Local models for Galois deformation rings and applications

We construct projective varieties in mixed characteristic whose singularities model, in generic cases, those of tamely potentially crystalline Galois deformation rings for unramified extensions of $\mathbb{Q}_p$ with small regular Hodge-Tate weights. We establish several significant facts about their geometry including a unibranch property at special points and a representation theoretic description of the irreducible components of their special fibers. We derive from these geometric results a number of local and global consequences: the Breuil-M\'ezard conjecture in arbitrary dimension for tamely potentially crystalline deformation rings with small Hodge-Tate weights (with appropriate genericity conditions), the weight part of Serre's conjecture for $U(n)$ as formulated by Herzig (for global Galois representations which satisfy the Taylor-Wiles hypotheses and are sufficiently generic at $p$), and an unconditional formulation of the weight part of Serre's conjecture for wildly ramified representations

A Liquid-Liquid Thermoelectric Heat Exchanger as a Heat Pump for Testing Phase Change Material Heat Exchangers

The primary objective of the Phase Change HX payload on the International Space Station (ISS) is to test and demonstrate the viability and performance of Phase Change Material Heat Exchangers (PCM HX). The system was required to pump a working fluid through a PCM HX to promote the phase change material to freeze and thaw as expected on Orion's Multipurpose Crew Vehicle. Due to limitations on ISS's Internal Thermal Control System, a heat pump was needed on the Phase Change HX payload to help with reducing the working fluid's temperature to below 0degC (32degF). This paper will review the design and development of a TEC based liquid-liquid heat exchanger as a way to vary to fluid temperature for the freeze and thaw phase of the PCM HX. Specifically, the paper will review the design of custom coldplates and sizing for the required heat removal of the HX

QoS-Aware Utility-Based Resource Allocation in Mixed-Traffic Multi-User OFDM Systems

This paper deals with the joint subcarrier and power allocation problem in a downlink multi-user orthogonal frequency division multiplexing system subject to user delay and minimum rate quality-of-service (QoS) requirements over a frequency-selective multi-carrier fading channel. We aim to maximize the utility-pricing function, formulated as the difference between the achieved spectral efficiency and the associated linear cost function of transmit power scaled by a system-dependent parameter. For a homogeneous system, we show that the joint resource allocation can be broken down into sequential problems while retaining the optimality. Specifically, the optimal solution is obtained by first assigning each subcarrier to the user with the best channel gain. Subsequently, the transmit power for each subcarrier is adapted according to water-filling policy if the global optimum is feasible, else it is given by a nonwater-filling power adaptation. For a heterogeneous system, an optimal solution needs exhaustive search and hence, we resort to two reduced-complexity sub-optimal algorithms. Algorithm-I is a simple extension of the aforementioned optimal algorithm developed for a homogeneous system, while Algorithm-II further takes into consideration the heterogeneity in user QoS requirements for performance enhancement. Simulation results reveal the impacts of user QoS requirements, number of subcarriers and number of users on the system transmit power

Structural Results and Approximation Algorithms in Minor-free Graphs

Planarity has been successfully exploited to design faster and more accurate approximation algorithms for many graph optimization problems. The celebrated theorem of Kuratowski completely characterizes planar graphs as those excluding K_5 and K_{3,3} as minors. Kuratowski's theorem allows one to generalize planar graphs to H-minor-free graphs: those that exclude a fixed graph H as a minor. The deep results of Robertson and Seymour reveal many hidden structures in H-minor-free graphs, that have been used extensively in algorithmic designs. Relying on these structures, we design (i) an (efficient) polynomial time approximation scheme (PTAS) for two different variants of the traveling salesperson problem (TSP) and (ii) simple local search PTASes for r-dominating set and feedback vertex set problems. We then present several results concerning structures of planar graphs. Specifically, we make progresses on two conjectures on existence of large induced forests in planar graphs

Design and Development of a Sub-Zero Fluid System for Demonstration of Orion's Phase Change Material Heat Exchangers on ISS

NASA's Orion Multipurpose Crew Vehicle's Exploration Mission 2 is expected to loiter in Lunar orbit for a relatively long period of time. In low Lunar orbit (LLO) the thermal environment is cyclic - extremely cold in the eclipse and relatively hot near the subsolar point. Phase change material heat exchangers (PCM HXs) are the best option for long term missions in these environments. A PCM HX allows a vehicle to store excess waste energy by thawing a phase change material such as n-pentadecane wax. During portions of the orbit that are extremely cold, the excess energy is rejected, resolidifying the wax. Due to the inherent risk of compromising the heat exchanger during multiple freeze and thaw cycles, a unique payload was designed for the International Space Station to test and demonstration the functions of a PCM HX. The payload incorporates the use of a pumped fluid system and a thermoelectric heat exchanger to promote the freezing and thawing of the PCM HX. This paper shall review the design and development undertaken to build such a system