Articles of analysis

All derivations are directional derivatives

This question already has an answer here: Leibniz rule and Derivations 1 answer

Can I research in complex analysis, PDE and differential geometry without exposure to mathematical physics?

I love mathematics, but physics is far away from my interest. I see that recent mathematics research is strongly connected to mathematical physics which is something doesn’t interest me! I love mathematical entities and structures, stuff like relativity and quantum mechanics are not even close to my area of interest, so I wonder if I […]

Backwards heat equation (stability analysis)

Problem Consider the backwards heat equation of the form $$ \left\{ \begin{aligned} u_{t} &= \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= f(x), \end{aligned} \right.\tag{*}\label{*}$$ Establish whether solution is unique and analyze its stability. Attempt of proving uniqueness My attempt to prove uniqueness is provided in this post. Attempt […]

Measuring the set-theoretical complexity of sets/spaces encountered in general analysis

In analysis, it is common to encounter subsets of $\mathbb R$ (or even $\mathbb R^n$) which appear to be “well-behaved”, especially with regard to properties like being measurable, compactness, etc. It is core to descriptive set theory (DST) that one is able to impose a classification of such subsets by closure properties of varying degree. […]

Lipschitz condition on a second order nonlinear ODE?

Preliminaries: Let the matrix norm be $$\sqrt{\sum_{j=1}^n\sum_{i=1}^n a_{ij}^2}=||\mathbf A||.$$ I am trying to prove uniqueness and existence of a second order nonlinear ODE (Ordinary Differential Equation). So I need to show f a function is continuous and satisfies a Lipschitz condition. Take for example the second order nonlinear ODE $$x”=-\cos(x).$$ Now simplify to a first […]

Determine whether this map is an isomorphism

Assume all the normed spaces are over $\mathbb{F}=\mathbb{C}$ or $\mathbb{R}$. Let $c$ be the space of all convergent sequences equipped with the supremum norm. For $g\in\ell^1$, define the map $$T_g:c\to \mathbb{F},\quad T_g(f)=g(1)\lim_{k\to\infty}f(k)+\sum_{k=1}^{\infty}g(k+1)f(k).$$ It is easy to see that $T_g\in c^*$. Now let $$\phi:\ell^1\to c^*,\quad \phi(g)=T_g.$$ Show that $\phi$ is an isometry and determine whether $\phi$ […]

Understanding angle-preserving definition

My book (Real and complex analysis, by Rudin) gives the following definition: Let $A(z) = \frac z{|z|}$. Then we say $f$ preserves angles at $z_0$ if $$\lim_{r \to 0}e^{-i\theta} A[f(z_0 + re^{i\theta} )- f(z_0)] \ \ \ \ (r > 0)$$ exists and it is independent of $\theta$. It then adds The requirements is that […]

Visualizing Balls in Ultrametric Spaces

I’ve been reading about ultrametric spaces, and I find that a lot of the results about balls in ultrametric spaces are very counter-intuitive. For example: if two balls share a common point, then one of the balls is contained in the other. The reason I find these results so counter-intuitive is that I can easily […]

Is this integral finite? (convergent)

I came to this problem and I got very curious to know… intuitively I would say this integral is not finite but maybe it is. Let us consider $\mathbb{R}^2$ and only the region $R=\{(x,y)\in \mathbb{R}^2: \, |x|>1, |y|>1\}$. Is the following integral finite? $$\int_{R} \frac{1}{(\max \{x,y\})^2} dxdy.$$ Any ideas? 🙂 Thanks a lot 🙂

Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$.

Let $I$ be a generalized rectangle in $\Bbb R^n$ Suppose that the function $f\colon I\to \Bbb R$ is continuous. Assume that $f(x)\ge 0$, $\forall x \in I$ Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$. My idea is that For $(\impliedby)$ Since $f\colon I\to \Bbb R$ is identically zero, $$f(I)=0$$ […]