On the Multilinear Restriction and Kakeya conjectures

We prove $d$-linear analogues of the classical restriction and Kakeya conjectures in $\R^d$. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called "joints" problem, as well as presenting some $n$-linear analogues for $n<d$.Comment: 38 pages, no figures, submitte

Flexibility Ranking To Schedule Events

A system and a method are disclosed, to identify suitable time slots to schedule events. The method involves providing flexibility ranking to time slots in a calendar based on events scheduled. When a new event with attendees is created, the system looks into each attendee’s calendar to rank the time slots based on flexibly. The time slot with maximum flexibility is identified as suitable. The system enabling flexibility ranking to schedule events further creates a quantifiable framework to analyze calendar conflicts. At the time of a calendar conflict, the system determines calendar density percentage of each attendee. The system then provides a time slot with maximum flexibility for all the attendees. This method saves time, by eliminating a need to manually find a suitable time slot for meetings

Finite bounds for Holder-Brascamp-Lieb multilinear inequalities

A criterion is established for the validity of multilinear inequalities of a class considered by Brascamp and Lieb, generalizing well-known inequalities of Holder, Young, and Loomis-Whitney. This is a companion to a recent paper by the same authors, containing a different proof and discussing certain variants.Comment: 15 page

Adjoint Brascamp-Lieb inequalities

The Brascamp-Lieb inequalities are a generalization of the H\"older, Loomis-Whitney, Young, and Finner inequalities that have found many applications in harmonic analysis and elsewhere. In this paper we introduce an "adjoint" version of these inequalities, which can be viewed as an $L^p$ version of the entropy Brascamp-Lieb inequalities of Carlen and Cordero-Erausquin. As applications, we reprove a log-convexity property of the Gowers uniformity norms, and establish some reverse $L^p$ inequalities for various tomographic transforms. We conclude with some open questions.Comment: 43 page

Effects of low earth orbit on the optical performance of multi-layer enhanced high reflectance mirrors

Two mirror designs developed for space applications were flown along with a standard mid-infrared design on the leading and trailing edges of the Long Duration Exposure Facility (LDEF). Preliminary observations of induced changes in optical performance of ZnS-coated mirrors and impact-related microstructural and microchemical effects are described in the proceedings of the First LDEF Post-Retrieval Symposium. In this paper, effects of the induced environment and meteoroid/debris impacts on mirror performance are described in more detail. Also, an analysis of reflectance spectra using the results of Auger and secondary ion mass spectroscopy (SIMS) profiling measurements are used to identify an optical-degradation mechanism for the ZnS-coated mirrors. Structural damage associated with a high-velocity impact on a (Si/Al2O3)-coated mirror was imaged optically and with scanning electron and atomic force microscopy (SEM and AFM). Scanning Auger and SIMS analysis provided chemical mapping of selected impact sites. The impact data suggest design and fabrication modifications for obtaining improved mechanical performance using a design variation identified in preflight laboratory simulations. Auger surface profile and SIMS imaging data verified the conclusion that secondary impacts are the source of contamination associated with the dendrites grown on the leading-edge ZnS-coated test samples. It was also found that dendrites can be grown in the laboratory by irradiating contaminated sites on a trailing-edge ZnS-coated sample with a rastered electron beam. These results suggest a mechanism for dendrite growth