Articles of analysis

Proving an asymptotic lower bound for the integral $\int_{0}^{\infty} \exp\left( – \frac{x^2}{2y^{2r}} – \frac{y^2}{2}\right) \frac{dy}{y^s}$

This is a follow up to the great answer posted to Let $ 0 < r < \infty, 0 < s < \infty$ , fix $x > 1$ and consider the integral $$ I_{1}(x) = \int_{0}^{\infty} \exp\left( – \frac{x^2}{2y^{2r}} – \frac{y^2}{2}\right) \frac{dy}{y^s}$$ Fix a constant $c^* = r^{\frac{1}{2r+2}} $ and let $x^* = x^{\frac{1}{1+r}}$. […]

Inverse of the Joukowski map $\phi(z) = z + \frac{1}{z}$

We know the Joukowski map $$\phi(z) = z + \frac{1}{z}$$ which maps the upper semidisc of radius $1$ in the lower half plane, and the lower semidisc of radius $1$ in the upper half plane. What is the inverse of this function ? We obtain $z ^{2}-zy + 1 = 0$ and this equation has […]

The closure of an open set in $\mathbb{R}^n$ is a manifold

I want to solve the following exercise from M. Spivak’s Calculus on Manifolds (p. 114): (a) Let $A \subseteq \mathbb{R}^n$ be an open set such that boundary $A$ is an $(n-1)$-dimensional manifold. Show that $N=A \cup \text{ boundary }A$ is an $n$-dimensional manifold-with-boundary. (It is well to bear in mind the following example: if $A=\{x […]

Rigorous hypothesis for Reynolds' transport theorem

I’m looking for rigorous hypothesis for the application of Reynolds’ transport theorem : $$ \frac{\mathrm{d}}{\mathrm{d}t}\left[ \int_{\Omega(t)} \phi({\bf x},t) \mathrm{d}{\bf x} \right] = \int_{\Omega(t)} \frac{\partial}{\partial t}\phi({\bf x},t) \mathrm{d}{\bf x} + \int_{\partial\Omega(t)} \phi({\bf x},t)\frac{\mathrm{d} {\bf x}}{\mathrm{d} t}.{\bf n}_b \mathrm{d}{\bf x}, $$ where $\Omega(t)$ is a piecewice smooth manifold with boundaries (a portion of a polyhedron for instance), […]

A problem in the space $C$

Let $E=C[a,b]$ provide with the $\max$ norm. Let $S\neq \emptyset$, let and $D(t,\lambda)$ be a continuous function (for each $\lambda\in S$), from $[a,b]$ to $\mathbb{R}$, such that $\displaystyle D=\sup_{\lambda}\int_{a}^b |D(t,\lambda)|dt<\infty$. We define for $f\in E$ $$A(f)=y(\lambda)=\int_{a}^b D(t,\lambda)f(t)dt$$ I have to prove that: $\|A\|=D$, where $\sup_{\|x\|=1}{\|A(x)\|}=\|A\|$. My Approach: I have already showed that $A$ is a […]

Sequence $a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n$.

Consider the sequence $a_1, a_2,\ldots,a_n$ with $a_1=1$ and defined recursively by $$a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n.$$ Find $\lambda>1$ such that $a_n=0$. The answer is $\lambda=\dfrac1{\cos\frac{\pi}{n+1}}$.

Duality of $L^p$ and $L^q$

If $X$ is an arbitrary measure space, I already know with proof that $L^p(X)$ and $L^q(X)$ are mutually duals as Banach spaces, when $1<p$ and $p$, $q$ are dual indices. I also know a different proof that only works when $X$ is sigma finite, but then it establishes also that the dual of $L^1(X)$ is […]

Positive Linear Functionals on Von Neumann Algebras

Let $\omega$ be a positive linear functional on $M$ which is a Von Neumann Algebra. Suppose $\omega$ is completely additive (i.e. $\omega$ applied to a strongly convergent sum of mutually orthogonal projections is the same as the convergent complex valued sum of $\omega$ applied to the projections). Then supposedly, $\omega$ is the pointwise convergent sum: […]

(Baby Rudin) To show the set of all condensation points of a set in Euclidean space is perfect

This is from Rudin’s Principal of Mathematical Analysis, Chapter 2, Problem 27. Let $E \subseteq \mathbb{R}^k$. Let $P$ be the set of all condensation points of $E$. Let $\{ V_n \}$ be a countable base of $\mathbb{R}^k$. Let $W$ be the union of those $V_n$ for which $E \cap V_n$ is at most countable. Prove […]

How to prove convexity?

Let us consider the function $$I(p):= \frac {\Gamma(2-p)\Gamma(3p)}{(p\Gamma(p))^2} $$ on the interval $(0,1),$ where $\Gamma(x)$ denotes the gamma function. How to prove its convexity there?