High-accuracy attitude control of satellites in elliptic orbits

A class of feedback attitude-control systems which efficiently result in high-accuracy (10^–4 radians), Earth-pointing motions of satellites of different configurations in elliptic orbits is developed. Gas jets provide the control torque. The state of the attitude motion is assumed to be completely known. Linear differential equations, which have time-varying coefficients and include terms for the gravity torque due to an oblate Earth and terms for the aerodynamic torque, are used to describe the attitude motion when the satellite is practically Earth-pointing. Pontryagin's maximum principle, the necessary conditions for exact solutions of optimal bounded-phase-coordinate problems, and guidelines obtained from the minimum-fuel attitude controls devised for single-axis systems are used in the development. The motions of somestable and someunstable satellites with the developed attitude-control systems are simulated on an analog computer. The overall control system, which is reliable as well as efficient, is simple to realize.This research was sponsored by the National Aeronautics and Space Administration under Grant No. G-133-61.     

The structure of -tensorial cochains of differential operators

Let M be a manifold. Let F = C^∞(M, R). Then the associative algebra of differential operators on is a two-sided -module. We prove that there is a natural isomorphism between the -tensorial Hochschild p-cochains of and the jets, taken on the diagonal, of smooth functions on the Cartesian product of p + 1 copies of M. There is an induced isomorphism of the corresponding associative differential graded algebras. The normalised -tensorial p-cochains correspond isomorphically to jets of those above functions which vanish on all the contiguous subdiagonals x_{j + 1} = X_j, j = 0,…, p − 1 of M^{(p + 1)}. This isomorphism may offer a useful alternative view of infinite-order jets of functions of several variables, taken on the diagonal as cochains of .     

Ein System zur graphischen Darstellung von Vorgangsknoten-Netzplänen mit Hilfe elektronisch gesteuerter Zeichenmaschinen

Zusammenfassung Als Grundlage für Plotterprogramme zum Zeichnen von Netzplänen wurde ein einfaches graphisches System entwickelt. Es geht von einem rechtwinkligen kartesischen Raster aus, auf dessen Gitterpunkten die Vorgangsknoten angeordnet werden können. Das System benutzt nur horizontale und vertikale Verbindungslinien. Die Zuordnung eines Vorgangs zu einem Gitterpunkt legt der Netzplanersteller in einem verkleinerten Entwurfsraster fest. Damit übernimmt er in der derzeitigen Ausbaustufe des Verfahrens die vollständige strukturelle Gestaltung des Netzplans selbst. Das hier beschriebene System GRANET (GRAphische NETzplantechnik) besteht darüber hinaus aus nur wenigen Grundelementen und -regeln, die in einfacher Weise für die Computerverarbeitung aufbereitet werden können. Die nach dem beschriebenen System vom Plotter gezeichneten Netzpläne sind in ihrer klaren und übersichtlichen Darstellung manuell gezeichneten qualitativ durchaus gleichwertig. In der vorliegenden Abhandlung werden im wesentlichen die Elemente und die Eigenschaften des Systems GRANET beschrieben. Ferner wird kurz über einige Kenndaten des nach ihm erstellten EDV-Programms und über dessen Einsatz im Rahmen eines Produktplanungssystems berichtet. Abschließend werden die Erweiterungsmöglichkeiten von Programm und System diskutiert.

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Summary An elementary graphical system has been developed representing a basis for plotter programs for the drawing of networks by means of computer controlled plotter. It is based on a rectangular Cartesian lattice, on to their lattice-points the activity knots may be placed. The system uses horizontal and vertical conjunction lines only. The assignment of an activity to a lattice point is performed by the network planner in a reduced design lattice. Therewith he himself designs in the present degree of completion of the method the entire structural configuration of the network. Apart form this the here described system GRANET (GRAphical NETwork techniques) consists of a few basic elements and rules only which can be converted into data for computer processing in a simple manner. The clear and perspicuous performance of plotter drawn networks produced according to the here described system are thoroughly equivalent to the quality of those drawn by hand. The present paper essentially describes in substance the elements and features of the GRANET-system. In addition to that it is shortly reported on some characteristic data of the provided edp-program and its application within a product planning system. Finally the extension possibilities of the program and system are discussed.

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Auszugsweise vorgetragen auf der DGOR-Jahrestagung am 19. 9. 74 in Würzburg.     

Recovery of the C ∞ jet from the boundary distance function

For a compact Riemannian manifold with boundary, we want to find the metric structure from knowledge of distances between boundary points. This is called the ??boundary rigidity problem??. If the boundary is not concave, which means locally not all shortest paths lie entirely in the boundary, then we are able to find the Taylor series of the metric tensor (C ^?? jet) at the boundary (see Lassas et?al. (Math Ann 325:767?C793, 2003), Uhlmann et?al. (Adv Appl Math 31:379?C387, 2003)). In this paper we give a new reconstruction procedure for the C ^?? jet at non-concave points on the boundary using the localized boundary distance function. A closely related problem is the ??lens rigidity problem??, which asks whether the lens data determine metric structure uniquely. Lens data include information of boundary distance function, the lengths of all geodesics, and the locations and directions where geodesics hit the boundary. We give the first examples that show that lens data cannot uniquely determine the C ^?? jet. The example include two manifolds with the same boundary and the same lens data, but different C ^?? jets. With some additional careful work, we can find examples with different C ¹ jets, which means the boundaries in the two lens-equivalent manifolds have different second fundamental forms.     

Double bubbles in<Emphasis Type="Bold">S</Emphasis><Superscript>3</Superscript> and<Emphasis Type="Bold">H</Emphasis><Superscript>3</Superscript>

We prove the double bubble conjecture in the three-sphereS ³ and hyperbolic three-spaceH ³ in the cases where we can apply Hutchings theory:

Aerodynamics of an isolated slot-burner from a tangentially-fired boiler

The aerodynamic development of fully turbulent isothermal jets issuing from rectangular slot-burners was modelled by obtaining a solution to the Reynolds averaged Navier–Stokes equations. A finite-volume method was used with the standard k–ε, RNG k–ε and Reynolds stress turbulence models. The slot-burners were based on physical models, which were designed to be representative of typical burner geometries found in tangentially-fired coal boilers. Two cases were investigated, in which jets from three vertically stacked rectangular nozzles discharged at 90° and then 60° to the wall containing the burner. The nozzle angle had little effect on jet centreline velocity decay, with the 60° nozzle showing a marginally higher rate of decay. The jets from the 60° nozzles were found to deviate slightly from their geometric axis slightly due to internal pressure redistribution in the flow at the nozzles. The simulations were validated against the physical models and were found to reproduce the flow field of the jets accurately with the Reynolds stress model producing the best results.     

Bi-Lipschitz Sufficiency of Jets

We give some theorems of bi-Lipschitz or C ¹ sufficiency of jets which are expressed by means of transversality with respect to some strata of a stratification satisfying the (L) condition of T. Mostowski. This enables us to prove that the number of metric types of intersection of smooth transversals to a stratum of an (a) regular stratification of a subanalytic set is finite.     

On the semicontinuity of ramification invariants in dimension 2

We consider a cyclic extension L/K of degree 2ⁿ of the field K = k[[T,U]] of characteristic 2. It is shown that for all sufficiently large N, the jets of order N of all curves which are not components of the ramification locus and for which the corresponding valuation of the function field has a unique extension, the valuations of the coefficients of the Inaba equation are positive, and the ramification jumps are maximal, is an open set. In the case of a general (not cyclic) extension, it is shown that the set of jets with a fixed value of the kth jump is the intersection of an open and a closed set. Bibliography: 4 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 321, 2005, pp. 13–35.     

Interpolation along manifolds in Hardy-Sobolev spaces

Traces of submanifolds of the sphere of Hardy-Sobolev spaces in the unit ball of ℂ ⁿ are studied. A complete characterization is obtained, in terms of Besov-type spaces of jets, for complex-tangential submanifolds. Similar results for Bergman-Sobolev spaces are also given. Supported by DGICYT grant PB95-0956-C02-02 and CIRIT grant 1996SGR00026, Spain.     

Movement of biological systems on different environments

This paper describes the movement of biological systems on different kinds of environments, which was simulated with the aid of a digital computer plotter. The biological system is represented by two areas A_a and A_d, called the feeding and discard areas respectively, and separated by a constant distance d. Each triplet (A_a,A_d,d) determines the position of M with respect to E, and the consecutive positions determine the movement of the biological system. The possible movements of the biological system vary according to the areas, the spaces to be covered, the rules that govern the succession of states, etc. The plotter makes it possible to visualize these motions, so that many different interactions between M and E may be studied. It also becomes possible to determine other important parameters, such as the general direction taken by the system M, the width of its path, the distance between successive states, the possibility of environmental regeneration, etc. Therefore, this line of work contributes not only to theoretical biomathematics, but also to a wide variety of practical applications.     

Some variations on the notion of connection

Distributions on manifolds are studied in terms of jets of submanifolds and are interpreted as «pre-connections» or «almost-fibrings»; the associated differential calculus is developed in detail. A comparison with connections on fibred manifolds is analysed. Moreover, «higher order pre-connections», defined as pre-connections dependent on jets of arbitrary order, are introduced and studied. It is shown that infinite jets play an essential role in the associated differential calculus.This work has been performed the visits of Prof. A. M.Vinogradov at the Department of Applied Mathematics, supported by Gruppo Nazionale per la Fisica Matematica of CNR (1989, 1990).This work has been partially supported by funds (40% and 60%) of MURST.     

Simulation of astrophysical phenomena on the basis of high-performance computations

A solution algorithm developed for three-dimensional magnetogasdynamic problems is used to simulate the accretion of interstellar matter onto a massive astronomical object with the formation of collinear jets. A kinetically consistent algorithm is well adapted to the architecture of high-performance computer systems with massive parallelism and has improved conditions for time discretization. The three-dimensional accretion of interstellar matter is computed in detail on a spatial grid consisting of 1 billion nodes, and the possibility of studying the formation of collinear jets is demonstrated.     

Some dimension-free features of vector-valued martingales

Summary Given any local maringaleM inR ^d orl ², there exists a local martingaleN inR ², such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR ² which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|² and |M| satisfy certain SDE's which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.Research supported by NSF grant DMS-9002732 and by AFOSR Contract F49620 85C 0144     

A comparison of heuristic best-first algorithms for bicriterion shortest path problems

A variety of algorithms have been proposed to solve the bicriterion shortest path problem. This article analyzes and compares the performance of three best-first (label-setting) algorithms that accept heuristic information to improve efficiency. These are NAMOA^∗, MOA^∗, and Tung & Chew’s algorithm (TC). A set of experiments explores the impact of heuristic information in search efficiency, and the relative performance of the algorithms. The analysis reveals that NAMOA^∗ is the best option for difficult problems. Its time performance can benefit considerably from heuristic information, though not in all cases. The performance of TC is similar but somewhat worse. However, the time performance of MOA^∗ is found to degrade considerably with the use of heuristic information in most cases. Explanations are provided for these phenomena.     

β-delayed proton decays of81Zr and85Mo

β-delayed proton decays of T_z = 1/2 series nuclei⁸¹Zr and⁸⁵Mo have been studied in detail by using p-γ coincidence measurement. The β-delayed proton spectra populating the first excited states of the daughter nuclei have been obtained. Half-life of⁸¹Zr has been measured tobe5.3s ± 0.5 s and that of⁸⁵Mo 3.2 s ± 0.2 S. By using statistical model calculations and systematic analyses, spins and parities for the ground states of⁸¹Zr and⁸⁵Mo have been tentatively assigned to be 3/2⁻ and 1/2⁻, respectively. Mass excess of⁸¹Zris -58.3 MeV± 0.2 MeV and that of⁸⁵Mo-59.1 MeV ± 0.4 MeV. Combining the measured half-lives with the calculated partial ones yields the branches of β-de-layed proton decay for⁸¹zr and⁸⁵Mo of (1.2 ± 0.2) × 10^-3 and (1.4 ± 0.2) × 10^-3, respectively.     

Distribution of Cardinalities of Row Spaces of Boolean Matrices of Order n

Let R(A) denote the row space of a Boolean matrix A of order n. We show that if n 7, then the cardinality |R(A)| (2^n–1 - 2^n–5, 2^n–1 - 2^n–6) U (2^n–1 - 2^n–6, 2^n–1). This result confirms a conjecture in [1].AMS Subject Classification (1991): 05B20 06E05 15A36Support partially by the Postdoctoral Science Foundation of China.Dedicated to Professor Chao Ko on the occasion of his 90th birthday     

Numerical analysis of flow-induced gas entrainment in roll coating

Gas entrainment by plane liquid jets which plunge into a liquid pool is analyzed by numerical simulations. The numerical model is based on the equations of incompressible newtonian fluids flow. The two-phase flow problem is described with the volume-of-fluid method. The dynamic behaviour of the interface is characterized by two similarity parameters, the capillary number Ca = uη/σ and the property number Γ = σ(ρ/η⁴g)^1/3 where u is the velocity, η the dynamic viscosity, σ the interface tension, ρ the density and g the gravitational constant. Numerical simulations are performed with the open source CFD code OpenFOAM. In the simulations the stability of the gas–liquid meniscus is tested for different sets of Ca and Γ. Critical values of Ca which indicate the beginning gas entrainment are deduced from the inspection of the simulation results. The findings of the numerical investigations agree well with corresponding experimental results.     

Stochastic differential equations with fractal noise

Stochastic differential equations in ?ⁿ with random coefficients are considered where one continuous driving process admits a generalized quadratic variation process. The latter and the other driving processes are assumed to possess sample paths in the fractional Sobolev space W^β₂ for some β > 1/2. The stochastic integrals are determined as anticipating forward integrals. A pathwise solution procedure is developed which combines the stochastic Itô calculus with fractional calculus via norm estimates of associated integral operators in W^α ₂ for 0 < α < 1. Linear equations are considered as a special case. This approach leads to fast computer algorithms basing on Picard's iteration method. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fast Wave Averaging for the Equatorial Shallow Water Equations

The equatorial shallow water equations in a suitable limit are shown to reduce to zonal jets as the Froude number tends to zero. This is a theorem of a singular limit with a fast variable coefficient due to the vanishing of the Coriolis force at the equator. Although it is not possible to get uniform estimates in classical Sobolev spaces (other than L²) by differentiating the system, a new method exploiting the particular structure of the fast coefficient leads to uniform estimates in slightly different functional spaces. The computation of resonances shows that fast waves may interact with a strong external forcing, introduced to mimic the effects of moisture, to create zonal jets.