Comparison of Current Gravity Estimation and Determination Models

This paper will discuss the history of gravity estimation and determination models while analyzing methods that are in development. Some fundamental methods for calculating the gravity field include spherical harmonics solutions, local weighted interpolation, and global point mascon modeling (PMC). Recently, high accuracy measurements have become more accessible, and the requirements for high order geopotential modeling have become more stringent. Interest in irregular bodies, accurate models of the hydrological system, and on-board processing has demanded a comprehensive model that can quickly and accurately compute the geopotential with low memory costs. This trade study of current geopotential modeling techniques will reveal that each modeling technique has a unique use case. It is notable that the spherical harmonics model is relatively accurate but poses a cumbersome inversion problem. PMC and interpolation models, on the other hand, are computationally efficient, but require more research to become robust models with high levels of accuracy. Considerations of the trade study will suggest further research for the point mascon model. The PMC model should be improved through mascon refinement, direct solutions that stem from geodetic measurements, and further validation of the gravity gradient. Finally, the potential for each model to be implemented with parallel computation will be shown to lead to large improvements in computing time while reducing the memory cost for each technique.Aerospace Engineering and Engineering Mechanic

A global optimization approach to solve multi-aircraft routing problems

"This chapter appears in Computational Models, Software Engineering and Advanced Technologies in Air Transportation edited by Dr. Li Weigang and Dr. Alexandre G. de Barros. Chap.12 pp.237-259. Copyright 2009. Posted by permission of the publisher."This paper describes the formulation and solution of a multi-aircraft routing problem which is posed as a global optimization calculation. The paper extends previous work (involving a single aircraft using two dimensions) which established that the algorithm DIRECT is a suitable solution technique. The present work considers a number of ways of dealing with multiple routes using different problem decompositions. A further enhancement is the introduction of altitude to the problems so that full three-dimensional routes can be produced. Illustrative numerical results are presented involving up to three aircraft and including examples which feature routes over real-life terrain data

Explicit convex and concave envelopes through polyhedral subdivisions with Unstable Equilibria

In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.

QPTAS and Subexponential Algorithm for Maximum Clique on Disk Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Clique on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics '90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show the rather surprising structural result that a disjoint union of cycles is the complement of a disk graph if and only if at most one of those cycles is of odd length. From that, we derive the first QPTAS and subexponential algorithm running in time 2^{O~(n^{2/3})} for Maximum Clique on disk graphs. In stark contrast, Maximum Clique on intersection graphs of filled ellipses or filled triangles is unlikely to have such algorithms, even when the ellipses are close to unit disks. Indeed, we show that there is a constant ratio of approximation which cannot be attained even in time 2^{n^{1-epsilon}}, unless the Exponential Time Hypothesis fails

A global study of enhanced stretching and diffusion in chaotic tangles

A global, finite-time study is made of stretching and diffusion in a class of chaotic tangles associated with fluids described by periodically forced two-dimensional dynamical systems. Invariant lobe structures formed by intersecting global stable and unstable manifolds of persisting invariant hyperbolic sets provide the geometrical framework for studying stretching of interfaces and diffusion of passive scalars across these interfaces. In particular, the present study focuses on the material curve that initially lies on the unstable manifold segment of the boundary of the entraining turnstile lobe.A knowledge of the stretch profile of a corresponding curve that evolves according to the unperturbed flow, coupled with an appreciation of a symbolic dynamics that applies to the entire original material curve in the perturbed flow, provides the framework for understanding the mechanism for, and topology of, enhanced stretching in chaotic tangles. Secondary intersection points (SIP's) of the stable and unstable manifolds are particularly relevant to the topology, and the perturbed stretch profile is understood in terms of the unperturbed stretch profile approximately repeating itself on smaller and smaller scales. For sufficiently thin diffusion zones, diffusion of passive scalars across interfaces can be treated as a one-dimensional process, and diffusion rates across interfaces are directly related to the stretch history of the interface.An understanding of interface stretching thus directly translates to an understanding of diffusion across interfaces. However, a notable exception to the thin diffusion zone approximation occurs when an interface folds on top of itself so that neighboring diffusion zones overlap. An analysis which takes into account the overlap of nearest neighbor diffusion zones is presented, which is sufficient to capture new phenomena relevant to efficiency of mixing. The analysis adds to the concentration profile a saturation term that depends on the distance between neighboring segments of the interface. Efficiency of diffusion thus depends not only on efficiency of stretching along the interface, but on how this stretching is distributed relative to the distance between neighboring segments of the interface

Improved Working Memory but No Effect on Striatal Vesicular Monoamine Transporter Type 2 after Omega-3 Polyunsaturated Fatty Acid Supplementation

Studies in rodents indicate that diets deficient in omega-3 polyunsaturated fatty acids (n-3 PUFA) lower dopamine neurotransmission as measured by striatal vesicular monoamine transporter type 2 (VMAT2) density and amphetamine-induced dopamine release. This suggests that dietary supplementation with fish oil might increase VMAT2 availability, enhance dopamine storage and release, and improve dopamine-dependent cognitive functions such as working memory. To investigate this mechanism in humans, positron emission tomography (PET) was used to measure VMAT2 availability pre- and post-supplementation of n-3 PUFA in healthy individuals. Healthy young adult subjects were scanned with PET using [11C]-(+)-α-dihydrotetrabenzine (DTBZ) before and after six months of n-3 PUFA supplementation (Lovaza, 2 g/day containing docosahexaenonic acid, DHA 750 mg/d and eicosapentaenoic acid, EPA 930 mg/d). In addition, subjects underwent a working memory task (n-back) and red blood cell membrane (RBC) fatty acid composition analysis pre- and post-supplementation. RBC analysis showed a significant increase in both DHA and EPA post-supplementation. In contrast, no significant change in [11C]DTBZ binding potential (BPND) in striatum and its subdivisions were observed after supplementation with n-3 PUFA. No correlation was evident between n-3 PUFA induced change in RBC DHA or EPA levels and change in [11C]DTBZ BPND in striatal subdivisions. However, pre-supplementation RBC DHA levels was predictive of baseline performance (i.e., adjusted hit rate, AHR on 3-back) on the n-back task (y = 0.19+0.07, r2 = 0.55, p = 0.009). In addition, subjects AHR performance improved on 3-back post-supplementation (pre 0.65±0.27, post 0.80±0.15, p = 0.04). The correlation between n-back performance, and DHA levels are consistent with reports in which higher DHA levels is related to improved cognitive performance. However, the lack of change in [11C]DBTZ BPND indicates that striatal VMAT2 regulation is not the mechanism of action by which n-3 PUFA improves cognitive performance. © 2012 Narendran et al

A Method to Find Community Structures Based on Information Centrality

Community structures are an important feature of many social, biological and technological networks. Here we study a variation on the method for detecting such communities proposed by Girvan and Newman and based on the idea of using centrality measures to define the community boundaries (M. Girvan and M. E. J. Newman, Community structure in social and biological networks Proc. Natl. Acad. Sci. USA 99, 7821-7826 (2002)). We develop an algorithm of hierarchical clustering that consists in finding and removing iteratively the edge with the highest information centrality. We test the algorithm on computer generated and real-world networks whose community structure is already known or has been studied by means of other methods. We show that our algorithm, although it runs to completion in a time O(n^4), is very effective especially when the communities are very mixed and hardly detectable by the other methods.Comment: 13 pages, 13 figures. Final version accepted for publication in Physical Review

Tropical Convexity

The notions of convexity and convex polytopes are introduced in the setting of tropical geometry. Combinatorial types of tropical polytopes are shown to be in bijection with regular triangulations of products of two simplices. Applications to phylogenetic trees are discussed. Theorem 29 and Corollary 30 in the paper, relating tropical polytopes to injective hulls, are incorrect. See the erratum at http://www.math.uiuc.edu/documenta/vol-09/vol-09-eng.html .Comment: 20 pages, 6 figure