Randomness and differentiability in higher dimensions

We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables. Firstly, we prove an effective form of the Rademacher's Theorem: we show that computable randomness implies differentiability of computable Lipschitz functions of several variables. Secondly, we show that weak 2-randomness is equivalent to differentiability of computable a.e. differentiable functions of several variables.Comment: 19 page

Partial positivity: geometry and cohomology of q-ample line bundles

We give an overview of partial positivity conditions for line bundles, mostly from a cohomological point of view. Although the current work is to a large extent of expository nature, we present some minor improvements over the existing literature and a new result: a Kodaira-type vanishing theorem for effective q-ample Du Bois divisors and log canonical pairs.Comment: v1: 24 pages; v2: 25 pages, minor changes, accepted for publication in the Robfest Proceedings in honor of Rob Lazarsfeld's 60th birthday, London Mathematical Society Lecture Notes Serie

Nonlinear Oscillations and Bifurcations in Silicon Photonic Microresonators

Silicon microdisks are optical resonators that can exhibit surprising nonlinear behavior. We present a new analysis of the dynamics of these resonators, elucidating the mathematical origin of spontaneous oscillations and deriving predictions for observed phenomena such as a frequency comb spectrum with MHz-scale repetition rate. We test predictions through laboratory experiment and numerical simulation.Comment: Main text: 5 pages, 6 figures. Supplemental material: 12 pages, 8 figure

Do Macroeconomic Variables Affect the Agricultural Trade Sector? An Elasticities Analysis

International Relations/Trade,

A Multiresolution Stochastic Process Model for Predicting Basketball Possession Outcomes

Basketball games evolve continuously in space and time as players constantly interact with their teammates, the opposing team, and the ball. However, current analyses of basketball outcomes rely on discretized summaries of the game that reduce such interactions to tallies of points, assists, and similar events. In this paper, we propose a framework for using optical player tracking data to estimate, in real time, the expected number of points obtained by the end of a possession. This quantity, called \textit{expected possession value} (EPV), derives from a stochastic process model for the evolution of a basketball possession; we model this process at multiple levels of resolution, differentiating between continuous, infinitesimal movements of players, and discrete events such as shot attempts and turnovers. Transition kernels are estimated using hierarchical spatiotemporal models that share information across players while remaining computationally tractable on very large data sets. In addition to estimating EPV, these models reveal novel insights on players' decision-making tendencies as a function of their spatial strategy.Comment: 31 pages, 9 figure

Regulation of Peripheral Molecular Clocks in Mammalian Tissues and In Vitro Skeletal Muscle Activation of AMP-Activated Protein Kinase via AICAR

Most organisms possess a common molecular machinery that governs cellular and tissue circadian rhythmicity through a roughly 24-hour transcription-translation feedback loop. It is estimated that up to 15 percent of human genes are influenced by the core clock machinery. It is likely, however, that the metabolic networks affected by the molecular clock differ according to body tissue. Recent evidence suggests that peripheral molecular clocks are governed to a greater extent by energy availability than by light and dark cycles. AMP-activated protein kinase (AMPK) acts as a cellular fuel gauge within the cell and is activated in response to exercise and fasting. AMPK can also be pharmacologically activated by 5-amino-1-β-D-ribofuranosyl-imidazole-4-carboxamide (AICAR). AMPK likely serves as an intermediary between metabolism and the molecular clock due to its activation of the rate-limiting enzyme in Nicotinamide adenine dinucleotide (NAD) biosynthesis, Nicotinamide phosphoribosyltransferase (NAMPT), and its role in PER and CRY degradation. The NAD-dependent histone deacetylase SIRT 1 inhibits the BMAL1-CLOCK complex in a NAMPT-dependent manner.The complex interplay between metabolism and peripheral clocks mediated by AMPK is beginning to be unraveled. AMPK’s tissue-specific influence on the molecular clock in skeletal muscles and other mammalian tissues requires further elucidation as it may provide insight into the etiology and treatment of metabolic disease. [excerpt

Refocusing two qubit gates with measurements for trapped ions

Dynamical decoupling techniques are the method of choice for increasing gate fidelities. While these methods have produced very impressive results in terms of decreasing local noise and increasing the fidelities of single qubit operations, dealing with the noise of two qubit gates has proven more challenging. The main obstacle is that the noise time scale is shorter than the two qubit gate itself so that refocusing methods do not work. We present a measurement and feedback based method to refocus two qubit gates which cannot be refocused by conventional methods. We analyze in detail this method for an error model which is relevant for trapped ions quantum information