scholarly journals Robust non-linear differential equation models of gene expression evolution across Drosophila development

2012 ◽  
Vol 5 (1) ◽  
pp. 46 ◽  
Author(s):  
Alexandre Haye ◽  
Jaroslav Albert ◽  
Marianne Rooman
1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramter ν occurs, but the ν = 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factor ε(ν) is required, so that the aspect ratio ε of a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion in ε gives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.


1980 ◽  
Vol 25 (92) ◽  
pp. 229-246 ◽  
Author(s):  
L. W. Morland ◽  
I. R. Johnson

AbstractSteady plane flow under gravity of a symmetric ice sheet resting on a horizontal rigid bed, subject to surface accumulation and ablation, basal drainage, and basal sliding according to a shear-traction-velocity power law, is treated. The surface accumulation is taken to depend on height, and the drainage and sliding coefficient also depend on the height of overlying ice. The ice is described as a general non-linearly viscous incompressible fluid, with illustrations presented for Glen’s power law, the polynomial law of Colbeck and Evans, and a Newtonian fluid. Uniform temperature is assumed so that effects of a realistic temperature distribution on the ice response are not taken into account. In dimensionless variables a small paramterνoccurs, but theν= 0 solution corresponds to an unbounded sheet of uniform depth. To obtain a bounded sheet, a horizontal coordinate scaling by a small factorε(ν) is required, so that the aspect ratioεof a steady ice sheet is determined by the ice properties, accumulation magnitude, and the magnitude of the central thickness. A perturbation expansion inεgives simple leading-order terms for the stress and velocity components, and generates a first order non-linear differential equation for the free-surface slope, which is then integrated to determine the profile. The non-linear differential equation can be solved explicitly for a linear sliding law in the Newtonian case. For the general law it is shown that the leading-order approximation is valid both at the margin and in the central zone provided that the power and coefficient in the sliding law satisfy certain restrictions.


2011 ◽  
Vol 18 (2) ◽  
pp. 345-364
Author(s):  
Tamaz Tadumadze

Abstract Variation formulas of solution are proved for a non-linear differential equation with constant delay. In this paper, the essential novelty is the effect of delay perturbation in the variation formulas. The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.


2004 ◽  
Vol 19 (supp02) ◽  
pp. 126-133
Author(s):  
G. DELFINO ◽  
P. GRINZA ◽  
P. MOSCONI ◽  
G. MUSSARDO

A unified analysis of the disorder operators for ghosts, complex boson and fermion fields is presented. Matrix elements on the asymptotic states of these operators can be exactly computed by solving the Form Factor functional equations. The two–point correlation functions of the disorder operators depend only on the statistics and can be expressed in terms of a solution of a non–linear differential equation of Painleve' type.


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