At work, we have recently had a "staff survey" performed, and yesterday we got some feedback on the results. I don't want to say anything about the content, but rather to speak about some basic statistics. It was stated that a change of about 2% (upon last year) was "statistically significant". The person giving the presentation then muttered about "standard deviation". I couldn't see what a standard deviation had to do with it.

Let us think how to model this problem. What was performed was a survey of \( N \) people, of whom \( n \) responded. Let us think about two possible ways to model how someone responds:

Read More →A well-known fact from the basic theory of von Neumann algebras is the polar decomposition of normal functionals: given \( \newcommand{\ip}[2]{\langle #1, #2 \rangle} \) \( \varphi\in M_{*} \) there is \( \omega\in M_{*}^+ \) and a partial isometry \( v\in M \) with \( \varphi = v\omega \). If \( v^* v \) is equal to the support of \( \omega \), then this decomposition is unique, and we write \( |\varphi| \) for \( \omega \).

I am mostly following Takesaki's book here; but the material is also nicely presented in a the MSc thesis of Zwarich. Neither of these sources quite gives a correct proof (IMHO) so I thought I would record here the main steps in proving existence.

Read More →We saw before that on a set \( X \) we can specify a (unique) topology by, for each \( x\in X \), specifying a collection of sets \( U_x \) which will satisfy that:

- Each \( V\in U_x \) contains \( x \) and will be open;
- Every open \( C \ni x \) will be such that there is \( V\in U_x \) with \( V\subseteq C \),

if and only if we have the conditions that:

- Given \( A_1,\cdots,A_n\in U_x \), there is \( A\in U_x \) with \( A\subseteq A_1\cap\cdots\cap A_n \);
- Given \( B\in U_y \) with \( x\in B \), there is \( A\in U_x \) with \( A\subseteq B \).

However, we might ask: what is the advantage of specifying the "basic open sets" about each point, rather than just specifying a base for the topology?

Read More →Motivated by some reading about quantum groups, I want to sketch how a semi-direct product of (topological) groups is the same as having an idempotent group homomorphism.

Firstly, let's remember what the (external) semi-direct product of groups is. I will follow the notation of the book of Kaniuth and Taylor. Let \( N,H \) be (topological) groups, and denote by \( \newcommand{\aut}{\operatorname{Aut}}\aut(N) \) the collection of continuous group automorphisms of \( N \). Suppose we have a group homomorphism \( \alpha:H\rightarrow\aut(N) \), written as \( h\mapsto \alpha_h \), which is continuous in the sense that \( N\times H\rightarrow N; (n,h)\mapsto \alpha_h(n) \) is continuous.

Read More →Reading a paper with my office mate, we ended up having a discussion about the notion of an "open neighbourhood base" in a topological space. For example, I might informally say that the weak topology on a Banach space \( E \) has, around a point \( x \), an open neighbourhood base is given by the sets \[ \{y\in E : |f_i(x-y)|<\epsilon \ (1\leq i\leq n) \} \] where \( f_i \) are members of \( E^* \) and \( \epsilon > 0 \).

This raises a natural question:

Read More →Suppose we have a set \( X \) and for each \( x\in X \) we have specified a collection \( U_x \) of subsets of \( X \), such that \( A\in U_x \implies x\in A \). When is there a topology on \( X \) such that \( U_x \) are the "basic open neighbourhoods" of \( x \)?

I have refreshed my website, now building it as a purely static site (instead of using Jekyll) built on top of Bootstrap. To keep the blog going, I have quickly written a Python script which re-creates what I need of Jekyll. Seems to be working, which is quite pleasing.

Read More →An aide-memoire for myself:

```
\documentclass[a4paper]{article}
\usepackage{graphicx,forloop}
\begin{document}
\pagestyle{empty}
\newcounter{pdfpagenumber}
\forloop{pdfpagenumber}{1}{\value{pdfpagenumber} < 115}{
\raisebox{-225ex}[0ex][0ex]{\makebox[90ex]{\includegraphics[width=12in,page=\arabic{pdfpagenumber}]{mtms.pdf}}}
\newpage
}
\end{document}
```

My new job came with a surprise: I get a Surface Pro with docking station as my work PC. This is actually very nice (I tend normally towards the "good enough" school of technology ownership). An Office365 subscription also comes with the job, and so 1TB (yes, a few years ago, a good hard-disk) of cloud storage from OneDrive for business.

Hmm, but... The Surface Pro only have GBs of free storage (thanks to a smallish SSD) and that's to be shared with applications I might want to install. But, surely, I can just sync the folders I want, and keep more in the cloud (swapping things about, perhaps, if needs be). Right? A bit of Internet searching suggests that, sure, that's an option. For normal consumer OneDrive. But not, it seems, for OneDrive Business. Until maybe mid-2018 when a new client comes out. YMMV of course.

Read More →