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Consider a triangulation of the 2-sphere thought of as the boundary of a 3-ball. A pairing of faces determines a quotient map in which paired faces are idenitified. A classical Euler characteristic criterion determines whether the quotient complex is a 3-manifold. In this talk we look at a family of cyclically presented groups that support face pairings and it is shown that under certain conditions the presentations correspond to 3- manifold spines.
In this talk we present some results related to the recent solution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava. We sharpen the constant in the $KS_2$ conjecture of Weaver that plays a key role in this solution. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger conjecture. The talk is based on a joint work with Casazza, Marcus, and Speegle.
Processor sharing is a mathematical idealization of round-robin scheduling algorithms commonly used in computer time-sharing. It is a fundamental example of a non-head-of-the-line service discipline. For such disciplines, it is typical that any Markov description of the system state is infinite dimensional. Due to this, measure-valued stochastic processes are becoming a key tool used in the modeling and analysis of stochastic network models operating under various non-head-of-the-line service disciplines.
In this talk, we discuss a new approach to studying the asymptotic behavior of fluid model solutions (formal functional law of large numbers limits) for critically loaded processor sharing queues. For this, we introduce a notion of relative entropy associated with measure-valued fluid model solutions. This approach is developed with idea that similar notions involving relative entropy may be helpful for understanding the asymptotic behavior of critical fluid model solutions for stochastic networks operating under protocols naturally described by measure-valued processes.
The title of Dr. Puha's talk is "From Queueing Theory to Modern Stochastic Network Models: A Mathematician's Perspective." Dr. Puha is also our colloquium speaker this week.
Abstract: The standard method for designing a secure computation protocol for a function f first transforms f into either a circuit or a RAM program and then applies a generic secure computation protocol that either handles boolean gates or translates the RAM program into oblivious RAM instructions respectively.
In this work, we show a large class of functions for which a different algorithmic approach to secure computation results in more efficient protocols. The first such examples of this technique was presented by Aggrawal, Mishra, and Pinkas (J. of Cryptology, 2010) for computing the median; later, Brickell and Shmatikov (Asiacrypt 2005) showed a similar technique for minimum spanning tree and a variant of shortest paths.
We generalize the technique in both of those works and show that it applies for a large class of problems including matroid optimizations, sub-modular optimization, convex hulls, and other scheduling problems. The crux of our technique is to securely reduce these problems to secure comparison operations. We then describe general properties under which simulation-based security can be achieved for honest-but-curious and covert adversary models.
In a shortest remaining processing time (SRPT) queue, the job that requires the least amount of processing time is preemptively served first. One effect of this is that the queue length is small in comparison to the total amount of work in the system (measured in units of processing time). In the case of processing time distributions with unbounded support, the queue length is so small that the sequence of queue length processes associated with a sequence of SRPT queues, rescaled with standard functional central limit theorem scaling and satisfying standard heavy traffic conditions, converge in distribution to the process that is identically equal to zero. This happens despite the fact that in this same regime the rescaled workload processes converge to a non-degenerate reflected Brownian motion. In particular, the queue length process is of smaller order magnitude than the workload process. In the case of processing time distributions that satisfy a rapid variation condition, we implement an alternative, unconventional spatial scaling that leads to a non-trivial limit for the queue length process. This result quantifies this order of magnitude difference between queue length and workload processes. We illustrate this result for Weibull processing time distributions.
Many fishes are capable of extraordinary athletic feats, including short sprints at blistering speeds or migrations across entire oceans. This performance is all the more impressive when one considers that these fishes are breathing water, which contains ~30X less oxygen than the air that we breathe. One of our lab’s primary interests is in understanding how this is achieved. We have used computational fluid dynamics and quantitative flow visualization to thoroughly examine how water passes through the gills of several fish species. Our results suggest that the flow patterns around the gills have dramatic and previously unrecognized effects on oxygen uptake. Such results provide new perspectives on the respiratory physiology of fishes and perhaps other aquatic animals.
In this thesis we construct compatible discretizations of Maxwell’s equations. We use the term compatible to describe numerical methods for Maxwell’s equations which obey many properties of vector Calculus in a discrete setting. Compatible discretizations preserve the exterior Calculus ensuring that the divergence of the curl and the curl of a gradient are zero in a discrete setting. This compatibility of discretizations with the continuum Maxwell’s equations guarantees that the numerical solutions are physically meaningful.
We focus on the construction of a class of discretizations called Mimetic Finite Differences (MFD). The MFD method is a generalization of both staggered finite differences and mixed finite elements. We construct a parameterized family of MFD methods with equivalent formal order of accuracy. For time-dependent problems, we exploit this non-uniqueness by finding parameters which are optimal with respect to a certain criteria, for example, minimizing dispersion error. Dispersion error is a numerical artifact in which individual frequencies in a wave propagate at incorrect speeds; dominating the error in wave problems over long time propagation. The novelty of this work is the construction of an MFD discretization for Maxwell’s equations which reduces dispersion error for transient wave propagation in materials that are modeled by a general class of linear constitutive laws. We provide theoretical analysis of these new discretizations including an analysis of stability and discrete divergence. We also provide numerical demonstrations to illustrate the theory.
In addition to applications in the time domain we consider equilibrium Magnetohydrodynamic (MHD) generators. MHD generators extract power directly from a plasma by passing it through a strong magnetic field. Used as a topping cycle for traditional steam turbine generator, MHD offers a theoretical thermal efficiency of 60% compared to 40% of traditional systems. However, this technology has high life cycle costs due to equipment failure. One source of failure is arcing: the formation of high density currents which damage the generator. In this work we develop, analyze, and simulate a model of these generators. We use these simulations to show the viability of detecting electrical arcs by measurements of their magnetic fields outside of the generator.