Jeffrey L. Jauregui

Penrose-type inequalities with a Euclidean background

The Riemannian Penrose inequality (RPI) bounds from below the ADM mass of asymptotically flat manifolds of nonnegative scalar curvature in terms of the total area of all outermost compact minimal surfaces. The general form of the RPI is currently known for manifolds of dimension up to seven. In the present work, we prove a Penrose-like inequality that is valid in all dimensions, for conformally flat manifolds. Our inequality treats the area contributions of the minimal surfaces in a more favorable way than the RPI, at the expense of using the smaller Euclidean area (rather than the intrinsic area). We give an example in which our estimate is sharper than the RPI when many minimal surfaces are present. We do not require the minimal surfaces to be outermost. We also generalize the technique to allow for metrics conformal to a scalar-flat (not necessarily Euclidean) background and prove a Penrose-type inequality without an assumption on the sign of scalar curvature. Finally, we derive a new lower bound for the ADM mass of a conformally flat, asymptotically flat manifold containing any number of zero area singularities.

Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass II

In this sequel paper, we give a shorter, second proof of the monotonicity of the Hawking mass for time flat surfaces under spacelike uniformly area expanding flows in spacetimes that satisfy the dominant energy condition. We also include a third proof which builds on a known formula and describe a class of sufficient conditions of divergence type for the monotonicity of the Hawking mass. These flows of surfaces may have connections to the problem in general relativity of bounding the total mass of a spacetime from below by the quasi-local mass of spacelike 2-surfaces in the spacetime.

Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass

We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called “time flat surfaces.” Such flows generalize inverse mean curvature flow, which was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality for one black hole. A flow of time flat surfaces may have connections to the problem in general relativity of bounding the mass of a spacetime from below by the quasi-local mass of a spacelike 2-surface contained therein.

Extensions and fill-ins with non-negative scalar curvature

Motivated by the quasi-local mass problem in general relativity, we apply the asymptotically flat extensions, constructed by Shi and Tam in the proof of the positivity of the Brown–York mass, to study a fill-in problem of realizing geometric data on a 2-sphere as the boundary of a compact 3-manifold of non-negative scalar curvature. We characterize the relationship between two borderline cases: one in which the Shi–Tam extension has zero total mass, and another in which fill-ins of non-negative scalar curvature fail to exist. Additionally, we prove a type of positive mass theorem in the latter case.

Epilepsy Treatment Simplified through Mobile Ketogenic Diet Planning

BACKGROUND The Ketogenic Diet (KD) is an effective, alternative treatment for refractory epilepsy. This high fat, low protein and carbohydrate diet mimics the metabolic and hormonal changes that are associated with fasting. AIMS To maximize the effectiveness of the KD, each meal is precisely planned, calculated, and weighed to within 0.1 gram for the average three-year duration of treatment. Managing the KD is time-consuming and may deter caretakers and patients from pursuing or continuing this treatment. Thus, we investigated methods of planning KD faster and making the process more portable through mobile applications. METHODS Nutritional data was gathered from the United States Department of Agriculture (USDA) Nutrient Database. User selected foods are converted into linear equations with n variables and three constraints: prescribed fat content, prescribed protein content, and prescribed carbohydrate content. Techniques are applied to derive the solutions to the underdetermined system depending on the number of foods chosen. RESULTS The method was implemented on an iOS device and tested with varieties of foods and different number of foods selected. With each case, the applications constructed meal plan was within 95% precision of the KD requirements. CONCLUSION In this study, we attempt to reduce the time needed to calculate a meal by automating the computation of the KD via a linear algebra model. We improve upon previous KD calculators by offering optimal suggestions and incorporating the USDA database. We believe this mobile application will help make the KD and other dietary treatment preparations less time consuming and more convenient.