# Posts

## Series Expansions in $\mathbb Q_p$

Here we give a practical representation of $p$-adic numbers in terms of a power series in powers of $p.$ This perspective will connect our understanding of the $p$_adics as a Cauchy completion and as an… Read More »Series Expansions in $\mathbb Q_p$

## Measures on $\mathbb Q_p$

Because $\mathbb Q_p$ is a locally compact abelian group it has a natural translation invariant measure, Haar measure, which allows us to formulate a natural theory of integration of (certain) real valued functions over subsets… Read More »Measures on $\mathbb Q_p$

## The Geometry of $\mathbb Q_p$ and $\mathbb Z_p$

Once we define distance appropriately, we’ll find a distance preserving embedding of $\mathbb Z_p$ into $\mathbb R^2$—that is we’ll be able to draw a picture of $\mathbb Z_p$ and determine the distance between points by… Read More »The Geometry of $\mathbb Q_p$ and $\mathbb Z_p$

## The Microcanonical Ensemble

The microcanonical ensemble is a probability space which models physical systems at a fixed energy. We imagine a physical system whose states are described by some set $\Omega$ (which we will assume is a measurable… Read More »The Microcanonical Ensemble