Riemann-Integral with Jordan-partition
I thougt about a modification of the construction of the Riemann-Darboux-integral in the following way: For a non-negative function $f\colon [a,b] \to \mathbb{R}$ take the upper and lower...
View ArticleWhat happens if we divide $(0,1]$ into the union of countably many subsets?
Let $(0,1] \subset \mathbb{R}$ be the usual interval, where the left endpoint is open and the right endpoint is closed. Assume that$(0,1]$ is the union of countably many subsets $\left\{V_j\right\}_{j...
View ArticleDoes integrability of $f>0$ imply integrability of $\ln (f)$?
In problem 36, chapter 18 of Spivak's Calculus, he asks the following:Prove that for all integrable $f>0$, we have $$\frac{1}{b-a}\int_a^b \ln f \le \ln \left(\frac{1}{b-a} \int_a^b f \right)$$I am...
View ArticleFind $(a-b)^3+(b-c)^3+(c-a)^3$ from $a+b+c=6$, $ab+bc+ca=11$ and $abc=6$...
I first started simplifying the expression:$$(a-b)^3+(b-c)^3+(c-a)^3=3(a-b)(b-c)(c-a)$$I went one more step further and wrote the expression $(a-b)(b-c)(c-a)$ as:$$ab^2-a^2b+b^2c-bc^2+c^2a-ac^2$$Now,...
View ArticleFind the values of $m$ and $n$ for which $\int_{0}^{1} x^n\exp{(-mx)}dx$...
I need to find the values of $m$ and $n$, so that the following integral converges:$$\int_{0}^{1} x^n\exp{(-mx)}dx$$My attempt:Case-1:If $n\ge0$ then the given integral is proper integral irrespective...
View ArticleShow that $\frac{1}{x}$ is not uniformly continuous on $(0, 1)$
I want to use the following criterion:Let $f \colon A \to \mathbb R$ be a function. If there are sequences $x_n, y_n$ in $A$ such that $|x_n - y_n| \to 0$, and $|f(x_n) - f(y_n)| \to a \neq 0 $, as $n...
View ArticleEquality involving non increasing rearrangements.
I am reading "Weakly Differentiable Functions" by William P. Ziemer, and I cannot prove an equality in a remark at page $28$ in the section on Lorentz spaces.The equality involved the so called...
View ArticleProof of $ \int_0^1 \frac{dx }{1+x^n}> \frac{1}{\sqrt[n]{2}} ,\forall...
This question is from AoPS Source, and I give a proof there as following:Proof:Use that result:$$\lim_{n\to\infty}\left(\int_0^1 \frac{dx}{1+x^n} \right)^n=\frac12.$$You can refer Limit for a proof.Let...
View ArticleClik here to view.
Help with this step in Calderon-Zygmund decomposition
I'm currently reading this pdf: https://uu.diva-portal.org/smash/get/diva2:1231351/FULLTEXT01.pdf On Singular Integral Operators, Marcus Vaktnäs. In the fourth line it says "so at least one of $\langle...
View ArticleSpivak's proof of Fubini's theorem
On Spivak "Calculus on Manifolds" p59 he provides a proof of Fubini theorem but there is a party i don't fully understand. We have in the hypothesis that $A\subset\mathbb{R}^n$ and...
View ArticleIntegral $\int^{\pi/2}_{0}...
I need to prove the following relation: $\int^{\pi/2}_{0} \left( \int^{\pi/2}_{0}f(1-{\sin\theta}{\cos\phi})\sin\theta \,\, d\theta \right) d\phi= {\pi/2}\int^1_0f(x)\,\,dx$.My guess is that this...
View ArticleSecond Mean Value Theorem of Integral Proof [closed]
This is from [wikipedia on MVT.]If $G: [a, b] \mathbb{R}$ is a positive monotonically decreasing function and $\phi : [a, b] \mathbb{R}$ is an integrable function, then there exists a number $x \in (a,...
View ArticleNeed some help on baby Rudin theorem 6.15
Following is theorem 6.15 of baby Rudin:If $a<s<b$, $f$ is bounded on $[a,b]$. $f$ is continuous at $s$, then $\alpha(x) = I(x-s)$, then $\int_a^b f d \alpha = f(s)$. $\alpha(x)= I(x-s)$ is the...
View ArticleCheck for continuity and uniform continuity of the given piecewise function
The function is$$f(x) =\begin{cases} 1+e^x, & \text{if } x < 0 \\2+\arctan x, & \text{if } x \geq 0\end{cases}$$Continuity:On the interval $(-\infty, 0)$, $f(x)=1+e^x$ is an elementary...
View ArticleSimple-ish inequality from entropy-inspired dynamical system: Does it have a...
This question introduces a dynamical system inspired by entropy where each step$$(p_1, \dots, p_n) \mapsto (\frac1H p_1 \ln p_1, \dots, \frac1H p_n \ln p_n)$$where $H = \sum_i p_i \ln p_i$ and...
View ArticleHaving trouble proving a real analysis proposition
Suppose $(X,\mathcal{M},\mu)$ is a measure space, and $(X,\mathcal{M}^*,\mu^*)$ is its smallest compeletion, i.e.,$\mathcal{M}\subset\mathcal{M}^*$;$\mu^*|_{\mathcal{M}}=\mu$;any $\mu$-null set belongs...
View ArticleThe difference between Riemann integrable function and Lebesgue integrable...
My professor asked me how to intuitively understand Lebesgue's dominated convergence theorem and what's the effect of the integrable dominated function. More specifically, when we are given a Lebesgue...
View ArticleNonlinear operators on power series
Is it possible to calculate the result of a nonlinear operator applied to a power series?In other words, are there closed form solutions for$$ N(\sum\limits_{n = 0}^\infty a_nx^n) $$whereby $N$ is a...
View ArticleWhat real roots exist for negative numbers? [closed]
I was thinking through the idea that a cube root of a negative number has a solution in the reals, but how a square root does not. More generally, I was wondering what types of "roots" like the square,...
View ArticleConditions for taking limit inside of a sequence of power series.
Given a sequence $f_n\in H(G)$ where $G$ is a region in the complex plane and $0\in G$. Suppose that for each $k\in\{0,1,2,...\}$ the sequence $f_n^{(k)}(0)$ converge as $n\to\infty$, and denote this...
View Article