Let $p\in\mathbb{R}\setminus\mathbb{Q}.$ Show that...
Let $p\in\mathbb{R}\setminus\mathbb{Q}.$ Show that $A=\{np-m:m\in\mathbb{Z},n\in\mathbb{N}\}$ is dense in $\mathbb{R}.$Note that I am taking $0\notin\mathbb{N}$ by convention.I do have an "almost...
View ArticleIf $\mu$ is a complex measure and $\|\mu\| = \mu(X)$, then $\mu$ is a...
Taken from Conway's A course in functional analysis Chapter 3 Section 7 Problem 2Problem Statement: Let $X$ be a set and $\Omega$ a $\sigma$-algebra of subsets of $X$. Suppose $\mu$ is a complex-valued...
View ArticleApproximation of complex numbers $a_{j_1,\ldots,j_k}$ with...
Let $k,n$ be fixed natural numbers and $a_{j_1,\ldots,j_k}$ be complex numbers with $\sum\limits_{1\le j_1,\ldots,j_k\le n}|a_{j_1,\ldots,j_k}|^2=1$.Observe that if we have a $n\times n$ matrix...
View ArticleA shortcut to $\int_{0}^{\infty}\left (...
How can we prove this canonically?$$\newcommand{\Ei}{\operatorname{Ei}}\newcommand{\Ci}{\operatorname{Ci}}\newcommand{\si}{\operatorname{si}}\newcommand{\Li}{\operatorname{Li}_4}\int_{0}^{\infty}\left...
View ArticleAn affine set $C$ contains every affine combinations of its points
Show that an affine set $C$ contains every affine combinations of its points. Proof by induction:From the definition of an affine set, we know that $\forall x_1,x_2\in C \text{ and } \theta_i\in R...
View ArticleBinary representation of the real numbers
I am solving the following exercise:for $n \in \mathbb{N}$ and $a_1,a_2, \ldots ,a_n \in \{0,1\}$ wedefine: $$ I(a_1, \ldots , a_n) := \left \lbrack \sum_{i=1}^n \frac{a_i}{2^i}, \sum_{i=1}^n...
View Articleare singletons always closed?
I am learning about metric spaces and I find it very confusing. Is this a valid proof that a singleton must be closed?If $(X,d)$ is a metric space, to show that $\{a\}$ is closed, let's show that $X...
View ArticleDifferentiating $x^{x^{x^{...}}}$
How do I differentiate $$ x^{x^{x^{...}}}$$ with respect to $x$? (Note that $x$ is raised infinitely many times.)My attempt: Let $y = x^{x^{x^{...}}}$. Taking logarithm of both sides we get $\ln y = y...
View ArticleMaximizing a sum of pairwise angle differences between unit vectors
I want to find the maximum (or a relatively tight upper bound on the maximum) of $$\sum_{i,j\in [n]} \left(\arccos\left(\left\langle \vec{v}_i, \vec{v}_j\right\rangle\right) - \arccos\left(\left\langle...
View ArticleProving that: $\lim\limits_{n\to\infty}...
Let $a$ and $b$ be positive reals. Show that$$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
View ArticleDoes $\exists y >0$ , such that $\forall x>2$, $\sum_{z=1}^{\infty}(...
By graphing the partial sums, the region $\sum_{z=1}^{\infty}( e^{-y(z-1/2)^x} -e^{-y(z)^x}) > 0.5$ when we set $y=0.87$, reaches arbitarily close to the $x=2$ line. Hence computationally sampling...
View ArticleApproximate any bounded monotone function $f$ on a closed interval by a...
I am reading Real Variable and Integration: with historical note by Benedetto, and he said that any monotone decreasing and bounded real-valued function $f$ on a closed interval [a,b] can be...
View ArticleEigenvalues with positive real parts imply instability
Given an autonomous ode $\dot{x}=F(x)$ with $0$ being its equilibrium point, and all eigenvalues of $DF(0)$ have non-zero real parts.I have learned that if the real parts of eigenvalues are all...
View ArticleCompactness of $\{(a,b,c,d)\in \mathbb R^4:a^2+b^2+c^2+d^2=1\}$
The set $$\{(a,b,c,d)\in \mathbb R^4:a^2+b^2+c^2+d^2=1\}$$ is compact.I want to know why this is true (or maybe, why this isn't, if that's the case).When doing olympiad inequality problems, we can...
View ArticleClik here to view.
Extension of $F_{\alpha}(x)=\int_{0}^{1}\frac{\sin(\pi x)}{t^x (1+\alpha...
ContextI have this integral representation$$F_{\alpha}(x)=\int_{0}^{1}\frac{\sin\left(\pi x\right)}{t^{x}\left(1+\alpha t\right)}\mathrm{d}t\qquad \text{for }\alpha>-1$$which converges only for...
View ArticleApplying L'Hopital to the log of an expression
The limit below is solved using multiple methods.$$\lim_{x\to\infty} \frac{x^n}{e^x}$$However, I am trying to solve it using the comment made below the question:Only one application of l'Hopital's rule...
View ArticleWhy is $\mu(X) < \infty$ important in Egoroff's theorem?
The following is Folland's proof of Egoroff's theorem.Without loss of generality we may assume that $f_n \rightarrow f$ everywhere on $X$. For $k, n \in \mathbb{N}$ let$$E_n(k) = \bigcup_{m=n}^\infty...
View ArticleComputing $ \int_{0}^{2\pi} \frac{\sin(nx)^2}{\sin(x)^2} \mathrm dx $
I would like to compute:$$ \int_{0}^{2\pi} \frac{\sin(nx)^2}{\sin(x)^2} \mathrm dx $$I think that :$$ \int_{0}^{2\pi} \frac{\sin(nx)^2}{\sin(x)^2} \mathrm dx=2n\pi $$I tried to use induction:$$...
View ArticleDefine an operator without assuming twice-differentiability of a function
Let $f: \mathbb{R}^d \to \mathbb{R}$ be continuously differentiable such that $f$ is $m$-strongly convex and $\nabla f$ is $M$-Lipschitz continuous. In other words$$ | \nabla f(x) - \nabla f(y) | \geq...
View ArticleAnalytic continuation of Dirichlet series of arithmetic function
I am trying to prove that the Dirichlet series of a multiplicative function with polynomial growth that vanishes at all but finitely many primes admits analytic continuation to $ Re(s)> 1/2$. Using...
View Article