About the Hölder continuity of the power of a function
Let $\Omega$ be a bounded domain and assume that $f: \Omega \subset \mathbb{R}^N \to \mathbb{R}$ is a function in the space $C^{2,\alpha}(\overline{\Omega})$, $0 < \alpha <1$, such that $f(x)...
View ArticleIs the best degree $n$ polynomial approximation an interpolation on $L^2[0,1]$?
The Question:Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous. Let $p:[0,1] \rightarrow \mathbb{R}$ be the $L^2$-closest degree $n$ polynomial to $f$. That is, $p$ minimizes $\int_0^1|f(x) - p(x)|^2...
View ArticleOne variable inequality with parameter
Using the power-mean inequality, one can prove that for all $p\geq 1$ and for all $x\geq 0$ we have that $$(x^{p+1}+1)^2\geq \left(\frac{x^2+1}{2}\right)^p(x+1)^2.$$The wolfram suggests that this...
View ArticleUnderstanding the Proof of Relative Openness Theorem in Rudin's Principles of...
I'm having trouble understanding Rudin's proof for the theorem stating:"Suppose $Y \subseteq X$. A subset $E$ of $Y$ is open relative to $Y$ if and only if $E = Y \cap G$ for some open subset $G$ of...
View ArticleSimplest proof of Taylor's theorem
I have for some time been trawling through the Internet looking for an aesthetic proof of Taylor's theorem.By which I mean this: there are plenty of proofs that introduce some arbitrary construct: no...
View ArticleRatio of central limit theorem
Suppose $\{X_n\}_{n\in\mathbb{N}}$ and $\{Y_n\}_{n\in\mathbb{N}}$ are two sequences each consisting of iid random variables. I would like to compute the following:$$T = \lim_{d \to \infty}...
View ArticleTheorem 3.54 (about certain rearrangements of a conditionally convergent...
Here's the statement of Theorem 3.54 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition:Let $\sum a_n$ be a series of real numbers which converges, but not absolutely. Suppose...
View ArticleI need feedback for my proof of the Dominated Convergence Theorem in IR.
I'm currently going through all of the theorems in my course on Measure Theory, and when proving the Dominated Convergence theorem I came up with a proof that is very convoluted compared to the way it...
View ArticleProof verification: Show that if a series is conditionally convergent, then...
Show that if a series is conditionally convergent, then the series obtained from its positive terms is divergent, and the series obtained from its negative terms is divergent.Please, help me to verify...
View ArticleAn idea for this difficult integral
I am being stuck in caculating this integral: $$J=\int_{-\tfrac{1}{2}}^{\tfrac{1}{2}}\dfrac{\arccos x}{\sqrt{1-x^2}(1+e^{-x})}dx$$ I tried to change to another variable: $x = - t$ then $dx = - dt$,...
View ArticleWhat is wrong in my solution for the following PDE $u_x^2+u_y^2=1$ with...
I am asked to solve the boundary value problem on a PDE about $u:\mathbb{R}_{\ge 0}^2\cap \{(x,y): x^2+y^2\ge 1\}\to\mathbb{R}$:$$\begin{cases}u_x^2+u_y^2=1 & x^2+y^2>1,\, x,y>0\\u(\cos...
View ArticleA question on measurable/non-measurable sets
Let $\sigma$ be a Borel measure on $\mathbb{R}$. Suppose we know that all Borel subsets $B \subseteq \mathbb{R}$ which contain points satisfying a certain fixed property (let's say property P) are...
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An "almost" geodesic dome
A regular $ n$-gon is inscribed in the unit circle centered in $0$.We want to build an "almost" geodesic dome upon it this way: on each side of the $n$-gon we build an equilateral triangle whose...
View ArticleEquidistribution Theorem for Vectors
I have seen the following question, where the equidistribution theorem can be generalized to 2 dimensions under certain conditions: Equidistribution in higher dimensions.I was wondering if the same...
View ArticleBuilding interval where $f$ is positive
Let $f:\Bbb R \to \Bbb R$ be a differentiable function such that $\exists x $ satisfying $f(x)>0$ and $f'(y) <0$ whenever $f(y)>0$. I want to show that $f$ is always positive before $x$.I...
View ArticleOn the bounding of a series with max or sup
I am interested in knowing if this is true or if not or if "it does depend".Say I have a series $$\sum_{k = 0}^{+\infty} a_k\cdot b_k$$Where $a_k, b_k$ are well defined sequences. Suppose $b_k$ is a...
View ArticleL2 boundedness of normalized graphon
Let $W: [0, 1]^2 \to [0, 1]$ be a graphon, i.e. a symmetric measurable function. Let $d(x) := \int_{[0, 1]} W(x, y) \, \mathrm{d}y$ be its degree function. Set$$W'(x, y) \: := \: \frac{W(x,...
View Articlecurve integral - intersection between plane and sphere
I am going to calculate the line integral$$ \int_\gamma z^4dx+x^2dy+y^8dz,$$where $\gamma$ is the intersection of the plane $y+z=1$ with the sphere $x^2+y^2+z^2=1$, $x \geq 0$, with the orientation...
View ArticleProof of the Intermediate Value Theorem in Knapp
Theorem 1.12 (Intermediate Value Theorem). Let $a<b$ be real numbers, and let $f : [a, b] →ℝ$ be continuous. Then $f$, in the interval $[a, b]$, takes on all values between $f (a)$ and $f (b)$....
View ArticleRudin Ch 4 excercise 3: the zero set of a continuous function is closed
Let $f$ be a continuous real function on a metric space $X$. Let $Z(f)$ (the zero set of $f$) be the set of all $p \in X$ at which $f(p) = 0$. Prove that $Z(f)$ is closed.My attemptLet $p$ be a limit...
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