40,173 research outputs found

    Numerical Studies of the Gauss Lattice Problem

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    The difference between the number of lattice points N(R) that lie in x^2 + y^2 ≤ R^2 and the area of that circle, d(R) = N(R) - πR^2, can be bounded by |d(R)| ≤ KR^θ. Gauss showed that this holds for θ = 1, but the least value for which it holds is an open problem in number theory. We have sought numerical evidence by tabulating N(R) up to R ≈ 55,000. From the convex hull bounding log |d(R)| versus log R we obtain the bound θ ≤ 0.575, which is significantly better than the best analytical result θ ≤ 0.6301 ... due to Huxley. The behavior of d(R) is of interest to those studying quantum chaos

    Comments on "Numerical studies of viscous flow around circular cylinders"

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    It is claimed by Hamielec and Raal(1) that their computations improve upon the extrapolation procedure of Keller and Takami(2) which is considered “inadequate” and “could presumably lead to appreciable errors.” However, the authors clearly do not understand the procedure of Keller and Takami or else do not understand the nature of Imai’s asymptotic solution, or both

    On the cometary hydrogen coma and far UV emission

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    Cometary hydrogen observations are reviewed with emphasis on observations of comet Bennett. The results are theoretically interpreted and a brief summary of ultraviolet observations other than Lyman alpha is given

    Spirillum swimming: theory and observations of propulsion by the flagellar bundle

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    The hydrodynamics and energetics of helical swimming by the bacterium Spirillum sp. is analysed using observations from medium speed cine photomicrography and theory. The photographic records show that the swimming organism's flagellar bundles beat in a helical fashion just as other bacterial flagella do. The data are analysed according to the rotational resistive theory of Chwang & Wu (1971) in a simple-to-use parametric form with the viscous coefficients C_s and C_n calculated according to the method of Lighthill (1975). Results of the analysis show that Spirillum dissipated biochemical energy in performing work against fluid resistance to motion at an average rate of about 6 X 10^(−8) dyne cm s^(-1) with some 62–72% of the power dissipation due to the non-contractile body. These relationships yield a relatively low hydromechanical efficiency which is reflected in swimming speeds much smaller than a representative eukaryote. In addition the C_n/C_s ratio for the body is shown to lie in the range 0–86-1-51 and that for the flagellar bundle in the range 1–46-1-63. The implications of the power calculations for the Berg & Anderson (1973) rotating shaft model are discussed and it is shown that a rotational resistive theory analysis predicts a 5-cross bridge M ring for each flagellum of Spirillum

    Three-Dimensional Ray Tracing and Geophysical Inversion in Layered Media

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    In this paper the problem of finding seismic rays in a three-dimensional layered medium is examined. The "layers" are separated by arbitrary smooth interfaces that can vary in three dimensions. The endpoints of each ray and the sequence of interfaces it encounters are specified. The problem is formulated as a nonlinear system of equations and efficient, accurate methods of solution are discussed. An important application of ray tracing methods, which is discussed, is the nonlinear least squares estimation of medium parameters from observed travel times. In addition the "type" of each ray is also determined by the least squares process—this is in effect a deconvolution procedure similar to that desired in seismic exploration. It enables more of the measured data to be used without filtering out the multiple reflections that are not pure P-waves

    Exact Boundary Conditions at an Artificial Boundary for Partial Differential Equations in Cylinders

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    The numerical solution of partial differential equations in unbounded domains requires a finite computational domain. Often one obtains a finite domain by introducing an artificial boundary and imposing boundary conditions there. This paper derives exact boundary conditions at an artificial boundary for partial differential equations in cylinders. An abstract theory is developed to analyze the general linear problem. Solvability requirements and estimates of the solution of the resulting finite problem are obtained by use of the notions of exponential and ordinary dichotomies. Useful representations of the boundary conditions are derived using separation of variables for problems with constant tails. The constant tail results are extended to problems whose coefficients obtain limits at infinity by use of an abstract perturbation theory. The perturbation theory approach is also applied to a class of nonlinear problems. General asymptotic formulas for the boundary conditions are derived and displayed in detail

    The Numerical Calculation of Traveling Wave Solutions of Nonlinear Parabolic Equations

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    Traveling wave solutions have been studied for a variety of nonlinear parabolic problems. In the initial value approach to such problems the initial data at infinity determines the wave that propagates. The numerical simulation of such problems is thus quite difficult. If the domain is replaced by a finite one, to facilitate numerical computations, then appropriate boundary conditions on the "artificial" boundaries must depend upon the initial data in the discarded region. In this work we derive such boundary conditions, based on the Laplace transform of the linearized problems at ±∞, and illustrate their utility by presenting a numerical solution of Fisher’s equation which has been proposed as a model in genetics

    Difference Methods and Deferred Corrections for Ordinary Boundary Value Problems

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    Compact as possible difference schemes for systems of nth order equations are developed. Generalizations of the Mehrstellenverfahren and simple theoretically sound implementations of deferred corrections are given. It is shown that higher order systems are more efficiently solved as given rather than as reduced to larger lower order systems. Tables of coefficients to implement these methods are included and have been derived using symbolic computations
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