Asymptotic Shape Research Articles

Precise Asymptotic Shapes of Solutions to Nonlinear Two-Parameter Problems

We consider the nonlinear two-parameter problem, which comes from a perturbed simple pendulum problem $$-u^{\prime\prime}(t)+\mu f(u(t)) = \lambda g(u(t)), t \in I: = (-T, T), u(t) > 0,\quad t \in I,\quad u(\pm T) = 0,$$ where μ, λ > 0 are parameters and T > 0 is a constant. For a given μ > 0, there exists a solution triple (\(\mu, \lambda(\mu), u_{\mu}) \in \rm{R}^{2}_{+}\times C^{2}(\bar{I}),\) which is obtained by a variational method, such that uμ is almost flat inside I and develops boundary layers as \(\mu \rightarrow \infty\). We establish the precise asymptotic formulas for \(\|u_{\mu}\|_{q}(1 \leq q \leq \infty), u^{\prime}_{\mu} (\pm T)\) and the variational eigencurve λ(μ) as \(\mu \rightarrow \infty\). By these formulas, we understand well not only the local behavior of uμ as \(\mu \rightarrow \infty\), but also the total shape of uμ. Furthermore, we find the precise asymptotics of \(\|u_{\mu}\|_{q}\) as \(q \rightarrow \infty\). By this, we understand well how \(\|u_{\mu}\|_{q}\) tends to \(\|u_{\mu}\|_{\infty}\) as \(q \rightarrow \infty\).

Large deviations for the chemical distance in supercritical Bernoulli percolation

The chemical distance D(x,y) is the length of the shortest open path between two points x and y in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $\mu$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{\mu(x)}\notin (1-\epsilon, 1+\epsilon) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.

Encoding of diffusion and T1 in the CPMG echo shape: single-shot D and T1 measurements in grossly inhomogeneous fields

Abstract We present a new approach of NMR measurements in the presence of grossly inhomogeneous fields where information is encoded in the echo shape of CPMG trains. The method is based on sequences that consist of an initial encoding sequence that generates echoes with contributions from at least two different coherence pathways that are then both refocused many times by a long string of closely spaced identical pulses. The generated echoes quickly assume an asymptotic shape that encodes the information of interest. High signal-to-noise ratios can be achieved by averaging the large number of echoes. We demonstrate this approach with different implementations of the measurements of longitudinal relaxation time, T 1 , and diffusion coefficient, D . It is shown that the method can be used for novel single-shot measurements.

Exponential growth rates in a typed branching diffusion

We study the high temperature phase of a family of typed branching diffusions initially studied in [Astérisque 236 (1996) 133–154] and [Lecture Notes in Math. 1729 (2000) 239–256 Springer, Berlin]. The primary aim is to establish some almost-sure limit results for the long-term behavior of this particle system, namely the speed at which the population of particles colonizes both space and type dimensions, as well as the rate at which the population grows within this asymptotic shape. Our approach will include identification of an explicit two-phase mechanism by which particles can build up in sufficient numbers with spatial positions near −γt and type positions near $\kappa \sqrt{t}$ at large times t. The proofs involve the application of a variety of martingale techniques—most importantly a “spine” construction involving a change of measure with an additive martingale. In addition to the model’s intrinsic interest, the methodologies presented contain ideas that will adapt to other branching settings. We also briefly discuss applications to traveling wave solutions of an associated reaction–diffusion equation.

Long-range asymptotics of a step signal propagating in a hereditary viscoelastic medium

The shape of the asymptotic long-range solution of the scalar linear viscoelastic signaling problem is found for a large class of creep compliances. If the asymptotic growth of the creep compliance at large times is given by a power law then the long-range asymptotic shape of an initial step pulse is given by a function depending exclusively on the value of the exponent in the power law. Two different cases arise corresponding to unbounded and bounded creep compliances. In the first case the asymptotic solution is the solution of a fractional wave equation with a scale factor. In the second case a different universal signal shape function is obtained.

Asymptotic Shapes of large Cells in Random Tessellations

We establish a close relationship between isoperimetric inequalities for convex bodies and asymptotic shapes of large random polytopes, which arise as cells in certain random mosaics in d-dimensional Euclidean space. These mosaics are generated by Poisson hyperplane processes satisfying a few natural assumptions (not necessarily stationarity or isotropy). The size of large cells is measured by a class of general functionals. The main result implies that the asymptotic shapes of large cells are completely determined by the extremal bodies of an inequality of isoperimetric type, which connects the size functional and the expected number of hyperplanes of the generating process hitting a given convex body. We obtain exponential estimates for the conditional probability of large deviations of zero cells from asymptotic or limit shapes, under the condition that the cells have large size.

Asymptotic behavior of solutions to the perturbed simple pendulum problems with two parameters

AbstractWe consider the perturbed simple pendulum equation –u ″(t) + μ |u (t)|p –1u (t) = λ sin u (t), t ∈ I ≔ (–T, T), u (t) > 0, t ∈ I, u (±T) = 0, where p > 1 is a constant,λ > 0 and μ ∈ R are parameters. The purpose of this paper is to clarify the structure of the solution set. To do this, we study precisely the asymptotic shape of the solutions when λ ≫ 1 as well as the asymptotic behavior of variational eigenvalue μ (λ) as λ → ∞. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Removal of residual phenols from coke wastewater by adsorption

After biological treatment, coke wastewater contains small amounts of phenolic compounds resistant to such treatment. The removal of phenols and COD from coke wastewater subjected to biological treatment was studied. The adsorbents used were granular activated carbon and the resins XAD-2, AP-246 and OC-1074. Equilibrium, kinetics and column assays were carried out, fitting the equilibrium data to Langmuir and Freundlich models and the kinetic data to the Lagergren equation. The best results were obtained with GAC, which presented higher adsorption capacities. In the equilibrium assays, the adsorption capacities ( Q) found were 1.48 mg g −1 for GAC versus 0.07 and 0.04 mg g −1 for resins AP-246 and OC-1074, respectively. In the kinetic assays, the values of the Lagergren adsorption parameter, q e, were 1.69, 0.15 and 0.14 mg g −1 for GAC, AP-246 and OC-1074, respectively. In the column assays, the dynamic capacity of GAC for up to 480 bed volumes was 1.82 mg mL −1. No saturation was obtained for this volume due to the asymptotic shape of the breakthrough curve, whereas for the same percolated volume, the resins AP-246 and OC-1074 were saturated. These two resins presented similar saturation capacities of around 1.1 mg mL −1.

A simple solution to the k‐core problem

AbstractWe study the k‐core of a random (multi)graph on n vertices with a given degree sequence. We let n →∞. Then, under some regularity conditions on the degree sequences, we give conditions on the asymptotic shape of the degree sequence that imply that with high probability the k‐core is empty and other conditions that imply that with high probability the k‐core is non‐empty and the sizes of its vertex and edge sets satisfy a law of large numbers; under suitable assumptions these are the only two possibilities. In particular, we recover the result by Pittel, Spencer, and Wormald (J Combinator Theory 67 (1996), 111–151) on the existence and size of a k‐core in G(n,p) and G(n,m), see also Molloy (Random Struct Algor 27 (2005), 124–135) and Cooper (Random Struct Algor 25 (2004), 353–375). Our method is based on the properties of empirical distributions of independent random variables and leads to simple proofs. © 2006 Wiley Periodicals, Inc. Random Struct. Alg.,, 2007

Encoding of diffusion and T1 in the CPMG echo shape: Single-shot D and T1 measurements in grossly inhomogeneous fields

We present a new approach of NMR measurements in the presence of grossly inhomogeneous fields where information is encoded in the echo shape of CPMG trains. The method is based on sequences that consist of an initial encoding sequence that generates echoes with contributions from at least two different coherence pathways that are then both refocused many times by a long string of closely spaced identical pulses. The generated echoes quickly assume an asymptotic shape that encodes the information of interest. High signal-to-noise ratios can be achieved by averaging the large number of echoes. We demonstrate this approach with different implementations of the measurements of longitudinal relaxation time, T 1, and diffusion coefficient, D. It is shown that the method can be used for novel single-shot measurements.

First-passage competition with different speeds: positive density for both species is impossible

Consider two epidemics whose expansions on $\mathbb{Z}^d$ are governed by two families of passage times that are distinct and stochastically comparable. We prove that when the weak infection survives, the space occupied by the strong one is almost impossible to detect. Particularly, in dimension two, we prove that one species finally occupies a set with full density, while the other one only occupies a set of null density. Furthermore, we observe the same fluctuations with respect to the asymptotic shape as for the weak infection evolving alone. By the way, we extend the Häggström-Pemantle non-coexistence result "except perhaps for a denumerable set" to families of stochastically comparable passage times indexed by a continuous parameter.

A study of the collisional fragmentation problem using the Gamma distribution approximation

The nonlinear fragmentation population balance formulation has been elevated in recent years from a prototype for studying nonlinear integro-differential equations to a vehicle for analyzing and understanding several physicochemical processes of technological interest. The so-called pure collisional fragmentation, which is the particular mode of nonlinear fragmentation induced by collisions between particles, is studied here. It is shown that the corresponding population balance equation admits large time asymptotic (self-similarity) solutions for homogeneous fragmentation and collision functions (kernels). The self-similar solutions are given in closed form for some simple kernels. Based on the shape of the self-similar solutions the method of moments with Gamma distribution approximation is employed for transient solution (from initial state to establishment of the asymptotic shape) of the collisional fragmentation equation. These solutions are presented for several sets of parameters and their behavior is discussed rather extensively. The present study is similar to the one has already been performed for the case of the much simpler linear fragmentation equation [G. Madras, B.J. McCoy, AIChE J. 44 (1998) 647].

Growth-Melt Asymmetry in Crystals and Twelve-Sided Snowflakes

It is in the lexicon of crystal growth that the shape of a growing crystal reflects the underlying microscopic architecture. Although it is known that in weakly nonequilibrium conditions the slowest growing orientations ultimately dominate the asymptotic shape, is the same true for melting? Here we observe and show theoretically that while the two-dimensional steady melt shapes of ice are bounded by six planes, these planes are not proper facets but instead are rotated 30 degrees from the prism planes of ice. Finally, the transient melting state exposes 12 apparent crystallographic planes thereby differing substantially from the transient growth state.

Asymptotic behavior of a fractional Fokker–Planck-type equation

It is proved that the asymptotic shape of the solution for a wide class of fractional Fokker–Planck-type equations with coefficients depending on coordinate and time is a stretched Gaussian for the initial condition being pulse function in the homogeneous and heterogeneous fractal structures, whose mean square displacement behaves like 〈 ( Δ x ) 2 ( t ) 〉 ∼ t γ and 〈 ( Δ x ) 2 ( t ) 〉 ∼ x - θ t γ ( 0 < γ < 1 ,- ∞ < θ < + ∞ ) , respectively.

Nucleon form factors in QCD

We calculate the electromagnetic and the axial form factors of the nucleon within the framework of light cone sum rules (LCSR) to leading order in QCD and including higher twist corrections. In particular we motivate a certain choice for the interpolating nucleon field. We find that a simple model of the nucleon distribution amplitudes which deviate from their asymptotic shape, but much less compared to the QCD sum rule estimates, allows one to describe the data remarkably well.

Answer to an open problem proposed by R Metzler and J Klafter

In a positive answer to the open problem proposed by R Metzler and J Klafter, it is proved that the asymptotic shape of the solution for a wide class of fractional Fokker–Planck equations is a stretched Gaussian for the initial condition being a pulse function in the homogeneous and heterogeneous fractal structures, whose mean square displacement behaves like (Δx)2(t) ~ tγ and (Δx)2(t) ~ x−θtγ(0 < γ < 1, − ∞ < θ < +∞), respectively.

Fixed points of two-dimensional maps obtained under rapid stimulations

Two-dimensional maps for the FitzHugh systems, both for the excitable and the limit cycle regimes, are analyzed in phase space and their stable fixed points are calculated. The asymptotic shapes of the solutions are analytically approximated. It is shown that for high stimulation rates the one-dimensional phase- or time-resetting maps are inadequate.

Optimal Proliferation Rate in a Cell Division Model

We consider a size structured cell population model where a mother cell gives birth to two daughter 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 cell, i.e., symmetric (the two daughter cells have the same size) or asymmetric. We use a min-max principle and a differentiation principle to find the variation of the first eigenvalue with respect to a parameter of asymmetry of the cell division. We prove that the symmetrical division is not always the best fitted division, i.e., the Maltusian parameter may be not optimal.

Stochastic analysis of nonlinear biodegradation in regimes controlled by both chromatographic and dispersive mixing

Nonlinear biodegradation in natural porous media is affected by the heterogeneity of the formation and dispersive mixing processes. We analyze these coupled effects by combining recent advances in analytical one‐dimensional modeling of bioreactive transport with stochastic concepts of dispersive mixing in heterogeneous domains. Specifically, we model bioremediation of a sorbing contaminant undergoing nonlinear biodegradation in heterogeneous aquifers applying the stochastic‐convective and the advective‐dispersive stream tube approaches, in which we use a semianalytical traveling wave solution for one‐dimensional reactive transport. The results of numerical simulations agree excellently with both models, which establishes that the traveling wave solution is an efficient and accurate way to evaluate the development of intrastream tube concentration distributions and that the advective‐dispersive stream tube approach is suitable to describe nonlinear bioreactive transport in systems controlled by local‐scale dispersion. In contrast with conservative transport the mean contaminant flux is shown to be significantly influenced by transverse dispersion, even for realistic Peclet values. Furthermore, asymptotic front shapes are shown to be neither Fickian nor constant (traveling wave behavior), which raises questions about the current practice of upscaling bioreactive transport. The error caused by neglecting local dispersion was found to increase with time and to remain significant even for large retardation differences between electron acceptor and contaminant. This implies that, even if reaction rates are dominated by chromatographic mixing, the dispersive mixing process cannot be disregarded when predicting bioreactive transport.

Random growth models with polygonal shapes

We consider discrete-time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half-space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases. In many cases, exact stability is observed. That is, for small perturbations, the shapes of the deterministic and random processes agree exactly. We give a complete characterization of such cases, and show that they are prevalent among threshold growth CA with box neighborhood. We also design a nontrivial family of CA in which the shape is exactly computable for all values of its probability parameter.