Connecting consumers with relevant products is a very important problem in both online and offline commerce. In physical retail, product placement is an effective way to connect consumers with products. However, selecting product locations within a s
tore can be a tedious process. Moreover, learning important spatial patterns in offline retail is challenging due to the scarcity of data and the high cost of exploration and experimentation in the physical world. To address these challenges, we propose a stochastic model of spatial demand in physical retail. We show that the proposed model is more predictive of demand than existing baselines. We also perform a preliminary study into different automation techniques and show that an optimal product allocation policy can be learned through Deep Q-Learning.
For $beta > 1$, a sequence $(c_n)_{n geq 1} in mathbb{Z}^{mathbb{N}^+}$ with $0 leq c_n < beta$ is the emph{beta expansion} of $x$ with respect to $beta$ if $x = sum_{n = 1}^infty c_nbeta^{-n}$. Defining $d_beta(x)$ to be the greedy beta expansion of
$x$ with respect to $beta$, it is known that $d_beta(1)$ is eventually periodic as long as $beta$ is a Pisot number. It is conjectured that the same is true for Salem numbers, but is only currently known to be true for Salem numbers of degree 4. Heuristic arguments suggest that almost all degree 6 Salem numbers admit periodic expansions but that a positive proportion of degree 8 Salem numbers do not. In this paper, we investigate the degree 6 case. We present computational methods for searching for families of degree 6 numbers with eventually periodic greedy expansions by studying the co-factors of their expansions. We also prove that the greedy expansions of degree 6 Salem numbers can have arbitrarily large periods. In addition, computational evidence is compiled on the set of degree 6 Salem numbers with $text{trace}(beta) leq 15$. We give examples of numbers with $text{trace}(beta) leq 15$ whose expansions have period and preperiod lengths exceeding $10^{10}$, yet are still eventually periodic.
We discuss the unique capabilities of programmable logic devices (PLDs) for experimental quantum optics and describe basic procedures of design and implementation. Examples of advanced applications include optical metrology and feedback control of qu
antum dynamical systems. As a tutorial illustration of the PLD implementation process, a field programmable gate array (FPGA) controller is used to stabilize the output of a Fabry-Perot cavity.
