Almost dominated convergence from Tao's measure theory book
I am trying to solve the following exercise from Tao's measure theory book (Exercise 1.4.46).Let $(X, B, \mu)$ be a measure space, and let $f_1, f_2, \ldots: X \to C$ be a sequence of measurable...
View ArticleDetermine all the values of $x\in \mathbb{R}$ for which the series...
Determine all the values of $x\in \mathbb{R}$ for which the series below converges$$\sum_{n=1}^\infty \frac{x^n}{1+n|x|^n} $$This is a phd qualifying exam problem that I'm doing for practice. One of...
View ArticleA question about how substitution methods work in integration
In trigonometric substitution, it's common to say $t=\tan(x/2)$ for integrals involving $R(\sin x,\cos x)$. However, this is also done when $\tan(x/2)$ can be undefined for values where the integrand...
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Dog bone-shaped curve: $|x|^x=|y|^y$
EDITED: Some of the questions are ansered, some aren't.EDITED: In order not to make this post too long, I posted another post which consists of more questions.Let $f$ be (almost) the implicit...
View ArticleA type of differentiation under the integral sign
This question arises in a problem I'm working on in climate economics. Let $X\left( s \right)% MathType!MTEF!2!1!+-% feaahCart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn%...
View ArticleProving that a sequence converges, with the epsilon definition
I want to prove that the following sequence $x_n = \frac{3+\sqrt{n}}{2n-\sqrt{n}}$ converges and has a limit.$$\lim_{n \to \infty} \frac{3+\sqrt{n}}{2n-\sqrt{n}} \Rightarrow \lim_{n \to \infty}...
View ArticleHow does the size of $\{x: f(x) \text{ is locally constant}\}$ relate to the...
I have been wondering (though I am not asking in this post) whethera function $f:\mathbb{R}\to\mathbb{R}$ is locally constant almost everywhere $\iff$$\text{im}(f)$ is countable.where by "locally...
View ArticleIs it possible to show my cover has no finite subcover?
I am currently trying to show that the set $S = \mathbb{Q} \cap [0, 1]$ is not compact by showing the cover$$\mathcal{C} = \left\{ \left( \frac{p}{q} - \frac{1}{10^q!}, \frac{p}{q} + \frac{1}{10^q!}...
View ArticleDoes symmetricy of relation implies that it consists only of same elements in...
I am reading Dudley's "Real analysis and probability" paragraph 1.2.There are concept of relation $E$ which is set of partial orderings ($<x,y>=\{\{x,y\},x\}$).Inverse relation is...
View ArticleSmoothness of expectation of a piecewise function
Suppose $f(x)$ and $g(x)$ are piecewise smooth functions. For simplicity, we can assume that $f(x)$ has $m$ pieces, and $g(x):=\max_{i=1,2,\ldots, I}\left\{k_i~ x+b_i\right\}$.I have two questions:Is...
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Convergence of the sequence of the n-th orthocenter, incenter and centroid of...
Given a triangle $A_1B_1C_1$ from the triangle $A_nB_nC_n$ construct a triangle $A_{n+1}B_{n+1}C_{n+1}$ where $A_{n+1}$ is the orthocenter of $A_nB_nC_n$, $B_{n+1}$ is the incenter of $A_nB_nC_n$ and...
View ArticleTranslation of odd and even functions
Let $\varphi: \mathbb{R} \to \mathbb{R}$ be a periodic function of period $L>0$, that is,\begin{equation}\label{periodicitycondition}\varphi(x+L)=\varphi(x),\; \forall\; x \in \mathbb{R}....
View ArticleFor all $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such that $a'\in...
The $\mathbb R^n$ space is partitioned into three sets $A,B,C$.Given:$B$ is convex.$A,C$ are non empty.For all open segment $(a,c)$ where $a\in A$ and $c\in C$, there exists $a',c'\in (a,c)$ such that...
View ArticleIs my proof that lim sup $(c_n^{\frac{1}{n}})_{n=m}^{\infty} \leq$ lim sup...
Let $(c_n)_{n=m}^{\infty}$ be a sequence of positive numbers.$L$ := lim sup $(\frac{c_{n+1}}{c_n})_{n=m}^{\infty}$. Choose any $\epsilon > 0$, this implies that for some $N \in \mathbb{N}$ we have...
View Articlehow do I find all $k \in \mathbb{R} $ such that $\lim_{n \to \infty} a_n =...
I have a sequence $(a_n)_{n \in \mathbb{N}}$ of positive terms defined by the recurrence relation$\frac{a_{n+1}}{a_n} = \left(\frac{2n}{2n + k + 4}\right)^{2n},$where $k \in \mathbb{R}$ and $k \neq...
View ArticleContinuity on the torus
Let $\mathbb{T}^d$ be the $d$-dimensional torus, for all $r >0,\eta_r(x):=e^{-r|x|^2},x \in \mathbb{T}^d,$ we denote by $\mathscr{F}^{-1}\eta_r$ the inverse Fourier transform defined by...
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Upper bound of polynomial integral
In my recent research, I encountered the following integral inequality$$\int_0^1kx^{k-1}(1+x)^{2k+1}\mathrm{d}x<2^{2k},$$where $k$ is a positive integer. This inequality can be transformed to...
View ArticleShow that $f(x) = x^3 - 6x$ has no minimum on $[0,2]_{\mathbb{Q}}$.
I am doing Exercise 2.3.2 in Tom Lindstrom's real analysis book Spaces. It readsShow that the function $f:\mathbb{Q} \to \mathbb{Q}$ defined by $f(x) = x^3 - 6x$ is continuous at all $x \in...
View ArticleSupport of translation in $\mathbb {R}^n$
Let $u$ a numerical function measurable in $\mathbb {R}^n$. We define the translation of $u$ by the vector $y$ as $(\tau_y u)(x)=u(x-y)$. Show that$$supp \, (\tau_y u)=y+ supp \, (u).$$The definition...
View ArticleIntegral function of bounded variation function derivative
Let $f: [a,b] \to \mathbb{R}$ be bounded variation. So $f’$ exists almost everywhere, and let$g(x):=\int_a^x f’(y)dy$.(Due to the fact that it is possible that $f\notin AC([a,b])$ it is not necessarily...
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