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 […]

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 […]

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 […]

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 […]

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 […]

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:

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 […]

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 […]

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 $x=0$ […]

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 […]

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