On nonlocal problems for fractional differential equations in Banach spaces

In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results

Determination of Dark Matter Halo Mass from Dynamics of Satellite Galaxies

We show that the mass of a dark matter halo can be inferred from the dynamical status of its satellite galaxies. Using 9 dark-matter simulations of halos like the Milky Way (MW), we find that the present-day substructures in each halo follow a characteristic distribution in the phase space of orbital binding energy and angular momentum, and that this distribution is similar from halo to halo but has an intrinsic dependence on the halo formation history. We construct this distribution directly from the simulations for a specific halo and extend the result to halos of similar formation history but different masses by scaling. The mass of an observed halo can then be estimated by maximizing the likelihood in comparing the measured kinematic parameters of its satellite galaxies with these distributions. We test the validity and accuracy of this method with mock samples taken from the simulations. Using the positions, radial velocities, and proper motions of 9 tracers and assuming observational uncertainties comparable to those of MW satellite galaxies, we find that the halo mass can be recovered to within $\sim$40%. The accuracy can be improved to within $\sim$25% if 30 tracers are used. However, the dependence of the phase-space distribution on the halo formation history sets a minimum uncertainty of $\sim$20% that cannot be reduced by using more tracers. We believe that this minimum uncertainty also applies to any mass determination for a halo when the phase space information of other kinematic tracers is used.Comment: Accepted for publication in ApJ, 18 pages, 13 figure

Localizing Region-Based Level-set Contouring for Common Carotid Artery in Ultrasonography

 This work developed a fully-automated and efficient method for detecting contour of common carotid artery in the cross section view of two-dimensional B-mode sonography. First, we applied a preprocessing filter to the ultrasound image for the sake of reducing speckle. An adaptive initial contouring method was then performed to obtain the initial contour for level set segmentation. Finally, the localizing region-based level set segmentation automatically extracted the precise contours of common carotid artery. The proposed method evaluated 130 ultrasound images from three healthy volunteers and the segmentation results were compared to the boundaries outlined by an expert. Preliminary results showed that the method described here could identify the contour of common carotid artery with satisfactory accuracy in this dataset

Huber Principal Component Analysis for Large-dimensional Factor Models

Factor models have been widely used in economics and finance. However, the heavy-tailed nature of macroeconomic and financial data is often neglected in the existing literature. To address this issue and achieve robustness, we propose an approach to estimate factor loadings and scores by minimizing the Huber loss function, which is motivated by the equivalence of conventional Principal Component Analysis (PCA) and the constrained least squares method in the factor model. We provide two algorithms that use different penalty forms. The first algorithm, which we refer to as Huber PCA, minimizes the $\ell_2$-norm-type Huber loss and performs PCA on the weighted sample covariance matrix. The second algorithm involves an element-wise type Huber loss minimization, which can be solved by an iterative Huber regression algorithm. Our study examines the theoretical minimizer of the element-wise Huber loss function and demonstrates that it has the same convergence rate as conventional PCA when the idiosyncratic errors have bounded second moments. We also derive their asymptotic distributions under mild conditions. Moreover, we suggest a consistent model selection criterion that relies on rank minimization to estimate the number of factors robustly. We showcase the benefits of Huber PCA through extensive numerical experiments and a real financial portfolio selection example. An R package named ``HDRFA" has been developed to implement the proposed robust factor analysis

Causality bounds on scalar-tensor EFTs

We compute the causality/positivity bounds on the Wilson coefficients of scalar-tensor effective field theories. Two-sided bounds are obtained by extracting IR information from UV physics via dispersion relations of scattering amplitudes, making use of the full crossing symmetry. The graviton $t$-channel pole is carefully treated in the numerical optimization, taking into account the constraints with fixed impact parameters. It is shown that the typical sizes of the Wilson coefficients can be estimated by simply inspecting the dispersion relations. We carve out sharp bounds on the leading coefficients, particularly, the scalar-Gauss-Bonnet couplings, and discuss how some bounds vary with the leading $(\partial\phi)^4$ coefficient and as well as phenomenological implications of the causality bounds.Comment: 72 pages, 15 figure