I had a dozen eggs in the fridge that I need to use up by Wednesday; now I’m down to four. I made Jaffa cakes last week (not bad, but not memorable, and definitely not worth the effort: my main complaint is that the cake portion is bland and has a too-tough texture) and I [...]

I’m reading Santosh Vempala’s survey “Geometric Random Walks: A Survey,” and already I’m puzzled at one of the very first definitions he gives. Define a Markov chain as a state space sigma algebra pair \((K, \mathcal{A})\) with transition probability measures given by \(P_u\) for each \(u \in K.\) A distribution \(Q\) on \((K, \mathcal{A})\) is [...]

Synopsis: Aldiko’s not appropriate if you plan on constantly rotating the collection of ebooks on your mobile device. Instead, a combination of Calibre, Moon+Reader, and DropBox seems to do the trick. Consider using Moon+Reader as your default reader unless you need Aldiko’s capabilities for dealing with large ebook collections, as Moon+Reader has a more polished [...]

.. duh. This is in relation to the symmetrized matrix AM-GM inequality conjectured by Recht and Re. The most obvious/direct way of maybe attempting to prove this inequality is to show the semidefinite inequality \[ \left(\frac{1}{n}\sum\nolimits_{i=1}^n\mat{A}_i\right)^n \succeq \frac{1}{n!} \sum_\sigma \mat{A}_{\sigma(1)}\mat{A}_{\sigma(2)}\times\cdots\times \mat{A}_{\sigma(n)}. \] Unfortunately, the right hand side isn’t even a positive matrix in general. The [...]

Here’s an old IMO question (or maybe it’s off one of the short or long lists) that I haven’t made any progress on: Show that for any natural numbers \(a\) and \(n\) the sum \[ 1 + \frac{1}{1+a} + \frac{1}{1 + 2a} + \cdots + \frac{1}{1+n a} \] is not an integer. An obvious and [...]

My top picks, with links if you want to tag along: Ledoux and Talagrand (available from Talagrand’s webpage). Specifically, the chapters on Rademacher and Gaussian sums, symmetrization and contraction properties, and tail bounds for sums of random vectors. Also the bits on scalar martingale arguments. Sourav Chatterjee’s thesis and its matrix counterpart. He came up [...]

Let \(\mathbb{P}_{WOR(\ell)}\) denote a probability under the model where a set \(I\) of \(\ell\) indices is sampled without replacement from \(\{1,2,\ldots,n\}\), and \(\mathbb{P}_{WR(\ell)}\) denote a probability under the model where the \(\ell\) indices are chosen with replacement. Then is it the case that, for any collection \(\vec{v}_i\) of \(n\) vectors in a vector space, \[ [...]