A big confusion about Lemma 16.2 in Munkres' Analysis on Manifolds
In Munkres's Analysis on Manifolds, page 137 Lemma 16.2, it states: Step 1. Let $D_{1},D_{2},\cdots$ be a sequence of compact (rectifiable) subsets of (the open set) $A$ whose union is $A$, such that...
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Which continuity class $C^k$ fulfills the following function?
Which continuity class $C^k$ fulfills the following function:$$f(x)=\begin{cases} 0,\quad |x|\geq 1; \\ x\ \ln(x^2)\ e^{\frac{x^2}{x^2-1}},\quad |x|<1;\end{cases}$$I am confused about what happens...
View ArticleProving the closure of a set in $R^2$
I’m having trouble calculating the closure of this set,$A = \left\{ (x, y) \in \mathbb{R}^n : 0 < y < x^2 \wedge 0 < x < 1 \right\}.$My solution is as follows: let's define another set$B =...
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How to prove that a multidimensional function has a unique maximum value
I have a $N$-dimensional function denoted as $f(\boldsymbol{x})=a (\sum_{i=1}^N \theta_i x_i)^{b-1}-\sum_{i=1}^N \frac{c}{x_i^2} $, where $a>0$, $b\in(0,1)$, $c>0$ are constant parameters.How can...
View ArticleConjecture regarding the series $\sum_{k=0}^\infty\frac{\tan\left(...
In 2018 this question was posted on AoPS:Prove that$$\sum_{k=0}^\infty\frac{\tan\left( \frac{2k+1}{2}\pi\color{red}{\sqrt{15}}\right)}{(2k+1)^3}=-\frac{\pi^3}{32\color{red}{\sqrt{15}}} $$Before...
View ArticleStrategies to find suitable function to apply MVT
1.Let $a, b$ be two positive numbers, and let $f:[a, b] \rightarrow \mathbf{R}$ be a continuous function, differentiable on $(a, b)$. Prove that there exists $c \in(a, b)$ such that$$\frac{1}{a-b}(a...
View ArticleLimit of $4^n\cdot (x_n-2)$, where $x_{n+1}=\sqrt{x_n+2}$
Let’s define a sequence $x_1=4$ and $x_{n+1}=\sqrt{x_n+2}$ for $n\geq 2$. We know that this series converges to 2. But what is the limit of $4^n\cdot (x_n-2)$?I know that $4^n\cdot (x_n-2)$ is a...
View ArticleAre strictly increasing functions with a three-point property necessarily...
Let $I$ be an interval of the real line. Consider two functions $f,g: I \to \mathbb{R}$ such that $f,g$ are strictly increasing and satisfy the following three points property: for all $a,b,c \in I$,...
View ArticleComputing the derivative: $\partial_x |\max(f_1(x),f_2(x), f_3(x))|$
Let $f_i$ with $i=1,2,3$ be $C^1$ function from $\mathbb{R}$ to $\mathbb{R}$. I want to compute the derivative of$$h(x) = (|\max(f_1(x),f_2(x), f_3(x))|-x^2)^2.$$so$$h'(x) = 2 g(x)(\partial_x...
View ArticleExplicit subsequence of $(\sin n )$ [duplicate]
The existence of a subsequence of the sequence $(\sin n)$ coming from the Bolzano Weierstrass theorem ,as we know any infinite bounded sequence has a convergent subsequence. But I can't find such...
View ArticleExercise on real numbers [closed]
Let $A$ be the set of real numbers defined by :$$A = \left\{nm+\frac{1}{nm} \; \middle|\; n \in \mathbf{N}^*, m\in \mathbf{N}^* \right\}$$Show that : $\max(A) = 2$.Show that : $\inf(A) = 1$.Deduce from...
View ArticleIs the adjoint $L^*$ of a bounded linear function $L$ on $C(X;\mathbb{R})$...
Let us start with some definitions: let $X$ be a compact metric space, abbreviate $C(X) := C(X;\mathbb{R})$, let $L$ be a bounded linear function $L:C(X)\to C(X)$, and equip $C(X)$ with, say, the...
View ArticleReal function resembling a sequence of numbers given by $(f(n)+P(n))\pmod{Q(n)}$
A while ago I have asked this question:here about the nature of a certain series which turns out it is divergent in most cases. Now I'm interested about the numbers yielded by nominators of such...
View ArticleAbout acceleration of the convergence of a sequence over another one.
Assume $u_n\to u$ in $L^2(\Omega)$, that is $\|u_n-u\|_{L^2(\Omega)}\to 0$. Let $\phi_\delta=\delta^{-d} \phi(\frac{x}{\delta})$ be a nice molifier $\phi\in C_c^\infty{B_1(0)}$, $0\leq \phi\leq 1$ and...
View ArticleClosed form for an integral with exponential
I would like to know if it is possible to write the following integral in a closed form:$$\int_0^1 \frac{\rho^{c+ t\alpha}}{(1-\rho^{2 \alpha})^d} e^{-\frac{1+\rho^{2 \alpha}}{1-\rho^{2 \alpha}} a}...
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Could this function be a traveling solution to the 1D wave equation?
I am trying to understand when solutions of the wave equation could have issues, so I made a continuous function which some extreme behaviors: has compact-support given by a smooth bump function term,...
View ArticleAsymptotics of the Maximum Diagonal Sum of Orthogonal Matrices for Large $n$
QuestionLet $A$ be a real square matrix $[a_{ij}]$ and let $D(A)=\sum_{i\geq j}a_{ij}$ . Find the maximum of $D(A)$ for the group of orthogonal matrices of given order $n$ ; and show that , as...
View ArticleA continuous function can be bounded from above by a smooth 'adapted' function?
Consider a continuous function $f:\mathbb{R}\to\mathbb{R}$. Is there a standard strategy that produces a function $g\in C^{\infty}(\mathbb{R},\mathbb{R})$, such that$f(t)\le g(t)$$\forall t$;$g(t)$ can...
View Article$E h(X_n,Y_n)\rightarrow E h(X,Y)\forall h\in C_b(\mathbf{R}^2)\iff E...
I want to show that $\mathbb{E} h(X_n,Y_n)\xrightarrow{n\to\infty}\mathbb{E} h(X,Y)$ for any bounded continuous function $h\in\mathcal{C}_b(\mathbf{R}^2)$. Show that it suffices to show this for...
View ArticleA confusion about Lemma 16.2 in Munkres' Analysis on Manifolds
In Munkres's Analysis on Manifolds, page 137 Lemma 16.2, it states: Step 1. Let $D_{1},D_{2},\cdots$ be a sequence of compact (rectifiable) subsets of (the open set) $A$ whose union is $A$, such that...
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