Articles of reference request

Representing Elementary Functions in a CAS

I’ve looked through several books about computer algebra. They are surprisingly scarce about how to actually represent elementary functions. Basically, as far as I understood elementary functions are roughly $$\textbf{Quot}\left(\mathbb K[x, \exp x, \exp x^2 \log x, \ldots] \right)$$ that is rational functions over field $\mathbb K$, with variables (or kernels) $x$, $\exp x$, $\exp […]

Is characterisation of degree 2 nilpotent matrices (i.e. $M^2=0$) known?

$M$ is $n\times n$ real (or complex) matrix. Also $M$ is nilpotent of degree 2, i.e. $M^2=0.$ Question. How does $M$ look like? I just calculated that $2\times 2$ matrix must have following form $$\begin{bmatrix} gh & \pm g^2 \\ \mp h^2 & -gh \end{bmatrix}.$$ I wanted to compute conditions on $3\times3,4\times 4$ and look […]

Reference for Topological Groups

Topological groups were a topic that were covered minimally at my undergraduate institution but it’s a topic that I’m finding a need quite a bit in the number theory I’m reading (class field theory). Is there a recommended source that covers the general theory preferably with lots of examples and exercises?

Book suggestion for probability theory

I need a good rigorous book to learn probability theory. So far, I’ve been suggested Gnedenko’s Theory of Probability; Shiyayev’s Probability; Feller’s An Introduction to Probability Theory and Its Applications. . Which one would you reccomend the most and why? Are there other books worth mentioning?

Reference to a basic result implying existence and uniqueness of the base-$b$ representation

Edit (Jan 17, 2016): Now crossposted at MO. I’m looking for a reference to the following elementary results (or to generalizations of them): Lemma 1. Let $x_1, \ldots, x_n$ be positive real numbers such that $x_1 + \cdots + x_i < x_{i+1}$ for every $i \in [\![1, n-1]\!]$, and let $y$ be an element in […]

Next book in learning General Topology

I have just finished the book “C Adams & R Franzosa – Introduction to Topology. Pure and Applied”. My aim is to reach to the level of the book “G E Bredon – Topology and Geometry”. Bredon’s book is not only too advanced to study after Adams’, but also I don’t think that it is […]

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I’m in the last period of the undergraduate course and I have interest in this field, but my university don’t have a specialist in this area. So, I want a book that can introduce me in this field without many difficulty and that permite me study still […]

Request for Statistics textbook

I am looking for a textbook on Statistical Analysis. Unfortunately most of the books I have seen, such as Statistics by DeGroot et al., are quite the opposite of the terse and lean textbooks I prefer (such as any book by Milnor). Can someone suggest to me an introductory or perhaps even intermediate statistics textbook […]

Compactness of Sobolev Space in L infinity

I want to show that if $u_{m} \rightharpoonup u$ in $W^{1,\infty}(\Omega)$ then $u_{m} \rightarrow u$ in $L^{\infty}(\Omega)$. I know that I can’t directly use the compactness of Rellich Kondrachov Theorem since I am taking $p = \infty$. From Morrey’s Inequality I have $||u||_{C^{0,\alpha}} \leq ||u||_{W^{1,p}}$ where $\alpha = 1 – \frac{n}{p}$. Is it possible to […]

Is there a classic Matrix Algebra reference?

I’m looking for a classic matrix algebra reference, either introductory or advanced. In fact, I’m looking for ways to factorize elements of a matrix, and its appropriate determinant implications. Your help is greatly appreciated.