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Title

Summary

Reference

Dwnld

[21]

Barycentric Hermite interpolation

Numerical stability of barycentric Hermite interpolation even
when derivatives are of high order. New
method to update barycentric coefficients in linear time.

Sadiq and DV, submitted.

pdf

[20]

Metric Entropy and the Optimal Prediction of Chaotic Signals

We show that all current methods for predicting chaotic
signals are necessarily suboptimal. We make a firm and
significant connection between the problem of predicting
chaotic signals and a major result in modern information
theory. We investigate ideas that point the way to the
derivation of an optimal general purpose predictor for chaotic
signals.

DV, Kirill Serkh, and Xuan Liang, submitted.

pdf

[19]

Finite Difference Weights Using The Modified Lagrange Interpolant

A new method for computing finite
difference weights. Basic result saying that the order of accuracy
of a finite difference formula cannot be boosted by more than one
beyond its typical order of accuracy. Algebraic condition for boost
in order of accuracy.

Burhan Sadiq and DV, submitted.

pdf
code 
[18]

Exact and Asymptotic Conditions on Traveling Wave Solutions of
the NavierStokes Equations

Some conditions derived are exact and must hold for any
traveling wave solution or periodic solution irrespective of the
Reynolds number (Re). Other conditions are asymptotic in the limit of
large Re. The latter are verified numerically.

Y. Charles Li and DV, Physics of Fluids, vol. 21 (2009), 101703.

pdf

[17]

Complex singularities and the Lorenz
attractor.

We suggest that the mathematical analysis of the Lorenz system
could be a problem in analytic function theory. A complete and
consistent formal development of singular solutions
of the Lorenz system is given.
Convergence is proved using a technique that is simpler, more
transparent and more powerful than that of Hille.

DV and S. Şahutoğlu, SIAM Review, vol. 52 (2010), p. 294314..

pdf

[16]

Stable manifolds and the transition to turbulence in pipe flow

A numerical method is devised for disturbing the laminar
motion in such a way that it hits a given traveling wave while
transitioning to turbulence. The connection of exact
solutions of the NavierStokes equation to experiment is
discussed.

DV and P. Cvitanović, J. Fluid Mech., vol. 627
(2009), p. 215233.

pdf

[15]

Heteroclinic connections in plane Couette flow

A numerical method for locating heteroclinic connections,
illustrations of the change in streak patterns along some
connections, and phase space portraits.

J. Halcrow, J.F. Gibson, P. Cvitanović and DV,
J. Fluid Mech., vol. 611 (2009), p. 365376.

pdf

[14]

The critical layer in pipe flow at high Reynolds number

Computations up to Re=75000. Details of
numerical method for locating traveling
waves and relative periodic orbits.

DV, Phil. Trans. Roy. Soc., vol. 367 (2009), p. 561576.

pdf

[13]

The dynamics of transition to turbulence in plane Couette flow

Possible role of lower branch solutions in transition to turbulence.

[in]
Mathematics and computation: a contemporary view.
The Abel Symosium 2006. SpringerVerlag. Berlin. 2008.

pdf

[12]

Recurrent motions within plane Couette turbulence.

Relative periodic orbits that correspond to coherent motions
in the nearwall region of turbulent boundary layers.
Importance of spanwise drift. Outline of numerical method for
finding relative periodic or traveling wave or equilibrium
solutions of the NavierStokes equation.

DV, J. Fluid Mech., 580 (2007), 339358.

pdf

[11]

Shuffling cards for blackjack, bridge, and other card games

The number of shuffles to mix a deck depends upon the way the
cards are dealt (in bridge). Determining the total variation
distance is NP complete. Excellent estimates are computed and
partially justified in a variety of situations.

Under revision

pdf

[10]

Stable manifolds and homoclinic points near resonances in the
restricted threebody problem

Proof of existence of homoclinic points near resonances

DV, Celestial Mechanics and Dynamical Astronomy,
94 (2006), 213235.

pdf

[9]

Normal approximations for descents and inversions of permutations
of multisets

In a long list of numbers, each instance where a larger number
precedes (immediately) a smaller number is an inversion
(descent). Central limit theorems for descents and
inversions. Application to DNA sequences.

M. Conger and DV, J. Theor. Probab., 20 (2007), 309325.

pdf

[8]

Linear stability analysis of resonant periodic motions in the
restricted threebody problem

Resonant periodic motions are alternately stable and unstable.

DV, J. Dynamics and Differential Equations, 17 (2005),
271292.

pdf

[7]

Riffle shuffles of decks with repeated cards

Descent polynomials for decks with repeated cards.
Combinatorics and algorithms for descent polynomials.

M. Conger and DV, Annals of Probability, 34 (2006),
804819.

pdf

[6]

The fractal property of the Lorenz attractor

Detailed computations of the fractal structure of the Lorenz
attractor. Numerical method for extended precision computation
of periodic orbits in which most of the arithmetic uses
double precision.

DV, Physica D, 190 (2004), 115128.

pdf

[5]

Symbolic dynamics and periodic orbits of the Lorenz attractor

Computation of orbits with special symbol sequences. Numerical
demonstration of the density of periodic orbits in a strange
attractor. First computation of the fractal structure of the Lorenz
attractor deduced by Lorenz in 1963. Appendix describes an
essential extension to the numerical method of [3].

DV, Nonlinearity, 16 (2003), 10351056.

pdf

[4]

How many timesteps for a cycle? Analysis of the WisdomHolman
algorithm?


DV, BIT, 42 (2002), 194205.


[3]

The LindstedtPoincaré technique as an algorithm for
computing periodic orbits

Computation of periodic orbits reduces to a two point
boundary value problem (BVP). The method described has two
novelties: the nonlinear BVP is linearized before
it is discretized and the LindstedtPoincaré device
is used to find the period. The method is especially
accurate (as illustrated further in [5]).

DV, SIAM Review, 43 (2001), 479495.

pdf

[2]

Global errors of numerical ODE solvers and Lyapunov's theory
of stability

Linear upper bounds for global error of trajectories that
converge to stable structures such as fixed points, periodic
orbits, and invariant torii.

DV, IMA J. Numer. Anal., 21 (2001), 387406.

pdf

[1]

Random Fibonacci sequences and the number 1.13198824...

Asymptotic growth rate of random Fibonacci sequences.

DV, Math. Comp., 69 (2000), 11311155.

pdf

[0]

Condition numbers of random triangular matrices

Random triangular matrices are exponentially
illconditioned. Asymptotically exact formulas
for the condition number are determined in
several situations.

DV and L.N. Trefethen, SIAM J. Matrix. Anal. Appl., 19 (1998),
564581.

pdf
