Functional Definition and 380 Threads

If I understand what's going on (quite possibly I don't), I think my book is using bad (confusing) notation. Homework Statement As written: "Calculate ##\frac{\delta H[f]}{\delta f(z)} \ \text{where} \ H=\int G(x,y)f(y)dy##" and ##\frac{\delta H[f]}{\delta f(z)}## is the functional derivative...

Entering my third year of my bachelor of science majoring in maths/physics and having some trouble deciding what courses to do this semester. I know for sure I will be taking complex analysis and 3rd year quantum however am having trouble picking between 3 in particular for my final two courses...

When density functional theory is used to simulate a molecule adsorbed on a surface, it turns out that due to their interaction, a fraction of an electron is transferred from the surface to the molecule or vice versa. These interactions are normally categorised in interactions involving...

Homework Statement I have been given a functional $$S[x(t)]= \int_0^T \Big[ \Big(\frac {dx(t)}{dt}\Big)^{2} + x^{2}(t)\Big] dt$$ I need a curve satisfying x(o)=0 and x(T)=1, which makes S[x(t)] an extremum Homework Equations Now I know about action being $$S[x(t)]= \int_t^{t'} L(\dot x, x)...

What is a difference between linear operator and linear functional? Do I understand it correctly that linear operator is any operator that when applied on a vector from a vector space, gives again a vector from this vector space. And also obeys linearity conditions. And linear functional is a...

I have one slot to fill in in the coming term. The two candidates are Functional Analysis and Complex Analysis (both on the undergraduate level). Here are some questions: 1) Which one would you pick and why? 2) What other classes in the standard B.Sc. math curriculum rely on either of these...

This is more a conceptual question. So i am doing some self review of multi variate calculus and i am looking at functinal relations of the form F(x, y, z,...) = 0 In the book they talk about implicit differentiation. Now i fully understand how to do the mechanics of it, but i was trying to...

Hi, I need a functional analysis book. I have Kreyszig's book. I'm at continuous mapping but I have some problems with completeness and accumulation points. So I would like to read a lot excercises about these introductory stuff. What are your suggestions? Thanks.

Something I've always wondered about. It's neat to see a quantitative answer, finally. "To reach the new figure, Dr Lunter and his colleagues took advantage of the ability of evolution to discern which activities matter and which do not. They identified how much of our genome has avoided...

Let f : R -> R be a continuous function such that, f(x) - 2f(x/2) + f(x/4) = x^2 then, . f(3) = ? Answer to be calculated in terms of f(0). I am puzzled on how to approach such problems. Some insight would be greatly appreciated.

Hello there, so today I started doing my research on oscillations in a course on advanced mechanics. The experiment was to mathematically model the speed of sound in air and experimentally prove the usability of the model. To keep it simple and pose my question as directly as possible, my...

I need a measure/integration theory book that covers the basics. I had already calculus, complex analysis, ODEs and topics of PDEs/Sturm-Liouville problem. More specifically I need to learn functional analysis to be prepared for stochastic calculus. Any suggestions? Thank you.

I read that in any locally convex topological space X, not necessarily a Hausdorff space but with linear operations continuous, for any ##x_0\ne 0## we can define a continuous linear functional f:X\to K such that f(x_0)\ne 0. I cannot find a proof of that anywhere and cannot prove it myself...

Consider the following linear functional operator: $$Q_w[f(x)] = \lim_{h\rightarrow w} \lbrace \frac{f(x + h) - f(x)}{h} \rbrace $$ How does one solve the equation $$a_0(x)Q_0[f(x)] = a_1(x)Q_1[f(x)]$$ Spelt out that is: $$a_0(x)*f'(x) = a_1(x)(f(x+1) - f(x))$$ For the case of constant...

Homework Statement which functional group is present in aspatame molecule? Homework Equations The Attempt at a Solution why the carbonyl group COO- is not present in the diagram? I can find it in the diagram

What is the difference between a functional and a composite function? Also, look those implicit equations: ##F(x, y(x))=0##, ##F(t, \vec{r}(t))=0##, ##F(x, y(x), y'(x), y''(x))=0##, ##F(t, \vec{r}(t), \vec{r}'(t))=0##... Can be understood that ##F## is the functional?

Homework Statement Let [a,b] \subset \mathbb{R} be a compact interval and t0 \in [a,b] fixed. Show that the set S = {f \in C[a,b] | f(t_0) = 0} is not dense in the space C[a,b] (with the sup-norm). Homework Equations Dense set: http://en.wikipedia.org/wiki/Dense_set sup -...

Let \normalsize S[y] = \int ^{a}_{b} f[y, \dot{y}, x] dx be the functional i want to minimize. Why does \normalsize f (inside the integral) take this specific form? Would i not be able to minimize the integral, \normalsize S , if f had any other form instead of f = f[x, y, \dot{y}] ?

Hi guys, I'm not sure where to put this question, so I'll just put it here. If a mod knows of a better place, just point me to it, thanks. I'm looking at the functional differentiation equation: $$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}\equiv \int\frac{\delta F[f]}{\delta...

Homework Statement Solve: $$\frac{\delta F[f]}{\delta f(x)}=b(x)f(x)^2F[f]$$ For b(x) a fixed smooth function. Homework Equations $$\left.\frac{dF[f+\tau h]}{d\tau}\right|_{\tau=0}\equiv \int\frac{\delta F[f]}{\delta f(x)}h(x)dx$$ The Attempt at a Solution This isn't a homework problem...

Hi guys, I'm trying to study the functional approach to quantization in QFT. The QFT books seem to often "sweep things under the rug" and not be too rigorous when it comes to issues like integral convergence, and the like. So I was wondering if there was a more mathematically rigorous...

Homework Statement Suppose f is differentiable on \mathbb R and \alpha is a real number. Let G(x) = [f(x)]^a Find the expression for G'(x) Homework Equations I'm pretty sure that I got this one right, but I really want to double check and make sure. The Attempt at a Solution...

Please suggest me, how to generate weak form or functional of any partial diffrential equation ( mostely second order) in Finite Element Method. Thanks in advance.

Two questions, really: I’m finding it hard to wrap my head around the connections between k-space and real-space for d-wave symmetry, as well as the connections between “order parameter,” “gap,” “Cooper pair wave function,” and “superconducting wavefunction,” which are all mentioned at various...

Problem: Let $f:R\rightarrow R$ be a continuous function such that $$f(x)-2f\left(\frac{x}{2}\right)+f \left( \frac{x}{4} \right)=x^2$$ Find $f(3)-f(0)$. Attempt: I really don't know how should I approach this problem. I could only deduce that $f(0)=0$. Then I tried putting a few values for $x$...

Homework Statement We have functional ##I(y)=\int_{0}^{2}{y}'(2+e^x{y}')dx## where ##y\in C^1(\mathbb{R})## and ##y(0)=0##. Calculate the extreme value.Homework Equations The Attempt at a Solution I am having some troubles here... :/ From Euler-Lagrange equation we get ##\frac{\partial...

Homework Statement We have functional ##A(y)=\int_{-1}^{1}(4y+({y}')^2)dx## where ##y\in C^1(\mathbb{R})## and ##y(-1)=1## and ##y(1)=3##. a) Calculate ##A(y)## if graph for ##y## is line segment. b) Calculate the extreme value of ##A(y)## for that ##y##. That does it represent...

Homework Statement Find extremals of the functional ##\Phi(y,z)=\int^{\frac{\pi}{2}}_0((y')^2+(z')^2+2yz)dx## for ##y(0)=0##, ##y(\frac{\pi}{2})=1##, ##z(0)=0##, ##z(\frac{\pi}{2})=-1##Homework Equations The Attempt at a Solution Well I have a solution but I have problem how to start with it...

Homework Statement The function f satisfies \dfrac{f(x)}{f(y)} \leq 2^{(x-y)^2} x,y \in D where D denotes domain set of the function, then f(x) can be I have a set of options as well but I'm not posting it now. I will post it if required, later. The Attempt at a Solution I have dealt...

Homework Statement https://imagizer.imageshack.us/v2/622x210q90/833/sqaw.png I am having difficulty understanding the notation <h, f''(x0)h>

Hello, I was wondering if such a thing even exists, so here it goes... Let's say I have a function x(s) (it is real, smooth, differentiable, etc.) defined on (0,1). In addition, dx/ds = 0 on the boundary (s=0 and s=1). I can compute its Fourier transform (?) as a_p = \int_0^1 x(s)...

Homework Statement Let a.b,c,d be real numbers such that a ≠ b and c ≠ 0 , find f:R->R for which this statement holds: af(x+y) + bf(x-y) = cf(x) + dy , for all x,y real numbers. Homework Equations Well this is a functional equation, that I know. I have less experience with...

Hi everyone, :) Here's a question with my answer, but I just want to confirm whether this is correct. The answer seems so obvious that I just thought that maybe this is not what the question asks for. Anyway, hope you can give some ideas on this one. Problem: Let \(X\) be a finite...

Greetings, I want to become more fluent using functional derivatives. Does anyone have a link to sets of problems involving functional derivatives or anything like that (e.g., a worksheet from a class where they were used or something)? The lengthier the better, and ideally the solutions...

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 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...