Functional Definition and 380 Threads

My solution: Let ##f(x+y)=f(x)f(y)-[f(x)+f(y)]-1/2## İf ##y=x## we find functional equation that given us. So for ##x=y=1## then ##f(2)=-1/2## İf we evaluate ##x=1, y=2## at above equation ##f(3)=-3## My question is: What is the solution of that functional equation; I mean are there other...

I have no idea for this question. Can you give me some clue, please?

In https://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/fnleqn.htm the equation ζ(s)=ζ(1-s) is used, where ζ is the Riemann zeta function, which I find curious, for the following reasons this indicates a symmetry around Re(s)=1/2, which seems to be what the diagram at 20:27 of seems to...

Let ##X=\{(x_i)\in \ell^\infty | x_i=1 \text{ for } i\leq n, x_i=0 \text{ for } i>n, n\in \mathbb{N}\}##. For every sequence ##(x_i)## in ##X##, ##(x_i)## has the property that the first ##n## entries are 1s and the rest are 0s. So, every sequence in ##X## trivially converges to ##X## and hence...

I considered ##X=\mathbb{R}^n## and quickly realized any linear functional like ##f=a_1x_1+\cdots a_nx_n## would attain a maximum on the boundary. I regret to say that my knowledge of topology is still very limited, and did a lot of experimenting with a pen and paper without fruitful results...

By the Euler's equation of the functional, we have ## J(\mathrm u)=\int ((\mathrm{u})^{2}+e^{\mathrm{u}}) \, dx ##. Then ## J(\mathrm{u}+\epsilon\eta)=\int ((\mathrm{u}'+\epsilon\eta')^{2}+e^{\mathrm{u}+\epsilon\eta}) \, dx=\int...

Dear everybody, I am having some trouble proving the implication (or the forward direction.) Here is my work: Suppose that we have an arbitrary linear functional ##l## on a Banach Space ##B## is continuous. Since ##l## is continuous linear functional on B, in other words, we want show that...

Hi everyone! Do you have an idea which organs/parts of the body are ONLY functional on glucose? I would say the brain, pancreas, liver and kidney, but I have to take into account only those organs that are ONLY functional on glucose

I am not sure if this is correct, but here is my work by using the definition of the Gateaux differential: \begin{align*} &dS(y; \psi)=\lim_{\tau\rightarrow 0}\frac{S(y+\tau\psi)-S(y)}{\tau}=\frac{d}{d\tau}S(y+\tau\psi)\biggr\rvert_{\tau=0}\\...

Hi; This is in fact not a homework question, but it rather comes out of personal curiosity. If you look at the graph of the two functions in the image attached, what is the simplest functional representation for such a symmetrical pattern?

Let ##U## be a bounded open subset of ##\mathbb{R}^n##. Given a continuous function ##\phi : \overline{U} \to \mathbb{R}##, show that any real-valued function ##u## of class ##C^2(\overline{U})## such that ##\Delta u = \phi## in ##U## and ##u|_{\partial U} = 0## is a minimizer of the energy...

Hello, PF! It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background: - This plan is in preparation for my master’s thesis, I...

Hello! I have some electrons produced from a 3D gaussian source isotropically inside a uniform electric field. The electric field guides them towards a position sensitive detector and I end up with an image like the one below (with more electrons on the edge and fewer as you move towards the...

In the book Quantum Field Theory for the Gifted Amateur, they define the functional derivative as: $$ \frac{\delta F}{\delta f(x))} = \lim_{\epsilon\to 0} \frac{F[f(x') + \delta(x'-x)) ] - F[ f(x') ]}{\epsilon} $$ Why do they use the delta function and not some other arbitrary function?

Step 1: Reduce RHS into singletons F = {AB->C, C->A, BC->D, ACD->B, D->E, D->G, BE->C, CG->B, CG->D, CE->G} Step 2: Reduce LHS redundant attributes ACD->B has closure of A: A, C: C,A and D: D,E,G since A is in the closure of C we can remove A from ACD->B making it CD->B no other LHS could be...

Hey, I would like to do an exchange year at Imperial. I would like to follow as a physicist the Functional Analysis course. However, I have not heard the best things about this peculiar course. What is the audience opinion on that?

Amputated hind legs were sucessfully regrown and functional in adult frogs. "Here, we demonstrate long-term (18 months) regrowth, marked tissue repatterning, and functional restoration of an amputated X. laevis hindlimb following a 24-hour exposure to a multidrug, pro-regenerative treatment...

I have some doubts with respect on how the functional derivative for the kinetic energy in density functional theory is obtained. I have been looking at this article in wikipedia: https://en.wikipedia.org/wiki/Functional_derivative In particular, I'm interested in how to get the...

Let $ \mathbb{Z} $ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z} $ such that, for all integers $a$ and $b$, $f(2a)+2f(b)=f(f(a+b))$.

$$f(xf(y) + f(x)) + f(y^2) = f(x) + yf(x + y)$$ A tricky question, i think. First fact i found was: f(f(0)) = 0 So i separate it in two types of functions f(0) = 0 and f(0) = u. I was trying to analyzing both cases, with the cases where x = y and x = -y but is is rather extended way, so i...

I will consider first the case of ## \left [ J \right ] = \int f(x,y,y') ##, if it is right believe is easy to generalize... $$ \Delta J $$ $$\int (f(x,y+h,y'+h'))^2 - (f(x,y,y'))^2 $$ $$\int \sim [f(x,y,y') + f_{y}(x,y,y')h + f_{y'}(x,y,y')h']^2 - [f(x,y,y')]^2$$ to first order: $$\int \sim...

Be ##x = x(u,v) y = y(u,v)##, if ##F = \int f(x,y,y')dx## and the Jacobian's determinant different of zero, ##v = v(u)## ##{\Large {J = \int F[x,y,y']dx ---> \int F[x(u,v),y(u,v),\frac{y_{u} + y_{v}v'}{x_{u} + x_{v}v'}](x_{u} + x_{v}v')du}}## The last term in the bracket is confusing me, how to...

I'd like to show that, by minimizing this functional $$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$ I get the well known expression $$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$ I'm...

Pop culture interpretation: Machine learning reveals recipe for building artificial proteins Actual Study: Evolution-based design of chorismate mutase enzymes Using Boltzmann machine learning to design enzymes with evolutionary compatible statistical constraints is possible now. How much...

So in particular, how could the determinant of some general "operator" like $$ \begin{pmatrix} f(x) & \frac{d}{dx} \\ \frac{d}{dx} & g(x) \end{pmatrix} $$ with appropriate boundary conditions (especially fixed BC), be computed? And assuming that it diverges, would it be valid in a stationary...

Let $f:\mathbb R\to \mathbb R$ be a function satisfying $f(x+y+2xy) = f(x)+f(y) + 2f(xy)$ for all $x, y\in\mathbb R$. Then I need to show that $f(2017 x) = 2017 f(x)$ for all $x\in \mathbb R$. I am not sure where to start. All I could note is that $f(0)=0$ which one obtains by susbtituing...

Definition: Let ##G## be a graph. ##G## is a functional graph if and only if ##(x_1,y_1) \in G## and ##(x_1,y_2) \in G## implies ##y_1=y_2##. Problem statement, as written: Let ##G## be a functional graph. Prove that ##G## is injective if and only if for arbitrary graphs ##J## and ##H##, ##G...

Laplace transform of eq. [1] [4] F1(p)-exp{-pi*p}*F1(p) = F2(p) Rearranging eq. [4] [5] F1(p) = frac{1}{1-exp{-pi*p}}*F2(p) Inverse LT of eq. [5]

My only qualm is that the statement “Let G be a functional graph” never came into play in my proof, although I believe it to be otherwise consistent. Can someone take a look and let me know if I missed something, please? Or is there another reason to include that piece of information?

I read Aczel book "Lectures of functional equations an their applications". On page 223. (Sincov's equation) is equation : ##F(x,y)+F(y,z)=F(x,z)## and general solution of this ##F(x,y)=g(x)−g(y)## , but how I prove that this function satisfies conditions ##F(x,y)=F(0,y−x)## ??

Let us suppose we have a functional of f such that ##f=f((\vec{r}(t),t)## where ##\vec{r}(t) = a(t)\vec{x}(t)##. I am trying to derive an equation such that $$\left.\frac{\partial}{\partial t}\right|_r = \left.\frac{\partial }{\partial t}\right|_x + \left.\frac{\partial \vec{x}}{\partial...

##\frac {\delta I[f]} {\delta f(x_o)} = \int_a ^b \delta(x-x_o) \, dx## with a=-1 and b=+1 ## -1 \leq x_o \leq +1 ## vs ## -1 \lt x_o \lt +1 ##, 0 otherwise. Which is correct and does it matter when doing integration by parts?

In page 9 it says: "It is interesting to note that Riemann does not speak of the “analytic continuation” of the function beyond the halfplane Re s > 1, but speaks rather of finding a formula for it which “remains valid for all s.” [...]. The view of analytic continuation in terms of chains of...

Hi, I'm looking for a book that explains more deeply (and a little bit more formal) the functional calculus than the typical introductions that I find in QFT books (like Peskin or Hatfield). Is there any good book for physicists to learn the mathematics behind functional calculus? Thanks

Hi In the last sentence I mean you do include constant terms like I have done when taking the product above?

I am trying to show that given the following stochastic differential equation: ##\dot{x} = W(x(\tau))+\eta(\tau),## we have ##det|\frac{d\eta(\tau)}{dx(\tau')}| = exp^{\int_{0}^{T}d\tau \,Tr \ln([\frac{d}{d\tau}-W'(x(\tau))]\delta (\tau - \tau'))} = exp^{\frac{1}{2}\int_{0}^{T}d\tau...

I know R4N+ is a quaternary ammonium, R2C=N+R2 is an iminium, and R-C≡N+-R is a nitrilium, but what is an R2C=N+=CR2 cation called?

' Finally, if we care only about long distances, the effective action should be a local functional, meaning that we can write is as ##S_{eff}[A]=\int d^d x...## ' Where does this come from and what does it mean? This isn't at all familiar with me, and I don't recall ever seeing anything...

I'm can't seem to figure out how to functionally differentiate a functional such as Z(J)= e^{\frac{i}{2} \int \mathrm{d}^4y \int \mathrm{d}^4x J(y) G_F (x-y) J(x)} with respect to J(x) . I know the answer is \frac{\delta Z(J)}{\delta J(x)}= -i \int \mathrm{d}^4y J(y) G(x-y) but I'm struggling...

While I was studying Propositional Calculus from Elliott Mendelson's "Introduction to Mathematical Logic," I came across, on page 18, the proof of functional completeness of \neg, \wedge, \lor logical connectives as shown below: The part that does not make sense to me starts from "Then C_{k}...

Hi PF! Can someone help me understand the notation here (I've looked everywhere but can't find it): given a function ##f:G\to \mathbb R## I'd like to know what ##C(G),C(\bar G),L_2(G),W_2^1(G),\dot W_1^2(G)##. I think ##C(G)## implies ##f## is continuous on ##G## and that ##C(\bar G)## implies...

Generalized momentum is covariant while velocity is contravariant in coordinate transformation on configuration space, thus they are defined in the tangent bundle and cotangent bundle respectively. Question: Is that means the momentum is a linear functional of velocity? If so, the way to...

Suppose that we have a function f(x) such that f(ab) = f(a)+f(b) for all rational numbers a and b. (a) Show that f(1) = 0. (b) Show that f(1/a) = -f(a). (c) Show that f(a/b) = f(a) - f(b). (d) Show that f(an) = nf(a) for every positive integer a. For (a), if ab = 1 then a = 1/b and b = 1/a. Not...

Homework Statement Hi I am looking at the attached question part c) Homework Equations belowThe Attempt at a Solution so if i take ##\frac{\partial^{(n-1)}}{\partial_{(n-1)}} ## of (2) it is clear I can get the ##\frac{i}{h} (\lambda_2 +\lambda_4 )## like-term, but I am unsure about...

Homework Statement Hi, I am looking at the attached question, parts a) and b).Homework Equations The Attempt at a Solution so for part a) it vanishes because in the ##lim \epsilon \to 0 ## we have a complete derivative: ## \int d\phi \frac{d}{d\phi} (Z[J]) ## for part b) we attain part a)...

How many isomers are there with the following description? - Thioesters with the formula ${C}_{4}{H}_{8}OS$? I was able to draw 2 of them, but apparently, the answer key showed and stated that there are 4. I am confused about why the following two are possibliities: I thought that thioesters...

I can’t decide which of these two books I should start reading. They are structured very differently so it’s hard to compare them before reading them. Can anyone fill me in on there experience with these books or maybe what things one has that the other doesn’t.

I have a pair of correlated datasets that I collected in the lab for temperature and conductivity of a solution vs time. I want to determine the functional relation between the two. (see attached plot-an interesting lead/lag in the phase difference). If I were trying to determine this...

Shown below is the graph of The graph interests the axis at 3 points, B, C and D. Given the points of intersection, and the brackets below, form and expand an equation for the graph of ATTEMPT: I've assumed to involve the points of intersection for the x-axis...

Homework Statement Prove that snell's law ## {n_1}*{sin(\theta_1)} ={n_2}*{sin(\theta_2)} ## is derived from using euler-lagrange equations for the time functionals that describe the light's propagation, As described in the picture below. Given data: the light travels in two mediums , one is...