Given $f^\prime(p) = f^{\prime\prime}(p) = 0, f^{\prime\prime\prime}(p)\neq...
Let $ f:(a,b)\to \mathbb{R}$ and $p\in (a,b)$. Assume that $f$ is $3$ times differentiable at p, with $$f^\prime(p) = f^{\prime\prime}(p) = 0, f^{\prime\prime\prime}\neq 0.$$Prove that $p$ is not a...
View Article$U\in \mathbb{R}^2$ bounded, smooth edge, $0\in U$, $f\in C_0^{\infty}(U)$....
$U \in \mathbb{R}^2$ is a bounded region with a smooth edge, such that $0 \in U$, $f \in C_0^{\infty}(U)$$$V(x) = \begin{cases} \dfrac{x}{\|x\|^2} & \text{for } x \neq 0 \\ 0 & \text{for } x =...
View ArticleSpecial case of L’Hopital for functions differentiable at a single point...
I was given the following homework problem:Let $x \in \mathbb{R}$ and $f,g \colon \mathbb{R} \to \mathbb{R}$ be differentiable at $x\in \mathbb{R}$.If $f(x)=g(x)=0$ and $g'(x) \neq 0$, show...
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Calculating an isosurface of an integral along a line segment in 3d
The motivation behind question is inspired by "metaballs", a fun computer graphics exercise in rendering isosurfaces of a field. In the classic 3D metaballs problem, you have a series of points...
View ArticleIs the set $\{\sqrt n - \lfloor\sqrt n\rfloor : n\in \mathbb Z^+\}$ dense in...
I was reviewing basic Analysis and thought of this question:Is the set $\{\sqrt n - \lfloor\sqrt n\rfloor : n\in \mathbb Z^+\}$ dense in $[0,1]$?Lest we have any ambiguity with notational stuff,...
View ArticleDense set of numbers
Show that the set $\{\sqrt{n}-[ \sqrt{n}]; n\in \Bbb N\}$ is dense in $\Bbb [0,1]$. I showed that for every real number $a$ such that $\ 0<a<1 $ there is at least one number $x$ from the set such...
View ArticleLimit related to $f(n) = \frac{f(n-1)^2 (f(n-1)+1)^2}{f(n-2)(f(n-2)+1)^3}$
Let $f(1) = f(2) = 1$and for $n>2:$$$f(n) = \frac{f(n-1)^2 (f(n-1)+1)^2}{f(n-2)(f(n-2)+1)^3}$$So we get$f(3) = 1/2,$$f(4) = 0.07031..f(5) = 0.00335..f(6)= etc$and the sequence is going to $0$ in the...
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Question about Spivak's proof of the existence of a partition of unity in...
Below I include images of the proof provided in the book.In the second case, the one where $A = A_1 \cup A_2 \cup \dots$ where the $A_i$'s are compact and $A_i \subseteq A_{i+1}^\circ$,the author...
View ArticleA lower bound for a quadratic form on $\mathbb{R}^2$
Let $a, c > 0$ and $b \in \mathbb{R}$. Show that for all $(x,y) \in \mathbb{R}^2$,\begin{equation*}a x^2 - 2bxy + cy^2 \ge \frac{ac - b^2}{2}\min(a^{-1}, c^{-1}) (x^2 + y^2).\end{equation*}This...
View ArticleIntegrate: $\int \frac{\sin x}{\sin ^{-1}x}dx$
I just faced an integral:$$\int \frac{\sin x}{\sin ^{-1}x}dx$$All I can do is, consider the denominator as $z$ and substitute:$$x=\sin z$$$$dx=\cos z$$And finally, I have:$$\int \frac{1}{z}\sin (\sin...
View Article$\omega = \frac{x(x^2+y^2 - 1)}{(x^2+y^2 + 1)^2-4x^2}dy+ \frac{y(x^2+y^2 +...
I found such problem with solution.Problem:We have:$C1 = \{ (x, y) : (x-1)^2 + y^2 = \frac{1}{4} \}$$C2 = \{ (x, y) : x^2 +y^2 = 4 \}$$C3 = \{ (x, y) : (x + 1)^2 + y^2 = \frac{1}{4} \}$Those are...
View ArticlePage 39, Exercise 22 "Measure, Integration & Real Analysis - Sheldon Axler"...
Suppose $ B \subseteq \mathbb{R} $ and $ f: B \to \mathbb{R} $ is an increasing function. Prove that $ f $ is continuous at every element of $ B $ except for a countable subset of $ B $.Please provide...
View ArticleAre the set of squares (and also higher powers) of positive rationals dense...
Consider the set $S$ of squares of positive rationals. For example, $1/4$ is in $S$, but not $1/2$. I wonder, is $S$ dense in the positive reals? A similar question can be asked for 3rd powers, fourth...
View Article$-\boldsymbol{p}\log \boldsymbol{p} \prec H(\boldsymbol{p})\boldsymbol{p}$:...
For any probability vector $\boldsymbol{p}=(p_1,\dots,p_n)\ge 0$ with $\sum_{i=1}^np_i=1$, I guess that the following majorization relation holds$$ -\boldsymbol{p}\log \boldsymbol{p} \prec...
View ArticleStrong convergence in $L^1 \implies$ weak convergence in $H^{-1}$?
Let $U$ be a bounded open set in $\mathbb{R}^3$. Prove or disprove:Strong convergence in $L^1(U) \implies$ weak convergence in $H^{-1}(U).$Attempt: Suppose $f_n \to f$ in $L^1(U)$ strongly. As the...
View ArticleIdentifying elements of $H^{-1}$
Let $U$ be a bounded open set in $\mathbb{R}^n$ for $n \geq 3$.Let $f \in H^{-1}(U) \cap L^2(U)$.I think the action of $f$ on $g \in H_0^1(U)$ must be $$\big<f,g\big>=\int_U fg$$Indeed $f$ is...
View ArticleHow to prove from the axioms of complete ordered fields that every bounded...
Let $S$ be a subset of $\mathbb{R}$ that is bounded from both above and below. How does one prove, from the axioms of complete ordered fields, that $S \cap \mathbb{Z}$ is finite?
View ArticleUnderstanding an argument for a countable number of discontinuities of a...
I came across this argument but I'm having some trouble with it. $f$ is from $\mathbb{R} \to \mathbb{R}$, $f(a\pm 0)$ is the right and left handed limits of $f$ at $a$ respectively.Assume that $f$ is...
View ArticleDensity of the sequence $(\sqrt{n}-\lfloor \sqrt{n}\rfloor:n\in\mathbb{N})$...
Show that the set $\{\sqrt{n}-[ \sqrt{n}]; n\in \Bbb N\}$ is dense in $\Bbb [0,1]$. I showed that for every real number $a$ such that $\ 0<a<1 $ there is at least one number $x$ from the set such...
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Would We have the preimage of at least one interval taken out from $[0;1]$,...
This is the theorem: Let $X$ be separable metric space endowed with non-atomic Borel measure such that $\mu X = 1$.Using this theorem We can establish isomorphism between $X$ and $[0;1]$.Denote this...
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