# My Android ebook reading workflow (optimized for frequent turnover of ebooks)

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 [...]

# Symmetrized products of positive matrices are not positive

.. 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 [...]

# QOTD (from an old IMO): show that none of these sums are integers

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 [...]

# Reading List for Feb and March 2012

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 [...]

# Norms of sums of random vectors sampled without replacement

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, \[ [...]

# Sampling models and vector sums

What happens when you sample columns from a matrix whose columns all have roughly the same norm? It’s clear that you should have some kind of concentration: say if you sample $$\ell$$ of $$n$$ columns, then the Frobenius norm of the resulting matrix, appropriately scaled, should be a very good approximation of the Frobenius norm [...]