Blog of Matthew Daws

Neighbourhood bases, continued.

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?

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Semi-direct products

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.

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Neighbourhood base

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:

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 \( x \)?

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New site

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.

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PDF munging with LaTeX

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}

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OneDrive for Business, or pleasure.

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.

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Expectations of brilliance underlie gender distributions across academic disciplines

I blogged previously about statistical programming in Python. Here I want to say something about the data I used, which is from the paper:

Sarah-Jane Leslie, Andrei Cimpian, Meredith Meyer, Edward Freeland "Expectations of brilliance underlie gender distributions across academic disciplines" Science 347 (2015) 262--265. DOI: 10.1126/science.1261375

The abstract explains the results of the survey and data analysis the author perform:

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Probabilistic programming in Python

Later in the week I will give a talk to the Centre for Spatial Analysis & Policy group in Geography, at Leeds Uni. See the GitHub Repo for details.

I had a few aims:

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Profile image; rendered glass discs
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