Does sequence produced by Newton's method converge
$\min f=2x_1^2+x_2^2-2x_1x_2+2x_1^3+x_1^4$. Given open ball$B(r)=\{(x_1,x_2)^T:x_1^2+x_2^2<r^2\}$, where$r=\frac{3-\sqrt{3}}{6}$, $\nabla^2f$ is positive definite on $B(r)$.For what starting points...
View ArticleLimit of a continuous Sobolev function to its boundary
Let $U$ be an open bounded subset of $\mathbb R^n$ and $\partial U$ be Lipschitz.By this answer, for $u \in W^{1, p}(U)$,$$\lim_{r \rightarrow 0} \frac{1}{|B_r(x_0)\cap U|} \int_{B_r(x_0)\cap U} |u(y)...
View ArticleTheorem 8.14 in Apostol's MATHEMATICAL ANALYSIS, 2nd ed:
Here is Theorem 8.14 in the book Mathematical Analysis - A Modern Approach to Advanced Calculus by Tom M. Apostol, 2nd edition:Let $\sum a_n$, $\sum b_n$ be related as in Definition 8.12. Assume that...
View ArticleShow that $\{(x,y)\in\mathbb{R}^2\mid x>0, 0
Show that $M:=\{(x,y\in\mathbb{R}^2\mid x>0, 0<y<x^2\}$ is not a star domain.If I draw a picture, then it's easy to verify that due to the slope of $x^2$ we can always find a point $(a,b)\in...
View ArticleTest if the series $\sum\limits_{n=1}^{\infty}\frac{\sin...
$$\sum\limits_{n=1}^{\infty}\frac{\sin n}{n}(1+\frac{1}{2}+\dots+\frac{1}{n})$$Distributing we have$$\sum\limits_{n=1}^{\infty}\Bigl(\frac{\sin n}{n}+\frac{\sin n}{2n}+\dots+\frac{\sin...
View ArticleWeak derivative of a quotient
Let $w_1,w_2\in W^{1,p}(\Omega)$ with $p\in (1,\infty)$ such that $w_1,w_2>0$ a.e. on $\Omega$ and $\dfrac{w_1}{w_2},\dfrac{w_2}{w_1}\in L^{\infty}(\Omega)$. How can we prove that $\dfrac{w_1}{w_2}$...
View ArticleIs nonnegative $C_c^\infty$ dense in nonnegative $H_0^1$?
I have encountered the same problem as Replacing $C_c^\infty$ by $H_0^1$ in the definition of weak subsolution.I wonder how to prove this statement there.If $\phi\in H_0^1(\Omega)$ and $\phi \geq 0$,...
View ArticleDoes $\limsup_{n \to \infty} f(a_n) = f(x) \implies \limsup_{n\to \infty} a_n...
Hi I was working on the following question : Given a real series $(a_n)_n$ ,$x \in \mathbb{R}$ and $f : \mathbb{R} \mapsto \mathbb{R}$ a strictly increasing function. Does $\limsup_{n \to \infty}...
View ArticleProve that $f$ is integrable$\iff\sum_{k \in Z}2^km(E_k)
Let $E$ be a measurable set. Suppose $f \geq 0$ and let $E_k=\{x \in E_k|f(x) \in (2^k, 2^{k+1}] \} $ for any integer $k$. If $f$ is finite almost everywhere, then $\bigcup E_k = \{x \in E |f(x)>0...
View ArticleShow that the function $f(x) =\begin{cases} x, & \text{if }2\leq x\leq 3 \\2,...
Define $$f(x) =\begin{cases} x, & \text{if }2\leq x\leq 3 \\2, & \text{if } 3<x\leq 4 \end{cases}.$$Prove that the function $f:[2,4]\rightarrow\mathbb{R}$ is integrable.I want to use the...
View ArticleInequality on Sobolev norm
I have a problem with the following inequality. Suppose r and r' are Holder conjugate, i.e. $\frac{1}{r} + \frac{1}{r'} = 1$. Then we have$$|||u|^{\alpha -1} u||_{W^{1, r'}(\mathbb{R}^n )} \leq c...
View ArticleWhy does this implication about this supremum in Axler’s proof of the...
I have a question about the proof of the Heine-Borel theorem in Sheldon Axler’s Measure, Integration & Real Analysis. The theorem and the associated proof are reproduced below.I don’t understand...
View ArticleHow to show that integral of density(kernal) function $\phi_{\sigma}$ over...
I am currently studying a paper published in the "Journal of Approximation Theory" on Neural Networks. We have the following definitions:Definition(Sigmoidal functions):A function...
View ArticleShow that $S:c_0\to c_0$ given by $S( x_1, x_2,....) = (0,x_1,x_2,..)$ is an...
Let $S:c_{0}\to c_{0}$ be given by $S( x_1, x_2,....) = (0,x_1,x_2,..)$. Show that $S$ is an isometry,where $c_0$ is the subspace of $l_{\infty}$ consisting of sequences which converge to $0$.My...
View ArticleLimit of $L^p$ norm
Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$,...
View ArticleNewton's method to solve for $f=ax^2-x+1=0$ [closed]
Let $\{x_k\}$ be sequence generated by Newton's Method for solving$ax^2-x+1=0$, where $0<a\leq\frac{1}{4}$. Suppose $x_0<\frac{1}{2a}$.(a) show $x_k<x_{k+1}<\frac{1-\sqrt{1-4a}}{2a}$(b)...
View ArticleAhlfors function and holomorphic maps on an annulus shappe region
The Ahlfors function, which in the case of the annulus is a 2-to-1 mapping of the annulus onto the unit disk, which extends to be holomorphic on a neighborhood of the annulus and which maps each...
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Another stuck on proof of theorem 8.5 in baby rudin [closed]
At first following is Theorem 8.5 and its proof in Rudin.I don't know why we need the process from "Let A" to "desired conclusion".Actually without it, Rudin just show that $f(x)=0\ \ \ for \ \ \...
View Article$\lim {b_n}^{c_n} = e^{\lim c_n (b_n - 1)}$ when $b_n \rightarrow 1$ and $c_n...
The notes I'm reading say that:$\lim {b_n}^{c_n} = e^{\lim c_n (b_n - 1)}$ when $b_n \rightarrow 1$ and $c_n \rightarrow \infty$This is my attempt to prove this fact given that if $a_n \rightarrow...
View ArticleDoes the series $\sum\limits_{n=1}^\infty a_n$ converge if $a_n \geq 0$ and...
I'm given the following problem: Does the series $\sum\limits_{n=1}^\infty a_n$ converge if $a_n \geq 0$ and $\sqrt[n]{a_n}<1$ for $n \in \mathbb N$?(It's a bit unclear to me if "for $n \in \mathbb...
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