functional

1. Linear Functional

Consider the set of all functions differentiable at the fixed point s. Let L(f) be the functional defined on this set by associating with each function f the number f'(s). Show that L is a linear functional. My attempt: I know I have to show L(f+g) = L(f) + L(g) and I know the right side is...
2. Graph the function given below and determine the given functional values.(Updated)

I do not understand how both of those are graph I liked, I realize that the one lower near (3,-3) is a parabola and opens upward, but how do I get it from that? Also what do the three functions have to do with this? I have my exam tomorrow, this is the only question I really do not fundamentally...
3. Graph the function given below and determine the given functional values.

f(x)={(3-x)^2 -4 if x=<; f(-3), f(3), f(6) ------{ 5 if x>3; So does it want me to plug in the 3 function of x's given? I am not sure what it is asking for.

1. How many fatal casualties were caused by fires between 2010/11 and 2012/13? a. 882 b. 129624 c. 782 d. 556 e. 316 2. In 2012/13 how many more fatal casualties occurred in an area where a fire alarm was present than in an area where a fire alarm...
5. Functional equation Integration

Let a differentiable function f(x) satisfies f(x)\cdot f'(x) = f(-x)\cdot f'(x) and f(0) = 1 Then find the value of \displaystyle\int_{-2}^{2}\frac{1}{1+f(x)}dx
6. Expansion of a functional

Hi I'm given the functional S\left[y\right]=\int_{1}^{2}\sqrt{x^{2}+y^{\prime}}dx\qquad y(1)=0,\: y(2)=B and asked to find \Delta=S[y+\epsilon g]-S[y] to the second order in \epsilon, so with a Taylor expansion I get...
7. Meaning of an Extremum of a Functional

Consider the following minimisation problem: $$\int_0^3\left(0.5\dot{x}^2-x\right)\,\mathrm{dt}$$ Subject to $x_0=0$ and $\dot{x}=0$. Using the Euler lagrange equation one can get: $$\frac{d^2x}{dt^2}=-1$$ The solution of the ODE is: $$x=-0.5t^2$$ and $$\dot{x}=-t$$ The integral is then...
8. functional linear

I succeed in proving that the right side is contained in the left side of the equation, but I didn't succeed to prove the opposite direction. I think I should do here something with bases for each of the spaces here, but I don't know how. Please help me here...
9. functional form

Is it possible to have a function with L1_h (but not L1_L) as one of its inputs and yet en corporate L1_L in its output? L is a variable that can take on one of two states, low or high, denoted by L1_L or L1_h. If L is high then w is high, If L is low then w is low also. I feel like the...
10. Help solving a functional equation

I am stuck trying to solve the following functional equation: Find all functions f:N_{0}\rightarrow N_{0} satisfying the equation f\left(f(m)^{2} + f(n)^{2}\right)=m^{2} + n^{2} for all non-negative integers m and n. Note: The set N_{0} represents the set of all non-negative integers...
11. Solving Functional Equation for PDF

Hello to everyone, how to solve following functional equation having that "f" is a probability density function of variable "x"? f(x)/f(-x)=exp(b*(x-G)) Regards, Rafael
12. Stationary path of functional, possible Transversality condition?

Hi everyone, I have the following functional S[y] = \int_{0}^{v}\! \left( {\frac {\rm d}{{\rm d}x}}y \left( x \right) \right) ^{2}+ \left( y \left( x \right) \right) ^{2}\,{\rm d}x Where y(0) = 1, y(v) = v, v > 0, I then showed that the stationary path of the functional is y = cosh x + B...
13. Functional Transformation

If F(x)= sqrt(x) , describe the transformation from F(x) to G(x) if G(x) = -f(-x+1)-7
14. Functional Transformations

Why is f(x) = 1/2 (x^3) a vertical shrink? Isn't is supposed to be a horizontal stretch, because f(x) = f(cx), 0<c<1?
15. Functional Equation question

Show that if: F(xy) = F(x)F(y); F(30) = 1; F(10a+7) = 1 for any integer a 0< a Then there is either simply one solution, F(x) = 1, or there are more functional equations that fit the above prerequisites. Provide justification. I understand this is a surjective function and F(30/x) = F(7/x) =...
16. Kakutani Theorem (functional analysis)

Could someone assist in explaining the reasoning behind the highlighted text of the attached document. Thanks for any assistance. The attached document has two theorems from the book by Haim Brezis called "Functional Analysis, Sobolev Spaces and Partial Differential Equations"
17. Functional Analysis Proof: complete metric space

Define the space β1([a,b]) as the space of functions f:[a,b]↦R which are everywhere differentiable and whose derivative f′ is a bounded function. One equips this space with the metricd(f,g)=sup|f(x)−g(x)|+sup|f′(x)−g′(x)| Prove that this turns β1([a,b]) into a complete metric space.Please help...
18. Functional Analysis Proposition used in proof of Hahn Banach Extention Theorem

Can anyone see how the proof follows from the part "it follows from (7) that" and how does this prove continuity?? (Rock) Thanks
19. About the one functional equation

The function is given by the table. f: N\{1,2}-> No __n_| 3 | 4 | 5 | 6 | 07 | 08 | 09 | 10 |... f(n) | 0 | 2 | 5 | 9 | 14 | 20 | 27 | 35 | ... 1) which functional equation characterizes this function? 2) solve this difference equation. 3) express this function in an explicit form 4)...
20. Functional equations

One more ploblem: Find all functions f: Z\rightarrowZ \therefore f(f(x)+x+1)=x+f(y)+1 Thank you very much in advance