit and what title should it bear, but here is my problem: \psi(x) \exp (a\psi(x)^2) = C f(x) given a positive definite f(x), find ψ(x) and the constant C, subject to the condition \int \psi(x)\, dx = 1 I want to solve this numerically...

Homework Statement Let f be a linear functional and set A=f-1({0}) Show that A is a closed linear subspace. Homework Equations The linearity comes from the fact that if f(a)=0 and f(b)=0 then f(βa+γb)=βf(a)+γf(b)=0 But how do we know it is closed? Do we show every sequence in A is...

Homework Statement Dear all, Good day. I am currently working on the phase field modeling of ferroelectrics. For this reason, I need to find functional derivative of an expression as presented in attached picture 1. Then in picture 2, it shows the final form of equation that I am...

Homework Statement Suppose ##f(x)## and ##g(x)## \rightarrow 0 as x \rightarrow 0+. Find examples of functions f and g with these properties and such that: a.) ## \lim_{x\rightarrow 0+} { f(x)^{g(x)} = 0 } ## Homework Equations None The Attempt at a Solution Let ## f(x) = 2^x-1...

Dear all, I am stuck with the problem which is given below; In this problem the equilibrium equations of the given functional must be derived in u, v, and w directions from which the boundary terms must be found. I think that i have derived the equilibrium equations( 5 equations), but i...

Hello, I have been increasingly running into topics in my field where at least a basic faculty with real and functional analysis would be quite helpful and I would like to go about self-studying a bit in that area. I know that Rudin is the canonical text in the field, but I have also heard...

Homework Statement Let ##f:R^+ \rightarrow R## be a strictly increasing function such that ##f(x) > -\frac{1}{x} \, \forall \,x>0## and ##\displaystyle f(x)f\left(f(x)+\frac{1}{x}\right)=1 \, \forall \, x>0##. Find a. f(1) b. Maximum value of f(x) in [1,2] c. Minimum value of f(x) in [1,2]...

In my schools functional analysis course, under prerequisites, it says "real analysis would be a good preparatory course, but is not required". In the concurrent real analysis thread, it was mentioned that real analysis is a stepping stone to functional analysis. I'm curious about two things...

Homework Statement Let ##f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}## for all real x and y. If f'(0) exists and equal -1 and f(0)=1, find f(2). Homework Equations The Attempt at a Solution Substituting y=0, 2f(x/2)=f(x)+1. This doesn't seem to be of much help. I don't see how...

Good day! I have a question regarding the law of the ff: $$ \int_0^t h(s) e^{2\beta(\mu(s) + W_s)} $$ where $\beta >0;$ $h,\mu$ are continuous functions on $\mathbb{R}_+$ with $h\geq 0;$ and $W=\{W_s,s\geq 0\}$ is a standard Brownian motion. Thanks for any help.:D

Hello, While analysing the asymptotic value of a ratio of a bessel and a hankel function, I reduced it to something of the form [(1 + β/n)^ n * (1 + n/β)^ β] / 2^(n+β) ; n and β are integers and greater than 1 how do I show that the above expression is always less than 1, for n≠β...

I major in physics, but I'm also very interested in mathematics, especially analysis. Until now, I have taken mathematical analysis and real analysis. Now, I want to learn functional analysis by myself, and my teacher adviced me to read topology first. But I found it difficult to understand and...

When expanding a function (for example the determinant of the space-time metric g) as a functional of a perturbation from the flat metric ##h_{\mu \nu}##, i.e. ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} ## i would think that the thing to do is to recognize that ##g_{\mu \nu}## and thus also...

In Zee's book at page 12 in both editions he finds that he can write the amplitude $$\langle q_f|e^{-iHT} |q_i\rangle = \int Dq(t) e^{iS} $$ where T is the time between emission at ##q_i## and observation at ##q_f##. He then states that we often define $$Z = \langle 0 | e^{-iHT} |0...

Dear PF, I'm reading a book on DFT, and it says that only ground-state wave function is a unique functional of the ground-state density, n(r). However, if in DFT the potential, v(r), is a unique functional of n(r), then shouldn't all wave functions be functionals of n(r), because you can...

Suppose that $$X$$ is a f-space and $$Y$$ is a subspace of $$X$$ and $$Y^{c}$$ is a first category in $$X$$. Can we show $$Y=X$$?

Hello I was doing an exercise that said: "If $P$ is a continuous operator in a Hilbert space $H$ and $P^2=P$ then the following five statements are equivalent". The first statement was that P is an orthogonal projection. Now this was suposed to be equivalent, under the condition of $P^2=P$, to...

Hi all, A professor asked me to do something, but I'm not quite sure what he means -- He asked me to use Density Functional Theory (DFT) calculations of the band structure of a certain crystalline metal and adjust the matrix elements of a Tight Binding (TB) model to make a "minimal" TB model...

Consider the functional Tf = f(5) - i f(7). If we take the domain T to be C_0(ℝ) with supremum norm, is T a bounded linear functional? What if we take the domain to be C_c(ℝ) with L^2 norm || . ||_2?I know I should post what I have so far but this time I have no idea because I had to missed 2...

Lets define trace for each square matrix A its trace as sum of its diagonal elements, so tr_{n}(A)=\sum_{j=1}^{n}a_{j,j}. Now proove that trace is a linear functional for all square matrix. I would be happy to know what has to be true for anything to be a linear functional? If I...

Homework Statement Find all the polynomials P(x) for which P(x^2+2x+3)=[P(x+3)]^2 Homework Equations The Attempt at a Solution I don't really know how to solve functional equations systematically. I tried to to find a linear P(x) and found P(x)=x-2 through trial & error. I also tried...

I know complex analysis is of immense help in physics at it aids us in calculating certain integrals much more easily. But what about real analysis and functional analysis? Are these branches of mathematical analysis of much use in physics? If so, in what branches of physics and how?

I'm reading about path integrals in Peskin and Schroeder's Introduction to Quantum field theory and there is a few things in the text which I find puzzling. At page 283 in the section about correlation functions we are considering the object (equation 9.15) \int D\phi(x) \phi(x_1) \phi(x_2)...

Homework Statement as some of you might've done it this is the functional eqUATION FROM THE IMO 2012 / a + b + c = 0 f2(a)+f2(b)+f2(c)=2f(a)f(b)+2f(b)f(c)+2f(c)f(a). f:Z->Z http://www.cut-the-knot.org/arithmetic/algebra/2012IMO-4.shtml <- link of the problem and its SOLUTION now i worked with...

SOLVED Prove that the functional sequence has no uniformly convergent subsequence -check n \in \mathbb{R}, \ \ f_n \ : \ \mathbb{R} \rightarrow \mathbb{R}, \ \ f_n(x) =\cos nx We want to prove that {f_n} has no uniformly convergent subsequence. This is my attempt at proving that: Suppose...

Hey all, back with another question. I have the opportunity in the fall to choose 1 (maybe 2 if I'm lucky) of the following classes: Numerical analysis (undergrad numerical linear algebra, using matlab), Functional Analysis (as a directed study course with a prof), and the other is doing a...

Homework Statement F[y(x)]=\int [y(x)\frac{dy(x)}{dx}+y(x)^{2}]\,dx Homework Equations δ(x-x') I think this is the Kronecker Delta. It might be the Dirac Delta. The Attempt at a Solution I have the whole thing written in my notes, I just don't know how to make sense of it...

Here's an example from my homework. I already turned it in, though. I basically just copied what I could from my notes, but I have no idea how this is done. Could someone explain this to me? I can't find anything intelligible (at least to me) of this stuff on any website. My notes contain parts...

Hi folks ... I urgently need good books about Functional analysis and Topology. These must be comprehensive and thorough, undergraduate or graduate. Please, advise and provide your experiences with such books. I accept only thick books ;) e.g Introductory Functional Analysis with...

Homework Statement Consider the functional I:W^{1,2}(\Omega)\times W^{1,2}(\Omega)\rightarrow \mathbb{R} such that I(f_1,f_2)=\int_{\Omega}{\dfrac{1}{2}|\nabla f_1|^2+\dfrac{1}{2}|\nabla f_1|^2+e^{f_1+f_2}-f_1-f_2}dx. I would like to show that the functional is strictly convex by using the...

Author: Michael Reed, Barry Simon Title: Functional Analysis Amazon link https://www.amazon.com/dp/0125850506/?tag=pfamazon01-20 Level: Undergrad

Author: Essential Results of Functional Analysis Title: Robert Zimmer Amazon Link: https://www.amazon.com/dp/0226983382/?tag=pfamazon01-20

Author: Frigyes Riesz, Bela Sz.-Nagy Title: Functional Analysis Amazon link: https://www.amazon.com/dp/0486662896/?tag=pfamazon01-20

I don't know what to title this but will change it if $f(f(x))$ has a name. Anyway, I need to find $f(x)$ such that $f(f(x))=-x$. My friend gave me this challenge question and I haven't been able to figure it out. There are many examples where $f(f(x))=x$ for example f(x)=\frac{1}{x} but that...

Author: Elias Stein, Rami Shakarchi Title: Functional Analysis: Introduction to Further Topics in Analysis Amazon Link: https://www.amazon.com/dp/0691113874/?tag=pfamazon01-20 Prerequisities: Real Analysis by Stein and Shakarchi Level: Undergrad Table of Contents: Foreword Introduction...

Author: Serge Lang Title: Real and Functional Analysis by Lang Amazon Link: https://www.amazon.com/dp/0387940014/?tag=pfamazon01-20 Prerequisities: Undergrad analysis Level: Grad Table of Contents: General Topology Sets Some Basic Terminology Denumerable Sets Zorn's Lemma...

I am a new user for Quantum Espresso(QE). Recently I have installed Quantum Espresso in my system. Now i am struggling to give input file in QE. How to generate input file in QE?

Author: Erwin Kreyszig Title: Introductory Functional Analysis wih Applications Amazon link https://www.amazon.com/dp/0471504599/?tag=pfamazon01-20 Prerequisities: Being acquainted with proofs and rigorous mathematics. Rigorous Calculus and Linear algebra. Level: Undergrad Table of...

I've always been curious why points in polar coordinates are defined as (r,θ) when all equations (including parametric equations formed from them) are defined as r=f(θ). Considering that point in cartesian coordinates are defined as (x,y) where y=f(x). Also a,b=(r,θ) ∫1/2[f(θ)]2 further...

A few days ago on MMF the following question was posted with no one showing how to solve it so far: Given: $\displaystyle f(f(x))=x^2-x+1\, \forall x\in \mathbb{R}$ find $\displaystyle f(x)$. I have never known how to solve such equations, except by trial and error, and this one has me...

Hi, I am trying to write down the propagator for a scalar field theory, but I want to try and get it in the functional representation. My plan is to compute the following: \langle \psi (x', t') | \psi (x,t) \rangle which gives the amplitude to go from x' to x. Now I guess I have to...

I have a definite integral defined by \begin{equation}T\left(G\left(g\right)\right)=\int_{g_{1}}^{g_{2}}G(g)\mathrm{d}g\end{equation} where G is a continuous function of a variable g, and g_{1} and g_{2} are known numbers. I want to minimize T\left(G\left(g\right)\right), that is I want to...

Homework Statement Hey, can I just check these functional derivatives?: 1) \frac{\delta F[g]}{\delta g(y)} where F[g] = \int dx \left[ \frac{1}{\sqrt{1+(g'(x))^2}} - 2g(x) + 5 \right]\>. 2) \frac{\delta F[a,b,g]}{\delta g(y)} where F[a,b,g] = \int d^4x \left[ A(\partial_{\mu}...

Homework Statement Determine whether the following functional series is pointwise and/or uniformly convergent: \sum_{n=1}^\infty \frac{x}{n} (x\in\mathbb{R}) Homework Equations The Attempt at a Solution My answer to this seems very straightforward and I would be very grateful if...

I'm self studying Chaikin's Principles of Condensed Matter Physics. I'm trying to figure out how to go from (5.2.30) to (5.2.31). Homework Statement 5.2.30 is the one-loop approx. to the free energy. I'll denote G0^-1 from the book G ~ Integral(ln(G(phi(x))) 5.2.31 is (as far as...

Homework Statement Let C be a non-empty convex subset of a real normed space (X,\|\cdot\|). Denote H(f,a):=\{x\in X: f(x)\leq a\} for f\in X^* (dual space) and a\in\mathbb{R}. Show that the closure \bar{C} of C satisfies \bar{C}=\bigcap_{f\in X^*,a\in\mathbb{R}: C\subseteq H(f,a)}H(f,a)...

Folks, The total potential energy functional for an isolated finite element timoshenko beam is given as ## \displaystyle \Pi_e(w, \Psi)=\int_{x_e}^{x_{e+1}} \left[ \frac{EI}{2} \left (\frac{d \Psi}{dx}\right )^2 + \frac{ G A K_s}{2} \left ( \frac {dw}{dx} + \Psi \right )^2 +...\right]dx...

Current follows the path of least resistance or shortest path. I just want to prove this or rather reproduce it using calculus of variations. I just want to show it in a fancy way. I want help to form the FUNCTIONAL for it. Useful equations: I=dq/dt=nqvA R=rho*l/A Where v is drift velocity...