Tagged Questions
1
vote
1answer
23 views
Product and Quotient rule for Fréchet derivatives
Does anyone know whether the product/quotient rule for Fréchet derivatives still hold? For example, consider the evaluation operator:
$$\rho_x : (C[a,b],\|\cdot\|_\infty) \rightarrow ...
5
votes
1answer
93 views
Calculus on surfaces and chain rule
Define the surface gradient operator on any surface $S$ as
$$\nabla_S f = \nabla f - \nabla f \cdot \nu_S \nu_S$$
where $\nu_S$ is the outward unit normal on $S$.
Let $T:S_1 \to S_2$ be a $C^2$ ...
2
votes
2answers
84 views
Weak / Classical derivative
I know the definitions of both weak and classical derivative. But I am trying to see the classical derivative as a weak derivative. When we have
$\int f' \varphi = -\int f\varphi'$ for all $\varphi\in ...
0
votes
0answers
85 views
integral equations - i need help to expand the function
I have the following integral equation to solve:
$$\int_{0}^{2\pi} (\cos^2(x+y)+1/2) \phi (y) dy$$
So, I need to find $\lambda$ where $\lambda$ is the eigenvalues's function. Well, my main goal is ...
1
vote
1answer
37 views
What conditions do I need for this calculation to work? (Product rule differentiation of bilinear form)
Let $b(\cdot,\cdot):X \times X \to \mathbb{R}$ be a bilinear mapping where $X$ is a Banach space.
Consider functions $f:[0,T]\to X$ and $g:[0,T] \to X$. What assumptions do I need on $b$ so that this ...
-2
votes
2answers
91 views
Integral equation that's cant solve… Need a hand [closed]
Help me solve this integral equation, I'm having some troubles... I need to use the Fredholm method for second kind integral equations.
$$\phi(x)= \sin(x)+ \lambda \int_{0}^{\pi}\cos(2x+y)\phi ...
-4
votes
2answers
197 views
Integral eigenvectors and eigenvalues
I need to find the eigenvalues e eigenvectors of this integral.
a)
$$\int_{0}^{2\pi}(\cos^2(x+y)+1/2)\phi (y)dy$$
b)- Solved thanks
$$\int_{0}^{1}(x^2y^2-2/45)\phi (y)dy$$
0
votes
0answers
128 views
Find the eigenvalues and eigenvectors of an integral operator
I need to find the eigenvalues e eigenvectors of this integral.
$$\int_{0}^{1} K(x,y)\phi (y)dy,$$
where
$K(x,y)=x(1-y),\; 0 \le x\le y \le 1$
and
$K(x,y)=y(1-x),$ $0\le y\le x \le 1$
I ...
1
vote
1answer
30 views
Lipschitz condition normed vector space
Am I right that $g: C^{1}[0,2],||*||_{C^1[0,2]} \rightarrow \mathbb{R}$ with $g(f)=f'(1)$ satisfies a Lipschitz condition?
Cause $|f'(1)-h'(1)|=|g(f)-g(h)|\le \text{max} |f-h|+ \text{max} |f'-h'|$, ...
3
votes
1answer
72 views
$C^k_b$ with sup-norm not complete
Let $C^n_b=\{ f : I\rightarrow \mathbb{C}: f~n\textrm{-times continuously differentiable and } \|f\|_{n,\infty} < \infty\}$, where $\emptyset\neq I\subseteq\mathbb{R}$ denotes an open interval and ...
1
vote
0answers
29 views
The tightest bound on an integral
Consider a polynomial $p(x)$ where $p(x)>0$ for $x\in(0,1)$ and $p(0)=0$. Let $s(x)$ be an increasing analytic function such that $s(0)=0$ and $s(1)=1$. I am interested to bound the following ...
2
votes
1answer
30 views
A bound on an integral
Consider $f(z)$ and $g(z)$ where $f(z)$ is a polynomial such that $f(z)=0$
and $g(z)$ is an analytic function. I want to find the tightest bound the following integral:
$\int_0^1 f(z)g(z) dz$
I know ...
1
vote
1answer
55 views
Show that this is a diffeomorphism
I have a function $F:(0,2\pi) \times \mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}^2$
with $(\phi,r)\mapsto(r(\phi-\sin(\phi)),r(1-cos(\phi)))$ and want to show that this is a smooth(meaning ...
0
votes
0answers
26 views
Convexifying Functions
I have the following question:
Imagine you have a function $F(x,y(x),y'(x)), F:\mathbb{R} \times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} $ and this function is not convex.
Then you can ...
1
vote
1answer
69 views
Evaluation of a limit with integral
Is this limit
$$\lim_{\varepsilon\to 0}\,\,\varepsilon\int_{\mathbb{R}^3}\frac{e^{-\varepsilon|x|}}{|x|^2(1+|x|^2)^s}$$
with $s>\frac{1}{2}$, zero?.
The limit of a product is the product of limit, ...
4
votes
2answers
165 views
Interchanging closed operators and integrals
I am dealing with a problem in Evans PDE without measure theory knowledge...
We have contraction semigroup $\{S_t\}_{t \geq 0}$ on real Banach space $X$, i.e family of bounded linear operators from $ ...
1
vote
0answers
51 views
Verify solution: Is this gradient, correct?
For a function $$f(X)=\operatorname{tr}(X^TAX)+\|\operatorname{diag}(X^TX)-\alpha I\|_2,$$ where all entries are real and $\alpha$ is a real scalar, while $A$ is a p.s.d matrix and $X$ is a real ...
1
vote
0answers
52 views
How to deduce the results of response time by this trajectory approach?
First, we denote this:
And
And we get this right property( $last_i$ means the last node on $τ_i$):
And:
$Smin_i^h$ = $\sum_{h'=first_i}^{h-1} ({C_i^{h'} + L_{max})}$
$Smax_i^h$ = ...
1
vote
1answer
63 views
Curvature via hessian in Taylor expansion
In the case of a univariate function, the smaller the second derivative in its Taylor expansion, the smaller is the curvature of the univariate function.
Now, how is the curvature of the function ...
1
vote
1answer
60 views
Calculation of the Laplacian of a function in $\mathbb{R}^3$.
I have to calculate the Laplacian distributional sense) of the following function
$$\frac{e^{i\sqrt{\lambda}|x|}-1}{4\pi|x|}$$
with $\lambda>0$, $x\in\mathbb{R}^3$. I've procedded in the following ...
-1
votes
1answer
84 views
Showing $h(x) = \frac{1}{\epsilon}\int_x^{x+\epsilon} f(t)dt$ is differentiable for continuous $f$ and $\epsilon > 0$
Assume $f$ is continuous on $\mathbb{R}$ and $\epsilon>0$. Let $h(x) = \displaystyle\frac{1}{\epsilon}\int_x^{x+\epsilon} f(t)dt$. Show $h$ is differentiable and $h'$ is continuous. Compute ...
1
vote
0answers
72 views
Extremal condition calculus of variations
if I have a functional with a Lagrangian $L(t,x(t),y(t),x'(t),y'(t))$, meaning two functions x and y of one parameter t. And want to solve the minimization problem $$ \int_0^t L \, dt. $$ Then I get ...
0
votes
0answers
58 views
Minimum calculus of variation
Hi I am looking for a criterion that is sufficient to prove that a solution to a functional depending on two functions y(t) and x(t) is an extremum.
it is about the following functional$$ \int_0^b ...
0
votes
2answers
54 views
Please help to make me understand why I cant optimize this function: $U=x^{1/3}*y^{2/3}$ ?
If I want to maximize a production the function of which is given by $$L=-x^2+10x-2y^2+12y$$ I know I have to take the partial derivatives of of the function in respect to X and Y, so $$\frac ...
2
votes
2answers
43 views
How to prove that $|||y |||$ is continuous using the usual basis of $\Bbb R^{n}$
How to prove that $|||y |||$ is continuous on $\Bbb R^{n}$ by using the usual basis of $\Bbb R^{n}$
By the way, $||| \cdot |||$ is a norm on $\Bbb R^{n}$
I can show this by using triangle ...
0
votes
2answers
27 views
Prove that $D ⊂\Bbb R^{n}$ is compact iff whenever {$C_{α}$} is a collection of relatively closed subsets of $D$ with the property $∩ C_{α} = ∅$
Prove that $D ⊂\Bbb R^{n}$ is compact if and only if whenever {$C_{α}$} is a collection
of relatively closed subsets of $D$ with the property $∩ C_{α} = ∅$ , there is a finite
subcollection satisfying ...
2
votes
1answer
32 views
How to prove $H(g)=\{(x,y) : x\in I,\; y=g(x)\}\subseteq \Bbb R^{2}$ is connected, where $g: I \longrightarrow \Bbb R$ is a continuous function?
How to prove
$$H(g)=\{(x,y) : x\in I,\; \; y=g(x)\}\subseteq \Bbb R^{2}
$$
is connected, where $g: I \longrightarrow \Bbb R$ is a continuous function.
By the way $I$ is an interval of $\Bbb R$.
I ...
2
votes
1answer
50 views
How to prove that $\{x\in E: h(x)>d\}$ is open and $\{x\in E : \varphi_\beta(x)=d \mbox{ for all } x\in A\}$ is closed
Let $E\subset \mathbb{R}^{n}$ and $h:\varphi_{\beta}:E\to\mathbb{R}$ be continuous Where $\beta=A$ and $d\in\mathbb{R}$
How to prove that $\{x\in E: h(x)>d\}$ is open and $\{x\in E : ...
2
votes
1answer
97 views
Finding $u(x)$ using Green's Function
Let $(Lu)(x)= -\frac{d}{dx}\big(\frac{1}{x} \frac{du}{dx} )$ where $u(x)$ is twice differentiable function defined on $[1,2]$.
A) I need to find Green's function $G(x,t)$ such that for any $h(x)$ ...
2
votes
2answers
148 views
Does this integral go to zero?
I have $f_1,f_2$, $C^\infty$ functions with compact support and $f_3$ a smooth and bounded function; let $a\in\mathbb{R}^3$. I have to evaluate this limit
...
0
votes
1answer
86 views
Is this integral 0?
Let $\phi \in C_0^{\infty}(\Omega)$ with $\Omega = (0,1)\times(0,1)$. Let $u\in L_2(\Omega)$ defined by $u(x,y) = 1$ for $x>y, u(x,y) = 0$ for $x\leq y$
Is there a way we can conclude ...
2
votes
1answer
50 views
Is this function in $L^2(\mathbb{R}^6)$?
I have to prove that the following function in $L^2(\mathbb{R}^6)$
$$F(x,y)=\frac{f(x)}{x^2+y^2+\frac{2}{m+1}x\cdot y+\lambda}$$
with $f\in H^{\frac{1}{2}}(\mathbb{R}^3)$, $x,y\in\mathbb{R}^3$ and ...
2
votes
1answer
51 views
Is this estimation correct?
I have to estimate the following quantity
$$\frac{|e^{i\sqrt{\lambda+i\varepsilon}|x|}-e^{i\sqrt{\lambda}|x|}|^2}{|x|^2}$$
in $\mathbb{R}^3$ ($\lambda>0$) where
...
1
vote
1answer
82 views
How can I study the continuty of this function?
Let $f\in L^2(\mathbb{R}^3)$ with compact support; is the function
$$F(z)=\int_{\mathbb{R}^3}dx\bigg(f(x)\frac{e^{i\sqrt{z}|x|}}{4\pi|x|}\bigg)$$ continuous in the set
$$Q=\lbrace{z: \Re z\in [a,b], ...
0
votes
0answers
71 views
Is this a continuous function?
Let $f\in L^2(\mathbb{R}^3)$ with compact suppport and $z\in\mathbb{C}$ with $\Im\sqrt{z}>0$. Is the following function continuous for $z\in Q=\{z: \Re z\in [a,b], \Im z\in (0,1]\}$:
$$ ...
0
votes
1answer
43 views
Conditions of parameter $\lambda$ ensuring integral is 0
Let $1 \le p \le \infty$. I am seeking to find the values of the parameter $\lambda$ such that:
$$\displaystyle \lim_{\epsilon \to 0^+} \frac{1}{\epsilon^\lambda} \int_{0}^{\epsilon} f = 0 \ \ \forall ...
2
votes
1answer
47 views
The Lebesgue Theory basic Application , get stuck
Ok, I am working on a very easy question but I get stuck when I trying to justify my answer.
I know that, in order to use Lebesgue's dominated Convergence Theorem, there are two conditions that we ...
0
votes
1answer
28 views
Information needed about Local Extremas in differential calculus
We know a function $f \in C^2(R)$ has a Local Maximum in the origin $(0,0)$. What can you say about the differential: $d_{(0,0)}^2f(1,-1)<0$?
I've recently got this on a test and I'm not sure if ...
4
votes
1answer
151 views
Is integration a continuous functional on the Skorohod space?
I have read several times that integration is a continuous functional on the Skorohod space $D[0,1]$, i.e., the set of all cadlag functions on $[0,1]$ equipped with the Skorohod metric; in symbols, ...
3
votes
0answers
89 views
Continuity criteria for Radon-Nikodym derivative
I have been looking for results or theorems which give me regularity conditions of the Radon-Nikodym derivative, but I have not found any :(
For instance, we know that if $\nu\ll\mu$ then there ...
3
votes
1answer
65 views
Show continuity of a function?
Are there theorems or results to show that if for every $\varphi\in \mathcal{C}_0^\infty(\mathbb{R})$ we have,
$$\int_{\mathbb{R}} \varphi^k(x)\mu(dx) \leq C$$
Then $\mu(dx) = f(x)dx$ and $f\in ...
2
votes
0answers
36 views
Question about: $C^k$ for $k\geq 1$ dense in the space of Lipschitz functions. Approxmating sequence?
I know that $\mathcal{C}^k$ for $k\geq 1$ is dense in the space of Lipschitz funcions. My question in fact is: If $\{f_n\}_{n\geq 0}\subset \mathcal{C}^k$ such that $f_n \to f$ where $f$ is only ...
1
vote
1answer
141 views
The vector space formed on C[0,1] and the norm $(\int_{0}^{1}|f(t)|^2 dt)^{1/2} $
How does one show that $(C[0,1], ||.||_{2})$ where $||.||_2=(\int_{0}^{1}|f(t)|^2dt)^{1/2}$ and $C[0,1]$ is the space of all continous function which are mapped from $[0,1]\rightarrow \mathbb{R}$, is ...
0
votes
0answers
82 views
Integrating $\ln(1+|\ln|x||)$ in $B_1(0)$
I am trying to integrate $$\int_{B_1(0)} \ln(1+|\ln|x||).$$ $B_1(0)\subset \mathbb R^n$
Basically what I am trying to see is whether $\ln(1+|\ln|x||)$ belongs to $L^\infty (B_1(0))$ and ...
1
vote
1answer
110 views
Is there a continous function which does not have a derivative in any of its points? [duplicate]
Possible Duplicate:
Are Continuous Functions Always Differentiable?
Is there a continous function (continous in every one of its points) which is not differentiable in any of its points?
1
vote
1answer
75 views
Sum of reciprocals, reciprocals of the sum
I have two vectors $a$ and $b$. I have the two following quantities,
$\sum_i a_i \frac{1}{b_i}$ and $\sum_i a_i \frac{1}{\sum_j b_j}$. I know that for every $i$, $0\leq a_i \leq b_i \leq 1$. Which ...
2
votes
1answer
49 views
Non-regularity of non-elliptic operator
Let $\Omega\subset\mathbb{R}^d$ be open,and $P(D)=\operatorname{\sum_{|\alpha|\le N}}f_{\alpha}D^{\alpha}$ be an elliptic differential operator. Rudin proves in Functional Analysis Part II the ...
2
votes
1answer
43 views
Inequality between products of measures of sets
Let $X$ be a compact space and $\mu$ the Lebesgue measure, with $\mu(X)=1$. Let A and B be two subsets of $X$ with positive measure. What can I say about the relations between $\mu (A \cap B)$ and ...
1
vote
1answer
67 views
Diagonalization theorem and convergence
Let $\{f_{n}\}$ be a sequence of pointwise bounded continuous functions on a separable metric space $X$. There is a common diagonalization theorem (see Baby Rudin, Theorem 7.23) which states that if ...
2
votes
2answers
68 views
Bounded function on $\mathbb R$
Is it true that if $0< f(x)$ is a continuously differentiable function on $\mathbb R$ with $\int_{-\infty}^{\infty}|f(x)|^{2}dx<\infty$ then $|f(x)|$ must be bounded above on $\mathbb R$?
