Articles of analysis

Showing a metric space is bounded.

This is from a review packet: Let $d:\mathbb{R} \to \mathbb{R}$ be defined as $$d(x,y)=\frac{|x-y|}{1+|x-y|}.$$ i) Show that $(\mathbb{R},d)$ is a bounded metric space. ii) Show that $A=[a,\infty)$ is a closed and bounded subset of $\mathbb{R}$. iii) Show that $A=[1,\infty)$ is not compact. For (i) I think this is a true observation: $d(x,y)\leq1$ for an arbitrary […]

An open interval as a union of closed intervals

For $a<b, a,b\in\Bbb R$ $$(a,b)=\bigcup_{0<\delta<(b-a)/2} I_{\delta} \quad I_{\delta}:=[a+\delta,b-\delta] $$ Clearly the RHS is an (uncountable) infinite sum of closed intervals. I have no idea how to show it is open at two ends. (My hope is that if this is true, then it is trivial that: $$f(x)\in\mathscr C^r(a,b)\Longleftrightarrow f(x)\in\mathscr C^r[a+\delta,b-\delta]\quad \forall \delta\in(0,\frac{b-a}2) $$ which will […]

At large times, $\sin(\omega t)$ tends to zero?

While doing a calculation in quantum mechanics, I got a expression $\sin(\omega t)$, and my prof said if I consider the consider at large times, then i can assume that this goes to zero because at large times, the graph of $\sin$ oscillates very rapidly and so you can take it to be zero. When […]

Laplacian Boundary Value Problem

Given a domain $ M \subset \mathbb{R}^2 $ and a function $ f : \partial M \rightarrow \mathbb{R} $, let $ \omega $ solve the boundary value problem: $$ \Delta \omega = 0 \text{ in } M \\ \omega = f \text{ in } \partial M $$ I would like to find some bounds on […]

Development of a specific hardware architecture for a particular algorithm

How does a technical and theoretical study of a project of implementing an algorithm on a specific hardware architecture? Function example: %Expresion 1: y1 = exp (- (const1 + x) ^ 2 / ((conts2 ^ 2)), y2 = y1 * const3 Where x is the input variable, y2 is the output and const1, const3 const2 […]

Equivalence of measures and $L^1$ functions

Suppose we have two probability measures $\mu$ and $\delta$ on $(X, \mathcal{B})$ such that $ \delta <<\mu << \delta $. How can I prove that $f \in L^1(X,\mathcal{B}, \mu)$ iff $f \in L^1(X,\mathcal{B}, \delta)$? My idea was to use that the Radon Nikodym theorem. So we know there exist $g$ $\mu$-measurable and $g^{-1}$ $\delta$-measurable such […]

Proving $C()$ Is Not Complete Under $L_1$ Without A Counter Example

I’d like to show that $C([0,1])$ (that is, the set of functions $\{f:[0,1]\rightarrow \mathbb{R} \, \textrm{ and } \, f \, \textrm{is continuous} \}$ is not a complete mertric space under the $L_1$ distance function: $$ d(f,g) = \int_0^1 |f(x)-g(x)|dx $$ I can find counter examples (for example, here) but would rather prove it using […]

A variation of fundamental lemma of variation of calculus .

I have a question on a variation of the fundamental lemma . If $\int_\Omega f(x) g(x)=0$ and $f, g $ are $C^0\Omega$ functions and $\int_\Omega g(x)=0 $ then is it possible that there exist some constant $c$ such that $f(x)=c$ for all $x\in \Omega$ I tried to use mollification on one of the function and […]

Has the polynomial distinct roots? How can I prove it?

I want to prove that the polynomial $$ f_p(x) = x^{2p+2} – cx^{2p} – dx^p – 1 $$ ,where $c>0$ and $d>0$ are real numbers, has distinct roots. Also $p>0$ is an even integer. How can I prove that the polynomial $f_p(x)$ has distinct roots for any $c$,$d$ and $p$. PS: There is a similar […]

How to prove $r^2=2$ ? (Dedekind's cut)

Let a (Dedekind) cut $r=\{p \in \mathbb{Q} :p^2<2 \text{ or } p<0\}$ and a cut $2^*=\{t\in \mathbb{Q} : t<2\}$. I want to prove $r^2=2^*$. I could show that $r^2 \subset 2^*$ easily, but I couldn’t show that $2^* \subset r^2$. How to show that there is $p$, $p’ \in r$ such that $t \leq pp’ […]