Sufficiency of a Numerical Downstream Continuation

(First paragraph) Customarily one does not impose n-th order boundary conditions on the solution of initial/boundary value problems whose characterizing partial differential equations are also n-th order. However, conjecture that such problems are not well-posed, or that a solution might not exist, is not always justified [l]. Perhaps a physically more natural example is provided by problems of computational fluid dynamics. Here boundary conditions which correctly should be applied at an infinite distance downstream from the region of interest are for computational convenience often applied at a finite location [2]. Results of numerical experimentation on viscous flows governed by the Navier-Stokes equations indicate that downstream continuation achieved by applying a second derivative boundary condition at a finite location often provides the least restrictive method of closing the flow [3]

Alien Registration- Scott, Charlie H. (Portland, Cumberland County)

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On the Existence of Strong Solutions to Autonomous Differential Equations with Minimal Regularity

For proving the existence and uniqueness of strong solutions to dY/dt = F(Y), Y(0) = C, the most quoted condition seen in elementary differential equations texts is that F(Y) and its first derivative be continuous. One wonders about the existence of a minimal regularity condition which allows unique strong solutions. In this note, a bizarre example is seen where F(Y) is not differentiable at an equilibrium solution; yet unique non-global strong solutions exist at each point, whereas global non-unique weak solutions are allowed. A characterizing theorem is obtained

Sufficiency of a Numerical Downstream Continuation

(First paragraph) Customarily one does not impose n-th order boundary conditions on the solution of initial/boundary value problems whose characterizing partial differential equations are also n-th order. However, conjecture that such problems are not well-posed, or that a solution might not exist, is not always justified [l]. Perhaps a physically more natural example is provided by problems of computational fluid dynamics. Here boundary conditions which correctly should be applied at an infinite distance downstream from the region of interest are for computational convenience often applied at a finite location [2]. Results of numerical experimentation on viscous flows governed by the Navier-Stokes equations indicate that downstream continuation achieved by applying a second derivative boundary condition at a finite location often provides the least restrictive method of closing the flow [3]

A System Equivalence Related to Dulac\u27s Extension of Bendixson\u27s Negative Theorem for Planar Dynamical Systems

Bendixson\u27s Theorem [H. Ricardo, A Modem Introduction to Differential Equations, Houghton-Mifflin, New York, Boston, 2003] is useful in proving the non-existence of periodic orbits for planar systems dx/dt = F(x, y), dy/dt = G (x, y) in a simply connected domain D, where F, G are continuously differentiable. From the work of Dulac [M. Kot, Elements of Mathematical Ecology, 2nd printing, University Press, Cambridge, 2003] one suspects that system (1) has periodic solutions if and only if the more general system dx/d tau = B(x, y)F(x, y), dy/d tau = B(x, y)G(x, y) does, which makes the subcase (1) more tractable, when suitable non-zero B (x, y) which are C1(D) can be found. Thus, Bendixson\u27s Theorem can be applied to system (2), where otherwise it is unfruitful in establishing the non-existence of periodic solutions for system (1). The object of this note is to give a simple proof justifying this Dulac-related postulate of the equivalence of systems (1) and (2). (c) 2006 Elsevier Ltd. All rights reserved

On the Existence of Periodic and Eventually Periodic Solutions of a Fluid Dynamic Forced Harmonic Oscillator

For certain flow regimes, the nonlinear differential equation Y¨=F(Y)−G, Y≥0, G\u3e0 and constant, models qualitatively the behaviour of a forced, fluid dynamic, harmonic oscillator which has been a popular department store attraction. The device consists of a ball oscillating suspended in the vertical jet from a household fan. From the postulated form of the model, we determine sets of attraction and exploit symmetry properties of the system to show that all solutions are either initially periodic, with the ball never striking the fan, or else eventually approach a periodic limit cycle, after a sufficient number of bounces away from the fan

Probability Models for Blackjack Poker

For simplicity in calculation, previous analyses of blackjack poker have employed models which employ sampling with replacement. in order to assess what degree of error this may induce, the purpose here is to calculate results for a typical hand where sampling without replacement is employed. It is seen that significant error can result when long runs are required to complete the hand. The hand examined is itself of particular interest, as regards both its outstanding expectations of high yield and certain implications for pair splitting of two nines against the dealer\u27s seven. Theoretical and experimental methods are used in order to determine whether the calculation can be truncated after the dealer\u27s fifth card without significant loss of accuracy. Less than one tenth of one percent difference is observed between the fifth card truncated expectation and the experimentally obtained expectation