Functional Definition and 380 Threads

Although this problem is meant to be easy I can't quite work it out. Let U(A) denote the set of unitary elements of a C*-algebra A. I've already shown that if u is unitary in A then the spectrum of u: \sigma(u) \subset \mathbb{T} = \{z\in\mathbb{C}\,:\,|z|=1\} which was easy. Now...

A linear functional is a function g:V to F where V is a vector space over a field F such that if u and v are elements of V and a is an element of F, then g(u+v) = g(u) + g(v) and g(au) = ag(u) Let G be the space of all linear functionals on V. Then if \oplus_{1} and \otimes_{1} are...

I have a commutative Banach algebra A with identity 1. If A contains an element e such that e^2 = e and e is neither 0 nor 1 (I think this also means to say that it contains a non-trivial idempotent), then the maximal ideal space of A is disconnected. Currently I am trying to show this but I...

I found this functional equation for the Riemann zeta function in Table of Higher Functions, 6th ed. by Jahnke, Emde, & Losch on pg. 40: z(z+1)\frac{\zeta (z+2)\zeta (1-z)}{\zeta (z)\zeta (-1-z)}=-4\pi ^2 any suggestions as to how one might consider such an equation, much less derive it...

Let,s suppose we have a functional J and we want to obtain its extremum to obtain certain Physical or Math properties: \delta{J[f(x)]}=0 Yes you will say to me " You can apply Euler-Lagrange Equation to it and generate a Diferential equation to obtain f"..of course is easier saying than...

The (main) functional group in esters are: R_1OOR_2 Where R_1 and R_2 are carbon chain. When I though about it, I've never heard the systematic name of this functional group. What is this functional group called? I've seen esterbonding etc. but that doesn't sound systematic as ex...

Hello everybody I'm given a continuous function f (from the real numbers to the real numbers) which I know obeys the following functional equation: f(x)=f(x^{2}) How can I proof that this function is constant? I started out like this: Looking at a number x in [0,1[ I said to myself that...

I put together two questions : a) suppose there is a point mass with mass M..if it is moving, then from a certain oberver, the total energy is higher, via E=Mc^2...hence, following the generaly relativity qualitatively, where the energy density defines the curvature, the gravitation should be...

let be the next functional differential equation: \delta{F[\phi]}=G[\phi,\partial{\phi}] then its solution would be: F[\phi]=\int{D[\phi]G[\phi,\partial{\phi}] would it be correct?..thanks..

Regarding the Barber of Seville paradox, I am looking for something equivalent that is expressed in functional notation. For example, this is my attempt at a piecewise definition of such a function: For a function f : \mathbb{N} \mapsto \{0,1\} f(n)=\left\{\begin{array}{cc}0,&\mbox{ if...

we know that the functional integrals are important in quantum field theory,but we have the problem that except for the semiclassical approach,they can not be solved anyway..but if we used the formula:. \int{d[\phi]F[\phi]=\sum_{n=1}^{\infty}(-1)^{n}\phi^n{D^{n}F[\phi]} where D is the...

Hi, I am reading chapter 5 of Ryder regarding path integrals and vacuum to vacuum transition amplitudes in presence of source. I follow the math but don't have a clear physical picture. The formula is: Z[J]=\int Dq \: exp ( \frac{i}{h}\int dt(L+hJq+\frac{1}{2}i\epsilon q^2) ) Can...

Why do we treat a scalar field phi and its derivatives as being independent when trying to find a stationary solution for the action? Doesn't that give too general solutions? Where does the restriction that (d_mu phi) is dependent on phi come back in?

what are the requirements of a functional J[y] to exist in the form that its minimum will yield to a differential equation?..i mean let be the functional with condition: J[y]=\int_{a}^{b}dx(p(x)(y`)^{2}+V(x)y^{2}) \int_{a}^{b}y^{2}dx=C with c a constant... then what conditions should...

let be the functional equation f(x,y)=g(r(x)+h(y)) my question is..does this equation have a solution?..thanks...

Im having some difficulties proving some basic properties of the adjoint operator. I want to prove the following things: 1) There exists a unique map T^*:K\rightarrow H 2) That T^* is bounded and linear. 3) That T:H\rightarrow K is isometric if and only if T^*T = I. 4) Deduce that if T is...

Question 1 Prove that if (V, \|\cdot\|) is a normed vector space, then \left| \|x\| - \|y\| \right| \leq \|x-y\| for every x,y \in V. Then deduce that the norm is a continuous function from V to \mathbb{R}.

I saw this video today and I believe it is a presentation from SUN's "Project Looking Glass". Now this is the desktop of my dreams! http://abum.com/files/Movies/3d_desktop.wmv http://www.sun.com/software/looking_glass/

Given a set of m real functions of n variables, what is a necessary and sufficient condition for the functions to be functionally independent ? A set a functions f_i(x_1,...x_n)\quad i=1,...m are functionally independent, if the only function \phi(u_1,...u_m) such that \phi(f_1,...f_m)=0 is...

Let be F the functional given by J(y)=J(x,y,dy/dx) my question is,apart from the usual definition of functional derivative and Lagrange equation,..does a numerical method exist to calculate it,i mean am looking for a recursive method,you introduce an initial function y0(x) from this you...

Find f(x) if f(x) + f(\frac{x-1}{x}) = 1 + x

Hi, Does anyone of you have some links or references to books on Functional Density Theory ? I know the basics but i would like to learn a whole lot more. Thanks in advance marlon

This is about amines: The functional groups present are: C=O and NH2 Why does the N get protonated always but not C=O, since the oxygen has more lone pairs and more electronegative than N so shouldn't the oxygen be protonated more easily? I can't think of any good reasons... please help...

Lets say for example a hydrocarbon skeleton has two diff functional groups branched to the skeleton.. Let's say one of the functional groups pH is less than 9 to 10, and the other functional group's pH is greater than 2 to 4 that means that molecule to be electrically charged, the pH of the...

Does anyone know good online references about functional derivatives? Most of the documents contain only the definition, but I would like some more complete material containing, for example, rules for differentiate composite functionals and other details.

I'm trying to learn density functional theory by myself, but I'm a bit confused as to how to use it. Is the following methodology correct (I think it'd take forever to use LaTex to write the equations, so I have a link to small webpage that already has the equations laid out and numbered)...

Hi All, I have a problem, and it is so confusing to me. I put it here in the hope of getting some helps to make it clear. Thank you, the problem is given below. What is the greatest value of b for which any real valued function f that satisfies the following properties mus also...

schröder's equation is a functional equation. let's assume A is a subset of the real numbers and g maps A to itself. the goal is to find a nonzero (invertible, if possible) function f and a real number r such that f\circ g=rf. motivation: if there is an invertible f, then the nth iterate...

hi! there is a good functional analysis book by Erwin Kreyzig...which has hilbert spaces and banach spaces and stuff!

Is it possible for us to make a functional model of the human brain? I'm not interested in the phylosophical implications, but more in the resources, techonolgies we can use (that's why I posted here... )