Minimum-fuel, 3-dimensional flightpath guidance of transfer jets

Minimum fuel, three dimensional flightpaths for commercial jet aircraft are discussed. The theoretical development is divided into two sections. In both sections, the necessary conditions of optimal control, including singular arcs and state constraints, are used. One section treats the initial and final portions (below 10,000 ft) of long optimal flightpaths. Here all possible paths can be derived by generating fields of extremals. Another section treats the complete intermediate length, three dimensional terminal area flightpaths. Here only representative sample flightpaths can be computed. Sufficient detail is provided to give the student of optimal control a complex example of a useful application of optimal control theory

Exact Distance Oracles for Planar Graphs

We present new and improved data structures that answer exact node-to-node distance queries in planar graphs. Such data structures are also known as distance oracles. For any directed planar graph on n nodes with non-negative lengths we obtain the following: * Given a desired space allocation $S\in[n\lg\lg n,n^2]$, we show how to construct in $\tilde O(S)$ time a data structure of size $O(S)$ that answers distance queries in $\tilde O(n/\sqrt S)$ time per query. As a consequence, we obtain an improvement over the fastest algorithm for k-many distances in planar graphs whenever $k\in[\sqrt n,n)$. * We provide a linear-space exact distance oracle for planar graphs with query time $O(n^{1/2+eps})$ for any constant eps>0. This is the first such data structure with provable sublinear query time. * For edge lengths at least one, we provide an exact distance oracle of space $\tilde O(n)$ such that for any pair of nodes at distance D the query time is $\tilde O(min {D,\sqrt n})$. Comparable query performance had been observed experimentally but has never been explained theoretically. Our data structures are based on the following new tool: given a non-self-crossing cycle C with $c = O(\sqrt n)$ nodes, we can preprocess G in $\tilde O(n)$ time to produce a data structure of size $O(n \lg\lg c)$ that can answer the following queries in $\tilde O(c)$ time: for a query node u, output the distance from u to all the nodes of C. This data structure builds on and extends a related data structure of Klein (SODA'05), which reports distances to the boundary of a face, rather than a cycle. The best distance oracles for planar graphs until the current work are due to Cabello (SODA'06), Djidjev (WG'96), and Fakcharoenphol and Rao (FOCS'01). For $\sigma\in(1,4/3)$ and space $S=n^\sigma$, we essentially improve the query time from $n^2/S$ to $\sqrt{n^2/S}$.Comment: To appear in the proceedings of the 23rd ACM-SIAM Symposium on Discrete Algorithms, SODA 201

Computational studies of light acceptance and propagation in straight and curved multimodal active fibres

A Monte Carlo simulation has been performed to track light rays in cylindrical multimode fibres by ray optics. The trapping efficiencies for skew and meridional rays in active fibres and distributions of characteristic quantities for all trapped light rays have been calculated. The simulation provides new results for curved fibres, where the analytical expressions are too complex to be solved. The light losses due to sharp bending of fibres are presented as a function of the ratio of curvature to fibre radius and bending angle. It is shown that a radius of curvature to fibre radius ratio of greater than 65 results in a light loss of less than 10% with the loss occurring in a transition region at bending angles of pi/8 rad.Comment: 21 pages, 13 figure

Space Shuffle: A Scalable, Flexible, and High-Bandwidth Data Center Network

Data center applications require the network to be scalable and bandwidth-rich. Current data center network architectures often use rigid topologies to increase network bandwidth. A major limitation is that they can hardly support incremental network growth. Recent work proposes to use random interconnects to provide growth flexibility. However routing on a random topology suffers from control and data plane scalability problems, because routing decisions require global information and forwarding state cannot be aggregated. In this paper we design a novel flexible data center network architecture, Space Shuffle (S2), which applies greedy routing on multiple ring spaces to achieve high-throughput, scalability, and flexibility. The proposed greedy routing protocol of S2 effectively exploits the path diversity of densely connected topologies and enables key-based routing. Extensive experimental studies show that S2 provides high bisectional bandwidth and throughput, near-optimal routing path lengths, extremely small forwarding state, fairness among concurrent data flows, and resiliency to network failures

Minimal stretch maps between hyperbolic surfaces

This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured laminations, which is attained with probability one on a simple closed curve. Cataclysms are introduced, generalizing earthquakes by permitting more violent shearing in both directions along a fault. Cataclysms provide useful coordinates for Teichmuller space that are convenient for computing derivatives of geometric function in Teichmuller space and measured lamination space.Comment: 53 pages, 11 figures, version of 1986 preprin

Global system of rivers: Its role in organizing continental land mass and defining land‐to‐ocean linkages

The spatial organization of the Earth\u27s land mass is analyzed using a simulated topological network (STN‐30p) representing potential flow pathways across the entire nonglacierized surface of the globe at 30‐min (longitude × latitude) spatial resolution. We discuss a semiautomated procedure to develop this topology combining digital elevation models and manual network editing. STN‐30p was verified against several independent sources including map products and drainage basin statistics, although we found substantial inconsistency within the extant literature itself. A broad suite of diagnostics is offered that quantitatively describes individual grid cells, river segments, and complete drainage systems spanning orders 1 through 6 based on the Strahler classification scheme. Continental and global‐scale summaries of key STN‐30p attributes are given. Summaries are also presented which distinguish basins that potentially deliver discharge to an ocean (exorheic) from those that potentially empty into an internal receiving body (endorheic). A total of 59,122 individual grid cells constitutes the global nonglacierized land mass. At 30‐min spatial resolution, the cells are organized into 33,251 distinct river segments which define 6152 drainage basins. A global total of 133.1 × 106 km2 bear STN‐SOp flow paths with a total length of 3.24 × 106 km. The organization of river networks has an important role in linking land mass to ocean. From a continental perspective, low‐order river segments (orders 1‐3) drain the largest fraction of land (90%) and thus constitute a primary source area for runoff and constituents. From an oceanic perspective, however, the small number (n=101) of large drainage systems (orders 4‐6) predominates; draining 65% of global land area and subsuming a large fraction of the otherwise spatially remote low‐order rivers. Along river corridors, only 10% of land mass is within 100 km of a coastline, 25% is within 250 km, and 50% is within 750 km. The global mean distance to river mouth is 1050 km with individual continental values from 460 to 1340 km. The Mediterranean/Black Sea and Arctic Ocean are the most land‐dominated of all oceans with land:ocean area ratios of 4.4 and 1.2, respectively; remaining oceans show ratios from 0.55 to 0.13. We discuss limitations of the STN‐30p together with its potential role in future global change studies. STN‐30p is geographically linked to several hundred river discharge and chemistry monitoring stations to provide a framework for calibrating and validating macroscale hydrology and biogeochemical flux models