The best constant for the centered maximal operator on radial decreasing functions

We show that the lowest constant appearing in the weak type (1,1) inequality satisfied by the centered Hardy-Littlewood maximal operator on radial integrable functions is 1.Comment: corrected typo

A refined analysis of the low-mass eclipsing binary system T-Cyg1-12664

The observational mass-radius relation of main sequence stars with masses between ~0.3 and 1.0 Msun reveals deviations between the stellar radii predicted by models and the observed radii of stars in detached binaries. We generate an accurate physical model of the low-mass eclipsing binary T-Cyg1-12664 in the Kepler mission field to measure the physical parameters of its components and to compare them with the prediction of theoretical stellar evolution models. We analyze the Kepler mission light curve of T-Cyg1-12664 to accurately measure the times and phases of the primary and secondary eclipse. In addition, we measure the rotational period of the primary component by analyzing the out-of-eclipse oscillations that are due to spots. We accurately constrain the effective temperature of the system using ground-based absolute photometry in B, V, Rc, and Ic. We also obtain and analyze V, Rc, Ic differential light curves to measure the eccentricity and the orbital inclination of the system, and a precise Teff ratio. From the joint analysis of new radial velocities and those in the literature we measure the individual masses of the stars. Finally, we use the PHOEBE code to generate a physical model of the system. T-Cyg1-12664 is a low eccentricity system, located d=360+/-22 pc away from us, with an orbital period of P=4.1287955(4) days, and an orbital inclination i=86.969+/-0.056 degrees. It is composed of two very different stars with an active G6 primary with Teff1=5560+/-160 K, M1=0.680+/-0.045 Msun, R1=0.799+/-0.017 Rsun, and a M3V secondary star with Teff2=3460+/-210 K, M2=0.376+/-0.017 Msun, and R2=0.3475+/-0.0081 Rsun. The primary star is an oversized and spotted active star, hotter than the stars in its mass range. The secondary is a cool star near the mass boundary for fully convective stars (M~0.35 Msun), whose parameters appear to be in agreement with low-mass stellar model.Comment: 18 pages, 15 figures, 15 table

Lower bounds for the centered Hardy-Littlewood maximal operator on the real line

Let $10$ such that for each $f\in L^p(\mathbb{R})$, the centered Hardy-Littlewood maximal operator $M$ on $\mathbb{R}$ satisfies the lower bound $\|Mf\|_{L^p(\mathbb{R})}\ge (1+\varepsilon_p)\|f\|_{L^p(\mathbb{R})}$.Comment: accepted manuscrip

Pseudospectral Model Predictive Control under Partially Learned Dynamics

Trajectory optimization of a controlled dynamical system is an essential part of autonomy, however many trajectory optimization techniques are limited by the fidelity of the underlying parametric model. In the field of robotics, a lack of model knowledge can be overcome with machine learning techniques, utilizing measurements to build a dynamical model from the data. This paper aims to take the middle ground between these two approaches by introducing a semi-parametric representation of the underlying system dynamics. Our goal is to leverage the considerable information contained in a traditional physics based model and combine it with a data-driven, non-parametric regression technique known as a Gaussian Process. Integrating this semi-parametric model with model predictive pseudospectral control, we demonstrate this technique on both a cart pole and quadrotor simulation with unmodeled damping and parametric error. In order to manage parametric uncertainty, we introduce an algorithm that utilizes Sparse Spectrum Gaussian Processes (SSGP) for online learning after each rollout. We implement this online learning technique on a cart pole and quadrator, then demonstrate the use of online learning and obstacle avoidance for the dubin vehicle dynamics.Comment: Accepted but withdrawn from AIAA Scitech 201