Laplace-Beltrami Operator for Euclidean Space
Consider the space $\mathbb{R}^n$ and let $x_1,\ldots, x_n$ be the coordinates. Fix the orientation $dx_1\wedge dx_2\ldots\wedge dx_n$. Let $E^p$ denote the space of smooth $p$ forms and let $d$ denote...
View ArticleHow can I prove $d(A,B)=0 \Rightarrow \partial A \cap \partial B \neq...
The problem is the following:Let $(X,d)$ be a compact metric space, and $A$, $B \subseteq X$ disjoint sets such that$d(A,B):=inf\{d(a,b):a \in A, b \in B\} = 0$Prove that $\partial A \cap \partial B...
View ArticleProve that $f(x) = x$ has a unique solution in $[a,b]$
I see there is a post almost related to this, but I think the conditions are slightly different.Suppose $f$ is differentiable on $(a,b)$ and continuous on $[a,b]$, with $a\leq f(x) \leq b, \forall x...
View ArticleHölder regularity and Hölder condition
Is there any textbooks or papers introducing Hölder regularity and Hölder condition? When I read the paper(https://www.ams.org/journals/tran/2021-374-11/S0002-9947-2021-08489-5/home.html), I found that...
View ArticleZeros of two equations
Consider the equations$$ 1+\frac{1}{z^k}=0 \quad\mbox{and}\quad 1+\frac{1}{z^k}+\frac{1}{(z+1)^k}=0,$$where $k$ is a positive integer $\ge 4$. I would like to show for instance that the number of zeros...
View Articlenonempty subset $E$ of $R$ closed and bounded iff every continuous...
I need to show that a nonempty subset $E$ of $R$ is closed and bounded iff every continuous real-valued function of $E$ takes a maximum value.I believe that "if $E$ is closed and bounded, then every...
View ArticleCompound interest: why is there the exponential
In economy, it’s well known that compound interest at a constant interest rate provides exponential growth of the capital.Why exponential though?The general expression for the compound interest is just...
View Article$I_1=\int_{0}^{\frac{\pi}{2}}\frac{(\ln(\tan x))^2}{1-\sin 2x}dx$ &...
If $$I_1=\int_{0}^{\frac{\pi}{2}}\frac{(\ln(\tan x))^2}{1-\sin 2x}dx$$ and $$I_2=\int_{0}^{\frac{\pi}{2}}(\ln(1-\sin x))(\cot x)dx$$ then evaluate $$\left|\frac{I_1}{I_2}\right|$$My...
View ArticleDo strictly convex functions take asymptote?
Let $f$ be a non-negative strictly convex function i.e. $f(\frac12(u+v))<\frac{f(u)}{2}+\frac{f(v)}{2}$ for any $u\neq v$ and continuous.Is it possible that $f$ takes oblique asymptote that pasaes...
View ArticlePart of proof of the set of continuous integrable functions is dense in...
I want to prove: If $g$ belongs to $L(\Bbb R, \Bbb B, \lambda)$ and $\epsilon\gt 0$, then there exists a continuous function $f$ such that $\Vert g-f\Vert_1=\int \lvert g-f\rvert \,\text{d}\lambda \lt...
View ArticleShowing that the set of a sequence going to $+\infty$ has a minimum
Link to questionThe Question: Let $a_n$ be a sequence of real numbers s.t $a_n \rightarrow+\infty$ as $n \rightarrow +\infty$, and let $A$ be the set of all its terms, i.e, $A=\{a_n\}$. Prove that $A$...
View ArticleCan the fundamental theorems of real analysis be proven/developed without...
I've been reading about philosophical debates between mathematicians, and some seemed to reject the ideas of real analysis based on a school called "intuitionism", where they rejected things such as...
View ArticleLet $a_{n} > 0$ and $\sum a_{n}$ converges. Prove that $\forall p >...
QuestionLet $a_{n} > 0$ and $\sum a_{n}$ converges. Prove that $\forall p > \frac{1}{2}$, $\sum \frac{\sqrt{a_{n}}}{n^{p}}$ converges.Attempt.Since, $\lim_{n \to \infty}a_{n} = 0$. After certain...
View ArticleFinding the Radius and Interval of Convergence for a Power Series
Question:Hello, I'm currently studying power series in my calculus class and I've come across a problem that I'm having trouble with. I'm trying to find the radius of convergence and the interval of...
View ArticleLet $a_{n}$ be integers, such that $\exists N \in \mathbb{N} \ni a_{n} =...
QuestionLet $a_{n}$ be integers, such that $\exists N \in \mathbb{N} \ni a_{n} = (n-1) \forall n > N$. Prove that $\sum \frac{a_{n}}{n!}$ is rationalAttemptI am not able to find the direction in...
View ArticleFinding the Radius and Interval of Convergence for a Given Power Series
I am currently studying power series in my calculus course and I have come across a problem that I am having difficulty with. I am trying to find the radius of convergence and the interval of...
View ArticleProving a variant of closed range theorem on Hilbert space
I've been working on closed range theorem. There are a lot of materials on general Banach spaces, but not much on Hilbert spaces, so I was wondering if I could get some help.I'm trying to prove the...
View ArticleAssumption in Hörmander-Mikhlin multiplier theorem
The well-known Hörmander-Mikhlin multiplier theorem is the following statement:Let $m \in C^N(\mathbb{R}^n \setminus \{0\})$ with $N = [n/2] + 1$ andsuppose for each multi-index $\alpha$ with $|\alpha|...
View ArticleFind Radius and Interval of Convergence
The problem is to find the radius of convergence and the interval of convergence for the following power series:$$\sum_{n=0}^{\infty}\frac{x^n}{3^n(n^2+1)}$$My Attempt:I know that the radius of...
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$n$-th derivative of $\exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$
Let $\lambda$ and $\mu$ be two positive real numbers and let denote $f$ the function defined as:$$\forall x>0,~f(x):= \exp\left(-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right)$$According to WolframAlpha...
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