17,379 research outputs found
Lie Symmetries and Exact Solutions of First Order Difference Schemes
We show that any first order ordinary differential equation with a known Lie
point symmetry group can be discretized into a difference scheme with the same
symmetry group. In general, the lattices are not regular ones, but must be
adapted to the symmetries considered. The invariant difference schemes can be
so chosen that their solutions coincide exactly with those of the original
differential equation.Comment: Minor changes and journal-re
Tractrices, Bicycle Tire Tracks, Hatchet Planimeters, and a 100-year-old Conjecture
Geometry of the tracks left by a bicycle is closely related with the
so-called Prytz planimeter and with linear fractional transformations of the
complex plane. We describe these relations, along with the history of the
problem, and give a proof of a conjecture made by Menzin in 1906.Comment: 20 pages, 18 figure
Marginal cost-based pricing of distribution: a case study
This paper presents results of a software development project carried out by the “Electricity North West” (ENW) and “TNEI” to find economic use-of-system charges for the extra high-voltage (EHV) network. Several cost-based charging models which satisfy principles set by the Regulator, such as cost reflectivity, predictability, stability and transparency were developed. In this paper, the emphasis is put on the developed software and the comparison of nodal marginal charges obtained from the proposed pricing models
Lie point symmetries of difference equations and lattices
A method is presented for finding the Lie point symmetry transformations
acting simultaneously on difference equations and lattices, while leaving the
solution set of the corresponding difference scheme invariant. The method is
applied to several examples. The found symmetry groups are used to obtain
particular solutions of differential-difference equations
Comparison of Perron and Floquet eigenvalues in age structured cell division cycle models
We study the growth rate of a cell population that follows an age-structured
PDE with time-periodic coefficients. Our motivation comes from the comparison
between experimental tumor growth curves in mice endowed with intact or
disrupted circadian clocks, known to exert their influence on the cell division
cycle. We compare the growth rate of the model controlled by a time-periodic
control on its coefficients with the growth rate of stationary models of the
same nature, but with averaged coefficients. We firstly derive a delay
differential equation which allows us to prove several inequalities and
equalities on the growth rates. We also discuss about the necessity to take
into account the structure of the cell division cycle for chronotherapy
modeling. Numerical simulations illustrate the results.Comment: 26 page
Multiscale expansion and integrability properties of the lattice potential KdV equation
We apply the discrete multiscale expansion to the Lax pair and to the first
few symmetries of the lattice potential Korteweg-de Vries equation. From these
calculations we show that, like the lowest order secularity conditions give a
nonlinear Schroedinger equation, the Lax pair gives at the same order the
Zakharov and Shabat spectral problem and the symmetries the hierarchy of point
and generalized symmetries of the nonlinear Schroedinger equation.Comment: 10 pages, contribution to the proceedings of the NEEDS 2007
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