1
vote
0answers
10 views

find a $B_{n,j}$ such that $|A_{n,j}-L_j| \leq B_{n,j}$ $\forall n,j$ and $\sum_{j=0}^{\infty}B_{n,j}$ converges

We have $A_{n,j}= 3(-1)^j2^{n-j+1}\frac{(2(n-j)-4)!}{(n-j)!(n-j-2)!}\binom{j+2}{2}\frac{n^\frac{5}{2}}{8^n}$ and $L_j=(-\frac{1}{8})^j\binom{j+2}{2}\frac{3}{8\sqrt{\pi}}$ So I know $\lim_{n \to ...
1
vote
0answers
28 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
0
votes
0answers
31 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
0
votes
0answers
46 views

How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
11
votes
3answers
222 views

Suggestion for Computing an Integral

Let $$A=\left\{(x,y,z)\in \mathbb R^3:\dfrac{x^2}{2}+\dfrac{y^4}{4}+\dfrac{z^6}{6}\leq1\right\}.$$ Then I want to compute the following integral: ...
-1
votes
2answers
49 views

Differentiability at x=0 [closed]

Discuss the differentiability of the following function in $x$ = $0$: $ f:\mathbb{R} \to \mathbb{R}: x\mapsto \begin{equation} f(x)= \begin{cases} \sqrt{x} & \text{if } x \geq 0 \\-\sqrt{-x} & ...
5
votes
3answers
107 views

How to $\int_{0}^\infty {\sin^3(x)\over x}dx$

How to evaluate : $$\int_{0}^\infty {\sin^3(x)\over x}dx$$ I don't know how to do it. I tried to finish it using integration by parts, but it doesn't work? Can someone tell me how to evaluate the ...
-2
votes
0answers
33 views

Does an lp norm induce a ball topology? [closed]

Namely, does the metric $$||x - y||_p$$ induce the usual ball topology that a metric induces? I wasn't able to find any results regarding this on a quick Google search.
0
votes
1answer
42 views

Questionable Intervals

Find the numbers $X_1 , X_2 , \ldots , X_{10} $ such: $X_1$ is in the interval $[0,1]$. If we divide the interval $[0,1]$ in halves,each half consists of only one of $X_1$ or $X_2$. If we divide ...
1
vote
3answers
36 views

Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
2
votes
1answer
70 views

Positive and négative Parts

we denote by $u^+=\max(u,0)$ and $u^-=\max(-u,0)$ the positive and the negative parts of $u$ we have that $u=u^+-u^-$ my question is : what is $u'$ using $u^+$ and $u^-$ ? and what is ...
1
vote
1answer
39 views

Converge of an inversion to a mirrorring

I want to ask something about a mirroring and a inversion in $\mathbb{R}^n$. An inversion in a sphere with center $m$ and radius $\rho$ can be written as $$ v \ \longmapsto \ ...
1
vote
1answer
20 views

Question about convergence in $H^1_0$

Please how to prove that if $u_n\rightarrow u$ on $H^1_0$ we have that $||u_n||\rightarrow ||u||$ ? Please i need your help Thank you
0
votes
1answer
55 views

Density and convergence

I have a small question: Is it true that if the basis of a space $A$ is dense in a space $B$ ($B\subset A$) then if $u_n\rightarrow u$ in $A$ we have that $u_n\rightarrow u$ in $B$ ?
1
vote
1answer
33 views

Small question about convergence

I have a small question: if i have that $$\int_0^{+\infty}p(t)|u'_n(t)-u'(t)|^2dt\rightarrow 0$$ is it true that $$\int_0^{+\infty} p(t)|u'_n(t)|^2 dt\rightarrow \int_0^{+\infty} p(t)|u'(t)|^2 dt $$ ...
0
votes
0answers
23 views

Question about convergence

If i have that $$\int_0^{+\infty} a(t)|u_n(t)-u(t)|^2 dt \rightarrow 0 $$ how we can deduce that $$\int_0^{+\infty} a(t)|~|u_n(t)|-|u(t)|~|^2 dt \rightarrow 0 $$ where $a>0, a\in ...
2
votes
1answer
82 views
+50

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
4
votes
1answer
75 views

a complicated question about double improper integral

how to evaluate $$\iint_{y\ge x^2+1}{dx\,dy\over{x^4+y^2}}$$ My solution: the initial intergral $$ =2\int_0^\infty \left(\int_{x^2+1}^\infty {dy\over {x^4+y^2}}\right)\,dx = \int_0^\infty ...
2
votes
2answers
36 views

Proove of equality of integrals

I'm currently sitting on the following problem: Let f be in the set of the integrable functions(:=$L^¹(\mathbb{R}^n))$, A $\in \mathbb{R}^{n\times n}$ invertible. Therefore define g:=$\mathbb{R}^n ...
2
votes
3answers
178 views

Derivative and integral of the abs function

I would like to ask about how to find the derivative of the absolute value function for example : $\dfrac{d}{dx}|x-3|$ My try:$$ f(x)=|x-3|\\ f(x) = \begin{cases} x-3, & \text{if }x \geq3 \\ ...
0
votes
2answers
40 views

Study of a function

I have this function $\displaystyle g(s)=\frac{s^{2-\sigma}}{1+s^2}, ~\text{for all} ~s\in \mathbb{R}$ , i need to find the interval of $\sigma$ and the maximum of the function $g$. I calculate the ...
1
vote
1answer
34 views

a question about multivariable integral!

If $\lfloor x \rfloor$ denotes the greatest integer in $x$, evaluate the integral$$ \iint_{R} \lfloor x+y \rfloor ~ \mathrm{d}x~ \mathrm{d}y$$where $R= \{(x,y)| 1\leq x\leq 3, 2\leq y\leq 5\}$. This ...
1
vote
0answers
34 views

On $\sigma$-algebra generated by sets

Given $\mathcal{S}$ a collection of subsets of $X$ and $A\subset X$. To show that $\sigma(\mathcal{S}\cap A)=\sigma(S)\cap A$, where for any collection of $\mathcal C$ of subsets of $X$, $\mathcal ...
1
vote
1answer
20 views

$\sigma$-algebra generated by a set

I want to show that if $X$ is an uncountable set then $\mathcal{S}=\{\{x\}:x\in X\}$ generates the $\sigma$-algebra $\mathcal{A}=\{A\subset X: A$ is countable or $X\setminus A$ is uncountable$\}$. I ...
3
votes
2answers
54 views

Convergence radius of power series is infinite

Which function is given by a power series whose convergence radius is infinite? $$A. \ \ \ e^{-\frac{1}{x^2}}$$ $$B. \ \ \ \sin{\left(\frac{1}{x}\right)}$$ $$C. \ \ \ ...
0
votes
0answers
37 views

Green function of Sturm liouville problem

How to find the Green function of the following problem: $$\begin{cases}-(p(t)u')'+q(t)u=f(t,u), t>0\\u(0)=u(+\infty)=0\end{cases}$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in ...
2
votes
0answers
52 views

General solution of ODE

please what is the general solution of $$-(p(t)u')'+q(t)u=0$$ where $\displaystyle\frac{1}{p},\frac{1}{q}\in L^1((0,+\infty))$ Thank you
0
votes
3answers
52 views

Show that series converges

Show that if $ \{ p_n \} $ is a Cauchy sequence, then it has a subsequence $ \{ p_{n_k}\} $ such that the series $ \sum_{k=1}^\infty b_k $ converges, where $ b_k = d(p_{n_k}, \, p_{n_{k+1}}) $. My ...
0
votes
1answer
27 views

Big-O Analysis: Max Bounded by the Sum

I have been asked to show that: $$ \mathcal{O}(Max\{ f(n), g(n) \}) = \mathcal{O}(f(n) + g(n)) $$ I have seen explanations of similar problems, but this is the first time I have encountered the ...
1
vote
1answer
49 views

Show that series in Cauchy Sequence

Let $a_n = d(p_n, p_n+1)$ for $n = 1, 2,\cdots $. Show that if the series $\displaystyle \sum^{∞}_{n=1} a_n$ converges, then $\{p_n\}$ is a Cauchy sequence. My Approach: I thought of using the ...
1
vote
1answer
27 views

Question about convergence

I have that $v=v^+-v^-$, $v^+,v^-$ are the positive and the négative part of $v$ and i have this: i dont understand why if $v_n\rightarrow v_0$ in $L^p(\Omega)$ then $v_n^+\rightarrow v_0^+$ in ...
1
vote
1answer
46 views

proof of coarea formula for n dimensional hypersurface in $R^n$

$f:R^n \rightarrow R$ be continuous and summable. please give the proof for these formulas $\int_{R^n}f dx = \int_0^\infty(\int_{\partial B(x_0,r)}fdS)dr$ $\frac{d}{dr}\int_{ ...
0
votes
0answers
9 views

Taylors formula

I have a circle $K(a,\epsilon) \subset \Omega $ and for $ \parallel \Delta x \parallel \lt \epsilon $ we look at $ \Delta f = f(a+ \Delta x) - f(a) $ Now I look at the function $ F:[0,1]\rightarrow ...
0
votes
0answers
15 views

Using Taylors formula and epsilon argument

I have a circle $K(a,\epsilon) \subset \Omega $ and for $ \parallel \Delta x \parallel \lt \epsilon $ we look at $ \Delta f = f(a+ \Delta x) - f(a) $ Now I look at the function $ F:[0,1]\rightarrow ...
1
vote
1answer
31 views

Show that the set of 2 continuous functions is closed.

Let $f: \mathbb R \to \mathbb R $ and $g: \mathbb R \to \mathbb R$ be continuous functions. Show the set $ E = \{ x \in\mathbb R: f(x)=g(x)\} $ is closed. My approach A solution I found is the ...
1
vote
2answers
41 views

Show that the sequence converges to 0

Given a sequence $\{\frac{(-1)^n}{n}\}$ show directly from the definition that it converges to $0$. Definition of convergence of a sequence is: A sequence $\{p_n\}$ converges if for every ...
3
votes
1answer
60 views

Show that the limit of $ f(x)$ as $x\rightarrow 0$ is $0$

Show directly from the definition of a limit of a function that lim x->0 (x^(1/3) * sin(1/x)) = 0. The definition is The limit of f as x goes to p is q if ...
8
votes
1answer
65 views

Limit and Holder inequlity

Let $p\neq0$ and $j=1,2,\cdots,n$ and $x_j>0$ and $$\chi(p)=\left(\frac{1}{n}\sum_{j=1}^nx_j^p\right)^\frac{1}{p}.$$ Prove that $\chi$ is strictly increasing and the following statements hold ...
5
votes
2answers
85 views

What's the spectrum of this operator in $\ell^2$?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
0
votes
2answers
32 views

Pointwise and uniform convergence of this series

$$\sum_{n=1}^{\infty}\left(1- \frac{1}{2n}\right)^{-n^2}(x^2-1)^n$$ I've tried treating it as a power series centered around $x = 1$ and $x = -1$ and using root test I arrive to radius of convergence ...
0
votes
1answer
40 views

How do you formally prove $\limsup\limits_{n\rightarrow\infty}(|na_n|^{\frac{1}{n}}) = \limsup\limits_{n\rightarrow\infty}(|a_n|^{\frac{1}{n}})$

My definition of $\limsup$ is it is the supremum of the accumulation points of a sequence (i.e the supremum of the limits of all possible subsequences of a sequence). So if: ...
0
votes
2answers
39 views

Question about limit and continuity

I have that $u_0>0$ , $u_n=u_n^+-u_n^{\raise{1pt}{-}}$ and $u\mapsto u^{±}$ is continuous if $u_n\rightarrow u_0$ why we have that $u_n^+\rightarrow u_0$ and $u_n^{\raise{1pt}{-}}\rightarrow 0 $ ...
2
votes
2answers
64 views

Is the image of this operator on $\ell^2$ closed?

Suppose that $\ell^2 = \biggl\{(x_n)_n \in \mathbb{K}^{\mathbb{N}_0} \biggm| \sum_{n=1}^{\infty}|{x_n}^2| < +\infty \biggr\}$ is a Hilbert-space with the inproduct $\langle\cdot,\cdot\rangle_2: ...
0
votes
1answer
65 views

Small question about calculus

I have this lemme from this paper: "Multiplicity results for quasi-linear problems A.Ayoujil, A.R. El Amrouss, 2008" We consider the truncated problem $$(\mathcal ...
0
votes
1answer
18 views

Injetivity of a function

Let f: U $\rightarrow R^{m}$ differentiable in $U \in R^m$. If $|f(x)|$ is constant in $U$, then for all $x \in U$ $, f'(x)$ is not injective. Hint: Derive the function $||f(x)||^{2}$. And verify ...
1
vote
1answer
57 views

Show that there exists a sequence $\{x_n\}$ such that $f'(x)\to f'(c)$

Suppose that $f'(x)$ exists for all $x \in (a,b)$. Let $c \in (a,b)$, show that there exists a sequence $\{x_n\}$ in $(a,b)$ with $x_n \neq c$ and $x_n \to c$ such that $f'(x_n) \to f'(c)$ This is a ...
0
votes
2answers
71 views

Question on two metric spaces properties

Question: Let $X$ be a set and let $d_1$ and $d_2$ be two metrics on $X$. Assume that there exists a constant $C > 0$ such that $d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$. ...
1
vote
1answer
24 views

Is $ \gamma(t) = \left( A \cos(\sqrt{a} t),B \cos \! \left( \sqrt{b} t \right) \right) $ dense in the rectangle $ [- A,A] \times [- B,B] $?

What conditions must $ a $ and $ b $ satisfy in order for the curve $$ \gamma(t) = \left( A \cos(\sqrt{a} t),B \cos \! \left( \sqrt{b} t \right) \right) $$ to be dense in the rectangle $ [- A,A] ...
1
vote
2answers
47 views

Consider the sequence ${f_n}$ with $n \ge 2$, defined on $[0,1]$ by

Consider the sequence ${f_n}$ with $n \ge 2$, defined on $[0,1]$ by $$f_n(x) = \left\{ \begin{array}{c} n^2x, &0 \le x \le \frac{1}{n} \\ 2n - n^2x, &\frac{1}{n} < x \le \frac{2}{n} \\ ...
0
votes
2answers
32 views

Regarding measurable functions

Let $(X,\mathcal A)$ be a measurable space and let $f:X\to \mathbb R$ and $g:X\to \mathbb R$ be mesurable functions. Let $G$ be an open subset of $\mathbb R^2$. We want to show that $\{x\in ...