Asymptotic Shape Research Articles

Fitness optimization in a cell division model

We consider a size structured cell population model where a mother cell gives birth to two cells. We know that the asymptotic behavior of the density of cells is given by the solution to an eigenproblem. The eigenvector gives the asymptotic shape and the eigenvalue gives the exponential growth rate and so the Maltusian parameter. The Maltusian parameter depends on the division rule for the mother, i.e., symmetric (the two daughter cells have the same size) or asymmetric. We give some example where the symmetrical division is not the best fitted division, i.e., the Maltusian parameter is not optimal. To cite this article: P. Michel, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Exact solution of the Fokker-Planck equation for a broad class of diffusion coefficients

We consider the Langevin equation with a multiplicative noise term that depends on time and space. The corresponding Fokker-Planck equation in the Stratonovich approach is investigated. Its exact solution is obtained for an arbitrary multiplicative noise term given by g (x,t) =D (x) T (t) , and the behaviors of probability distributions, for some specific functions of D (x) , are analyzed. We show that the asymptotic shape of the random-walk model and power-law decay obtained from other approaches can be reproduced from our solutions, by employing two simple functions for g (x,t) . In particular, for D(x) approximately /x/(-theta/2) , the physical solutions for the probability distribution in the Ito, Stratonovich, and postpoint discretization approaches can be obtained and analyzed.

Asymptotic closeness to limiting shapes for expanding embedded plane curves

We show that for embedded or convex plane curves expansion, the difference u(x,t)-r(t) in support functions between the expanding curves γt and some expanding circles Ct (with radius r(t)) has its asymptotic shape as t→∞. Moreover the isoperimetric difference L2-4πA is decreasing and it converges to a constant \(\mathfrak{S} > 0\) if the expansion speed is asymptotically a constant and the initial curve is not a circle. For convex initial curves, if the expansion speed is asymptotically infinite, then L2-4πA decreases to \(\mathfrak{S}=0\) and there exists an asymptotic center of expansion for γt.

Shape and stability of a viscous thread

When a viscous fluid, like oil or syrup, streams from a small orifice and falls freely under gravity, it forms a long slender thread, which can be maintained in a stable, stationary state with lengths up to several meters. We discuss the shape of such liquid threads and their surprising stability. The stationary shapes are discussed within the long-wavelength approximation and compared to experiments. It turns out that the strong advection of the falling fluid can almost outrun the Rayleigh-Plateau instability. The asymptotic shape and stability are independent of viscosity and small perturbations grow with time as exp (C t(1/4)), where the constant is independent of viscosity. The corresponding spatial growth has the form exp [(z/L)(1/8)], where z is the down stream distance and L approximately Q(2) sigma(-2) g and where sigma is the surface tension divided by density, g is the gravity, and Q is the flux. We also show that a slow spatial increase of the gravitational field can make the thread stable.

Spikes in two coupled nonlinear Schrödinger equations

Here we study the interaction and the configuration of spikes in a double condensate by analyzing least energy solutions of two coupled nonlinear Schrödinger equations which model Bose–Einstein condensates of two different hyperfine spin states. When the interspecies scattering length is positive and large enough, spikes of a double condensate repel each other and behave like two separate spikes. In contrast, spikes of a double condensate attract each other and behave like a single spike if the interspecies scattering length is negative and large enough. Our mathematical arguments can prove such physical phenomena. We first use Nehari's manifold to construct least energy solutions, and then use some techniques of singular perturbation problems to derive the asymptotic behavior of least energy solutions. It is shown that the interaction term determines the locations of the two spikes and the asymptotic shape of least energy solutions.

Diffusion in an annihilating environment

In this paper we study the following system of reaction–diffusion equations: ∂ ϱ / ∂ t = Δ ϱ - V ϱ + λ δ 0 , ϱ ( 0 , x ) ≡ 0 , ∂ V / ∂ t = - ϱ V , V ( 0 , x ) ≡ 1 . Here ϱ ( t , x ) and V ( t , x ) are functions of time t ∈ [ 0 , ∞ ) and space x ∈ R d . This system describes a continuum version of a model in which particles are injected at the origin at rate λ , perform independent simple symmetric random walks on Z d , and are annihilated at rate 1 by traps located at the sites of Z d in such a way that the trap disappears with the particle. This lattice model was studied by a number of authors, who obtained the asymptotic size and shape of the front separating the zone of particles from the zone of traps as well as the asymptotic particle density profile to leading order, in the limit of large time. The continuum model has similar behavior but allows for a more detailed study. As t increases, the particle density ϱ ( t , · ) inflates and the trap density V ( t , · ) deflates on a growing ball with radius R * ( t ) centered at the origin. We derive the sharp asymptotics of the front position R * ( t ) , identify the shape of V ( t , · ) near the surface of the ball, and obtain the limiting profile of ϱ ( t , · ) inside the ball after appropriate scaling. We also identify the analogues of the total number and the age distribution of particles that are alive. It turns out that the cases d ⩾ 3 , d = 2 , and 1 exhibit different behavior.

Asymptotic behaviors of star-shaped curves expanding by $V=1-K$

We consider asymptotic behaviors of star-shaped curves expanding by $V=1-K$, where $V$ denotes the outward-normal velocity and $K$ curvature. In this paper, we show the followings. The difference of the radial functions between an expanding curve and circle has its asymptotic shape as $t\rightarrow+\infty$. For two curves, if the asymptotic shapes are identical, then the curves are also. The set of all asymptotic shapes is dense in $C(S^1)$.

Oxidation behaviour of an aluminium nitride–hafnium diboride ceramic composite

An aluminium nitride–hafnium diboride electroconductive particulate composite was produced via hot isostatic pressing without sintering aids. Oxidation kinetics studies were performed up to 1350 °C under a flow of pure oxygen using a microbalance. The reaction products were analysed using SEM and XRD techniques. This composite had a high oxidation resistance up to 1200 °C. The kinetic curves had an asymptotic or paralinear shape. The formation of a protective oxide scale containing hafnia (HfO 2) and aluminium borate (Al 18B 4O 33) phases embedded in a glassy phase was observed. The evaporation of B 2O 3 was limited by the formation of refractory aluminium borate. Above 1200 °C, morphological observations showed the formation of a Maltese cross structure associated with the cracking of the oxide scale along the edges resulting in sigmoidal oxidation kinetics and in a high oxidation rate of the ceramic composite.

Convex lattice polygons of fixed area with perimeter-dependent weights

We study fully convex polygons with a given area, and variable perimeter length on square and hexagonal lattices. We attach a weight tm to a convex polygon of perimeter m and show that the sum of weights of all polygons with a fixed area s varies as s(-theta(conv))eK(t)square root(s) for large s and t less than a critical threshold tc, where K(t) is a t-dependent constant, and theta(conv) is a critical exponent which does not change with t. Using heuristic arguments, we find that theta(conv) is 1/4 for the square lattice, but -1/4 for the hexagonal lattice. The reason for this unexpected nonuniversality of theta(conv) is traced to existence of sharp corners in the asymptotic shape of these polygons.

An Efficient Algorithm to Compute the Euclidean Distance Spectrum of a General Intersymbol Interference Channel and Its Applications

We present an efficient algorithm to compute the distance spectrum of a general finite intersymbol interference (ISI) channel, whose complexity is lower than those of existing methods. Closed-form expressions are derived for both input-output Euclidean distance enumerators and asymptotic distance spectrum shapes for 2-tap and 3-tap ISI channels. Coded and/or precoded ISI channels are also discussed.

Stretched Gaussian Asymptotic Behavior for Fractional Giona?Roman Equation on Biased Heterogeneous Fractal Structure in External Force Fields

We introduce a biased heterogeneous fractional Giona-Roman equation (BHFGRE) on biased heterogeneous fractal structure media describing systems involving external force fields. The BHFGRE is shown to obey generalized Einstein relation, and its stationary solution is the generalized Boltzmann distribution. It is proved that the asymptotic shape of its solution is a stretched Gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for the case of constant potentials and generic potentials.

Universality of stretched Gaussian asymptotic diffusion behavior on biased heterogeneous fractal structure in external force fields☆

A biased heterogeneous fractional Fokker–Planck equation (BHFFPE) is introduced to model the anomalous diffusion on biased heterogeneous fractal structure media describing systems involving external force fields. It is shown that the BHFFPE obeys the generalized Einstein relation, and that the asymptotic shape of its solution is a stretched Gaussian. The theoretic results obtained is consistent with GER.

Asymptotic shape for the chemical distance and first-passage percolation on the infinite Bernoulli cluster

The aim of this paper is to extend the well-known asymptotic shape result for first-passage percolation on to first-passage percolation on a random environment given by the infinite cluster of a supercritical Bernoulli percolation model. We prove the convergence of the renormalized set of wet vertices to a deterministic shape that does not depend on the realization of the infinite cluster. As a special case of our result, we obtain an asymptotic shape theorem for the chemical distance in supercritical Bernoulli percolation. We also prove a flat edge result in the case of dimension 2. Various examples are also given.

Reaction-limited sintering in nearly saturated environments

We study the shape and growth rate of necks between sintered spheres with dissolution–precipitation dynamics in the reaction-limited regime. We determine the critical shape that separates those initial neck shapes that can sinter from those that necessarily dissolve, as well as the asymptotic evolving shape of sinters far from the critical shape. We compare our results with past results for the asymptotic neck shape in closely related but more complicated models of surface dynamics; in particular, we confirm a scaling conjecture, originally due to Kuczinsky. Finally, we consider the relevance of this problem to the diagenesis of sedimentary rocks and other applications.

Improved light-cone sum rules for the electromagnetic form factors of the nucleon

We calculate the electromagnetic form factors of the nucleon within the light-cone sum rule approach. In comparison to previous work [Phys. Rev. D 65 (2002) 074011] we suggest to use a pure isospin-1/2 interpolating field for the nucleon, since the Chernyak–Zhitnitsky current leads to numerically large, unphysical, isospin violating contributions. The leading-order sum rules are derived for the form factors and the results are confronted with the experimental data. Our approach tends to favor the nucleon distribution amplitudes that are not far from the asymptotic shape.

A Seventh-Order Accurate and Stable Algorithm for the Computation of Stress Inside Cracked Rectangular Domains

A seventh-order accurate and extremely stable algorithm for the rapid computation of stress fields inside cracked rectangular domains is presented. The algorithm is seventh-order accurate since it incorporates basis functions, taking the asymptotic shape of the stress fields close to crack tips and corners into account at least up to order six. The algorithm is stable since it is based on a Fredholm integral equation of the second kind. The particular form of the integral equation represents the solution as the limit of a function which is analytic inside the domain. This allows for an efficient implementation. In an example, involving 112 discretization points on an elastic square with a center crack, values of normalized stress intensity factors and T-stress with a relative error of 10−6 are computed in seconds on a workstation. More points reduce the relative error down to 10−15, where it saturates in double precision arithmetic. A large-scale setup with up to 1024 cracks in an elastic square is also studied, using up to 740,000 discretization points. The algorithm is intended as a basic building block in general-purpose solvers for fracture mechanics. It can also be used as a substitute for benchmark tables. (Less)

The limit shape of the zero cell in a stationary Poisson hyperplane tessellation

In the early 1940s, D. G. Kendall conjectured that the shape of the zero cell of the random tessellation generated by a stationary and isotropic Poisson line process in the plane tends to circularity given that the area of the zero cell tends to $\infty$. A proof was given by I. N. Kovalenko in 1997. This paper generalizes Kovalenko's result in two directions: to higher dimensions and to not necessarily isotropic stationary Poisson hyperplane processes. In the anisotropic case, the asymptotic shape of the zero cell depends on the direction distribution of the hyperplane process and is obtained from it via an application of Minkowski's existence theorem for convex bodies with given area measures.

Influence of lateral acceleration on capillary interfaces between parallel plates

The paper considers limits for the refill of capillary liquid vanes of propellant management systems for low gravity applications under the influence of residual acceleration. The limit of existence of a connected capillary liquid interfaces in the vanes is analyzed. A mathematical approach to describe the characteristics of a liquid meniscus in the vane is derived under the assumptions of a quasi static interface and negligible viscous and inertia forces. Analytical considerations allow to determine the maximum width of the liquid column within the vane that can be stabilized by capillary forces. If the width of the capillary vanes exceeds this limit, the vane will be only partially filled. The consecutive equation is solved analytically with the approximation that the lateral acceleration does not distort the asymptotic shape of the free surface extended along the edges of the vanes from a circular cylindrical shap. The resulting relations can readily be analyzed and provide very clear insight into the di.erent limits for the meniscus to exist. The results of the approximate solution are compared with numerical solutions considering the actual shape of the distorted free surface. Although the global shape of the existence chart and the lower limits of existence are rather well predicted by the approximate solution, considerable deviations from the numerical solution can be observed for the upper limits with repect to Bond number and maximum width of the vanes. The numerical solution therefore should be preferred. Additionally, the numerical solution does not provide mere existence criteria but real static (geometric) stability criteria. Experimentally, the behaviour of liquid interfaces in vanes was studied under microgravity conditions in a drop tower, using a micro-g centrifuge to impose variable lateral accelerations on the liquid column between the vanes. The results of the experiment are shown to be in very good agreement with the analytical predictions. The analytical approach presented thus yields a powerful means to determine static stability limits of interface configurations subject to lateral forces for arbitrary geometries of the supporting capillary.

Stretched Gaussian asymptotic behavior for fractional Fokker–Planck equation on fractal structure in external force fields

In this paper, it is proven that the asymptotic shape of the solution of the fractional Fokker–Planck equation is a stretched Gaussian and that its solution can be expressed in the form of a function of a dimensionless similarity variable for generic potentials.

Asymptotic Finite Elements Introducing the Method of Integral Multiple Scales

The method of integral multiple scales (MIMS) is introduced and applied to linear and nonlinear beam models. Based on the method of multiple scales, MIMS is applied to the system Lagrangian and directly results in a system solution. An analytical solution approach is applied to a linear beam-string model to produce a system of linear differential equations that can be solved to produce an asymptotic solution. The true power of MIMS is then demonstrated through a finite element approach by using a set of parametric shape functions based on beam strings. Where the analytic methodology is limited to continuous systems, the finite element approach is easily applied to discontinuous systems providing an analysis method useful with distributed piezoelectric laminates. Both static and dynamic results are discussed. The use of the asymptotic shape functions in the MIMS asymptotic finite element method results in extremely high precision and provides a methodology that could provide a more efficient analytical tool for the development of highly compliant discontinuous systems.