What does an "exotic" derivation at a point $x_{0}\in \mathbb{R}^{n}$ look like?
for $x_{0}\in \mathbb{R}^{n}$ I denote by $G^{k}(x_{0})\: (1\le k\le \infty )$ the $\mathbb{R}$-algebra of germs of real-valued $C^{k}$-functions at $x_{0}$. Recall that a derivation at $x_{0}$ is a...
View ArticleNon trivial closure of a set
Let$$A=\{(x,y)\in\mathbb{R}^2 : x^3-x^2 -y^2 >0\}$$I would like to find its closure, however I do not see a reasonable way to find it.I first thought to relaxing the...
View ArticleProof verification: If $f$ is continuous and $|f(x)| \leq 1$, then $\lim_{n...
So I wanted to prove the following lemma:Lemma. Suppose $f:[a,b] \to \mathbb{R}$ is continuous and satisfy $|f(x)|\leq 1$ for all $x \in [a,b]$, where equality occur at only finitely many points. Then...
View ArticleCan this improper real integral be solved without complex numbers? [closed]
I'm taking a complex variables course, and we had this question on a homework:$$\int_0^\infty \dfrac{x \sin(2 x)}{x^2 + 3} dx$$I know how to solve this, but I was curious if this can possibly be solved...
View ArticleQuery about a Sobolev inequality (interpolation)
Let $\Omega$ be a domain in $\mathbb{R}^n$ with smooth boundary. Let $\phi \in C_c^{\infty}(\bar{\Omega})$ (yes, in the closure of $\Omega$ so support of $\phi$ can include the boundary of $\Omega$)....
View ArticleFinding a bounded and divergent sequence $(a_n)$ with...
I want to find a bounded and divergent sequence $(a_n)$ with $$\lim_{n\to\infty}a_{n+1}-a_n=0,$$and I prefer that $a_n=f(n)$ where the function $f$ is elementary, so that this answer is not relevant....
View ArticleTaylors theorem from $R^m$ to $R^n$
I am trying to understand the first order taylor approximation for functions of several variables from $R^m$ to $R^n$. But, I can’t find a single source online!$$f(x) = f(a) + Df(c)(x-a)$$Here $Df(c)$...
View ArticleLimit of integral of sum of cosine functions by CLT?
I want to show that $$\lim_{n\to \infty} (2\pi)^{-d}n^{d/2}d^{-2n}\int_{[-\pi, \pi]^d} (\cos(x_1)+\cdots +\cos(x_d))^{2n} dx_1\cdots dx_d =2(d/4\pi)^{d/2}$$ holds.How do I prove this?It seems that the...
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Example of a pointwise convergent sequence not convergent in $L^1$.
I'm trying to solve the following problem and I have this solution. So the solution shows that $f_n-f$ is not even in $L^1$ space, when we need $\|f_n-f\|_1\to 0$ to show that the sequence converges to...
View ArticleLagrange multipliers in Calculus of Variations
I am trying to learn about Calculus of Variations and I am beginning to see some constrained optimization problems in the domain of functionals, by using Lagrange multipliers. It seems that things work...
View ArticleFind lim sup and lim inf of $((-1)^n+1)+1/(2^n)$
Here is what I have, but I do not know if my understanding of lim sup and lim inf are correct:lim sup= 2because the smallest sup of the sequence would be $(1+1)+0=2$lim inf=0because the largest inf of...
View ArticleLet $\{a_{n}\}$ be a positive decreasing sequence such that $\lim_{n\to...
Let $\{a_{n}\}$ be a positive decreasing sequence such that $\lim_{n\to \infty}na_{n}$ goes to 0. Does it follow that $\sum a_{n}$ converges?ThoughtsI managed to solve the converse of it. But am not...
View ArticleShow that if $a, b \in \mathbb{R}$ and $a \neq b$, the intersection of the...
I know the $\epsilon$-neighbordhood for a $\in \mathbb{R}$ is defined as the set $V_\epsilon(a)$ of $x \in \mathbb{R}$ such that |x-a| < $\epsilon$. To prove that the intersection of two...
View ArticleFind all values $c\in\mathbb{R}$ for which the improper integral...
Find all values $c\in\mathbb{R}$ for which the improper integral $$\int_0^\infty\frac{x^c}{\sqrt{x^2+x}}dx$$ converges.If $c\le\frac{1}{2}$, then the integrand $f(x)$ is monotonically decreasing. Then...
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What is the probability that the absolute value of the roots of a polynomial...
Let $f(x) = 0$ be an equation of degree $n$. WLOG we can assume that the its coefficients are in $(-1,1)$. This is because we can divide each coefficient by the coefficient with the largest magnitude...
View ArticleTo which space $u$ should belong to have that both $u$ and $(-\Delta)^s u$...
The condition $u\in W^{2, \infty}(\mathbb R^n)$ guarantees that $u$, its first and second derivatives, are bounded. In particular, the $(-\Delta)u$ happens to be bounded as well.Take $s\in (0, 1)$....
View ArticleThe Closure of a Subset of a Metric Space is the Union of the Subset and Its...
I'm trying to prove the following statement. I can prove it with words, but I'm trying to see how the logic checks out.Let $(\Omega ,d)$ be a metric space. Assume the following definitions of the set...
View Article$f, g :\Bbb R\rightarrow\Bbb R$ be continuous functions whose graphs do not...
Given $f, g :\Bbb R\rightarrow\Bbb R$ be continuous functions whose graphs do not intersect. Then for which function below the graph lies entirely on one side of the $x$-axis(1). $f$(2). $g+f$(3).$...
View ArticleHomeomorphism between a subset of $\mathbb{R}^3$ and $\mathbb{R}^2$.
I consider the set, for $\varepsilon>0$ fixed :$$E=\{(x,y,z)\in\mathbb{R}^3 :x\in\mathbb{R},z\in\mathbb{R}, \exists d\in(0,\varepsilon), y= d-z\}$$We first observe that we have two free variables,...
View ArticleWhat is this sequence? What is the asymptotic order for this sequence?
I just came up with the sequence where a term is defined as the sum of its last predecessor with the average of all predecessors.$$a_{n+1} = a_{n} + \frac{a_1 + a_2 + ... + a_n}{n}\\$$When $a_1 = 1$,...
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