Why can't WolframAlpha compute the series $\sum_{n=1}^{\infty}...
Why can't WolframAlpha compute the series$$\sum_{n=1}^{\infty} \frac{\mu(n)}{e^{2n}-1}$$where $\mu$ is the Möbius function? The series is obviously absolutely convergent (summands dominated by...
View ArticleFunctions differentiable at a point from everywhere continuous nowhere...
I'm looking at an example about functions that are differentiable at a single point, and it starts with an everywhere continuous nowhere differentiable function $g:\mathbb{R}\rightarrow\mathbb{R}$,...
View ArticleUnderstanding a use of Tychonoff's theorem
I'm not well-versed in topology and encountered this as part of a larger argument about coloring the real numbers:"An arbitrary Cartesian product of compact topological spaces is compact (Tychonoff's...
View ArticleNorm of bounded operator on a complex Hilbert space.
It is fairly easy to show that for a bounded linear operator $T$ on a Hilbert space $H$$$\|T\|=\sup_{\|x\|=1,\|y\|=1}|\langle y, Tx \rangle |.$$If $H$ is a complex Hilbert space, can you show...
View ArticleIf $T$ is self-adjoint then $||T^n|| = ||T||^n$
Let $T$ be a bounded linear operator on a Hilbert space $H$. If $T$ is self-adjoint then $||T^n|| = ||T||^n$.It is easy to see that $||T||^n$ is an upper bound. Indeed, there exists a $C>0$ such...
View ArticleProve: Define $\nu$ on $\mathscr{A}$ by $\nu(A) = \int_Afd\mu$. Then $\nu$ is...
I am self-studying measure theory and I come across the following question:Prove: Let $(X,\mathscr{A},\mu)$ be a measure space, let $f$ belong to $\mathscr{L}^1(X,\mathscr{A},\mu,\mathbb{R})$, and...
View ArticleQuestion about existence of path-lifting property
I'm reading "Complex Made Simple" by David C. Ullrich and here i have a problem with the proof of a theorem:TheoremSuppose that $p : X \to Y$ is a covering map. If $\gamma : [0,1] \to Y$ is continuous,...
View ArticleMore on rationally independent subsets of $\mathbb{R}$.
Suppose that $\lambda_{1}, \lambda_{2}, \lambda_{3}\in\mathbb{C}\setminus\{0\}$ and that $\frac{\lambda_2}{\lambda_1}, \frac{\lambda_3}{\lambda_1}\in\mathbb{R}^{+}\setminus\mathbb{Q}$ such that the set...
View ArticleInfinite Integration Issues
Let me preface this by saying that I know I'm doing something wrong, I'm just here to find out what exactly.We know that integrating a function yields constants. Something like $e^x$ yields $+c$ when...
View ArticleShortcut for computing $ \lim_{x \to 0^+} \frac{1}{2x}\int_0^x \ln(t)t^2\,dt $
Suppose we want to find $$ \lim_{x \to 0^+} \frac{1}{2x}\int_0^x \ln(t)t^2\,dt $$ without computing the integral. What I would do is this :We know that $ 0 \leq t^2 \leq x^2$ for all $t \in (0,x)$Then...
View ArticleConfused about proof in textbook regarding subsequence convergence implying...
A proposition in my textbook essentially says $a_n \rightarrow a$ iff $a_{n_k} \rightarrow a$ for all subsequences $(a_{n_k})$ of $a_n$, but the proof is confusing me.The proof is as follows (condensed...
View ArticleFinding a bound for a sequence
Suppose that $A\succeq 0,$$0\le y$ and $\|y\|\le \|x_\rho\|, \mu \in \mathbb R^n,$ and $e$ be the vector of ones. My goal is to show $\{x_\rho\}$ is bounded when large enough $\rho$ goes to $+\infty$,...
View ArticleLet $f: \mathbb{R}^2 \to \mathbb{R}$ be a function such that for every...
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function and $M \in R$, such that for every $(x, y) \in \mathbb{R}^{2}$, the function $g(t):=f(x t, y t)$ is differentiable and...
View ArticleAre there other usual ways of justifying the passage of differentiation into...
I am recently reading Evan's Partial Differential Equations.In the book, the author sometimes passage differentiation into integrals (i.e. $\frac{\text{d}(\int_{\Omega}...
View ArticleContinuity of confluent hypergeometric function in terms of its parameters
The confluent hyper geometric function of the first kind (or the Kummer's function) is defined...
View ArticleLet $x_1=2,x_{n+1}=\ln|x_n|$, prove...
Let $x_1=2,x_{n+1}=\ln|x_n|$, prove $$z=\lim_{n\rightarrow+\infty}\frac{1}{n}\sum_{k=1}^nx_k$$exist and $z=\ln(-z)$.
View ArticleComparing a function and its derivative
Let $f : \mathbb{R} \rightarrow \mathbb{R} $ be defined by$$f(x)= \begin{cases} (1-x)^2\sin(x^2) & \text{if } x\in (0,1) , \\ 0 & \text{otherwise} \end{cases}$$And $f'$ be its derivative ....
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Approximating $\log x$ by a sum of power functions $a x^b$
Let's approximate $\log x$ on the interval $(0,1)$ by a power function $a x^b$ to minimize the integral of the squared difference$$\delta_0(a,b)=\int_0^1\left(\log x-a x^b\right)^2dx.\tag1$$It's easy...
View ArticleOrdinal Event Ordering $P(E)\geq P(E')\iff Q(E)\geq Q(E')$, so we can...
Let $(S,\Sigma, P)$ be a usual probability space. $S=[0,1]$ and $P$ is the usual Lebesgue measure.$E\in \Sigma$ is an event.Given $P(E)\geq P(E')\iff Q(E)\geq Q(E')$,Is it possible that $P$ and $Q$ are...
View ArticleHow to find closed subsets of $A,B$ such that $\lambda A+(1-\lambda)B$ is not...
How to find closed subsets of $A,B$ such that $\lambda A+(1-\lambda)B$ is not closed for a fixed $0<\lambda<1$.As is well-known that if $A,B$ are bounded closed subsets of $\Bbb R^n$, then for...
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