Articles of reference request

Dilemma for Studying Probability Theory while Waiting to Learn Measure Theory

I’m taking stochastic probability class but I’m now only taking analysis (with Rudin’s PMA) class. The stochastic probability class doesn’t depend heavily on the theoretic structures: rather, the professor wants to give the intution and that’s fine with me because I’ve taken set theory class and basic probability class before, so I can understand almost […]

Ring theory reference books

I am studying ring theory in this semester. I am new to this theory. Hence, I would like to have some recommendations on what books should be used for ring theory(beginner). If possible, I would like to have a book on theory and a lot of problems( include solution would be nicer,if possible). Can anyone […]

The structure of $(\mathbb Z/525\mathbb Z)^\times$

I am working on the following problem. Find the number of the elements of order $4$ in $(\mathbb Z/525\mathbb Z)^\times$. I tried to solve it in the following way: since $525=3\cdot5^2\cdot7$, we have by the CRT $$ (\mathbb Z/525\mathbb Z)^\times \cong (\mathbb Z/3\mathbb Z)^\times \times (\mathbb Z/25\mathbb Z)^\times \times (\mathbb Z/7\mathbb Z)^\times. $$ By the […]

$H^{-1}(\Omega)$ given an inner product involving inverse Laplacian, explanation required

Let $\Omega$ be a bounded domain and define $V=L^2(\Omega)$ and $H=H^{-1}(\Omega)$. Endow $H$ with the inner product $$(f,g)_{H} = \langle f, (-\Delta)^{-1}g \rangle_{H^{-1}, H^1}$$ where $(-\Delta)^{-1}g = \tilde g$ is the solution of $-\Delta \tilde g = g$ on $\Omega$, $\tilde g= 0$ on $\Gamma$. In Lions’ Quelques methodes… on page 192, he uses this […]

(Possible) application of Sarason interpolation theorem

This question is related to the following Wikipedia article on Nevanlinna–Pick interpolation. At the end it has been written as Pick–Nevanlinna interpolation was introduced into robust control by Allen Tannenbaum. I do not have the access of the paper or even if I had, without any basic knowledge on control theory, I would, perhaps, not […]

Good Textbook in Numerical PDEs?

I am currently taking a course on Numerical PDE. The course covers the following topics listed below. Chapter 1: Solutions to Partial Dierential Equations: Chapter 2: Introduction to Finite Elements:

Relationship among the function spaces $C_c^\infty(\Omega)$, $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Bbb{R}^d)$

I have seen the spaces $C_c^\infty(\Omega)$, $C_c^\infty(\overline{\Omega})$ and $C_c^\infty(\Bbb{R}^d)$ a lot in theorems regarding PDE where $\Omega$ denotes some open subset of $\Bbb{R}^d$. There is no doubt about the definitions of $C_c^\infty(\Omega)$ and $C_c^\infty(\Bbb{R}^d)$. But I’m not very clear about the relationships among these three spaces. Here are my questions: What is the definition for […]

Is $\mathbb{Z}/p^\mathbb{N} \mathbb{Z}$ widely studied, does it have an accepted name/notation, and where can I learn more about it?

Fix a positive integer $p$, possibly prime. For each natural number $n$, there is a ring $\mathbb{Z}/p^n \mathbb{Z}$ together with a distinguished ring homomorphism $$\pi_n:\mathbb{Z} \rightarrow \mathbb{Z}/p^n \mathbb{Z}.$$ For any given integer $k$, we can think of $\pi_n(k)$ as the remainder when $k$ is divided by $p^n$. Now clearly, $\pi_n$ will never be injective. If […]

$1992$ IMO Functional Equation problem

The problem states: Let $\Bbb R$ denote the set of all real numbers. Find all functions $f : \Bbb R \rightarrow \Bbb R$ such that $$f(x^{2}+f(y))=y+(f(x))^{2} \space \space \space \forall x, y \in \Bbb R.$$ My progress: 1. If we substitute $x=y=0$ in the given equation, then we get $$f(f(0))=(f(0))^{2}.$$ 2. We then substitute $x=0$ […]

Nimbers for misère games

Let $G$ be an impartial combinatorial game. I claim that there is a game $G’$ such that $G$ (without terminal positions; see below) under the misère play rule is equivalent to $G’$ under the normal play rule. I wonder if this construction is already known and written down somewhere in the literature (so that I […]