Asymptotic Shape Research Articles

Drive–response asymptotic shape synchronization for a class of two-dimensional chaotic systems and its application in image encryption

This paper proposes the concept of drive–response asymptotic shape synchronization for two-dimensional chaotic systems and its application in image encryption. It overcomes the limitations of traditional shape synchronization, eliminates the restrictions on the drive system’s continuous differentiability, and the response system’s specific format, while simplifying the controller design, thereby demonstrating enhanced applicability and ease of implementation. Moreover, based on the characteristics of asymptotic shape synchronization, we propose a novel image encryption scheme. In this scheme, the drive and response systems, initialized with distinct values, achieve identical chaotic attractors through rotation and translation parameters in rigid transformation, which are utilized for encryption and decryption processes, respectively. Our encryption algorithm employs DNA coding to construct a DNA plane and incorporates the concept of Rubik’s cube rotations for scrambling and diffusion. Experimental simulations and performance analysis verify the effectiveness of the proposed encryption scheme in withstanding various attacks, ensuring a high level of security.

Integrative framework of multiple processes to explain plant productivity–richness relationships

Plant diversity and productivity, two crucial properties that sustain ecosystem structures, functions, and services, are intrinsically linked to numerous ecological fields, making productivity–richness relationships (PRR) a central ecological concern. Despite extensive research from the Darwinian era to the 21st century, the various shapes of PRR and their underlying theories have sparked ongoing debates. While several processes, theories, and integrative models have been proposed to explain PRR, a comprehensive understanding of the types of PRR, the effects of these processes on plant productivity and richness, and the relationships between PRR shapes remains elusive. This paper proposes a new integrative framework that focuses on these aspects, aiming to elucidate the diverse shapes of PRR and their interconnections. We review recent integrative methods that explain the roles of processes and the varying shapes in PRR to support this new framework. The paper traces the distinct phases in PRR research, including the discovery of PRR shapes, tests of influencing processes, and integrative research. We discuss the application of the Structural Equation Model (SEM), Statistical Dynamical Model (SDM), and Differential Dynamical Model (DDM) in integrative research. This integrative framework can guide theoretical and applied ecologists in identifying, deriving, explaining, and predicting the interconnected but distinct shapes of PRR. The humped, asymptotic, positive, negative, and irregular shapes of PRR are interconnected, with one shape potentially transforming into another. The balance between the positive and negative effects of different processes determines the different shapes of PRR, ultimately leading to a globally positive effect of plant diversity on plant productivity and other ecosystem functions.

Root hydraulic properties: An exploration of their variability across scales.

Root hydraulic properties are key physiological traits that determine the capacity of root systems to take up water, at a specific evaporative demand. They can strongly vary among species, cultivars or even within the same genotype, but a systematic analysis of their variation across plant functional types (PFTs) is still missing. Here, we reviewed published empirical studies on root hydraulic properties at the segment-, individual root-, or root system scale and determined its variability and the main factors contributing to it. This corresponded to a total of 241 published studies, comprising 213 species, including woody and herbaceous vegetation. We observed an extremely large range of variation (of orders of magnitude) in root hydraulic properties, but this was not caused by systematic differences among PFTs. Rather, the (combined) effect of factors such as root system age, driving force used for measurement, or stress treatments shaped the results. We found a significant decrease in root hydraulic properties under stress conditions (drought and aquaporin inhibition, p < .001) and a significant effect of the driving force used for measurement (hydrostatic or osmotic gradients, p < .001). Furthermore, whole root system conductance increased significantly with root system age across several crop species (p < .01), causing very large variation in the data (>2 orders of magnitude). Interestingly, this relationship showed an asymptotic shape, with a steep increase during the first days of growth and a flattening out at later stages of development. We confirmed this dynamic through simulations using a state-of-the-art computational model of water flow in the root system for a variety of crop species, suggesting common patterns across studies and species. These findings provide better understanding of the main causes of root hydraulic properties variations observed across empirical studies. They also open the door to better representation of hydraulic processes across multiple plant functional types and at large scales. All data collected in our analysis has been aggregated into an open access database (https://roothydraulic-properties.shinyapps.io/database/), fostering scientific exchange.

Metropolis Monte Carlo sampling: convergence, localization transition and optimality

Among random sampling methods, Markov chain Monte Carlo (MC) algorithms are foremost. Using a combination of analytical and numerical approaches, we study their convergence properties toward the steady state, within a random walk Metropolis scheme. Analyzing the relaxation properties of some model algorithms sufficiently simple to enable analytic progress, we show that the deviations from the target steady-state distribution can feature a localization transition as a function of the characteristic length of the attempted jumps defining the random walk. While the iteration of the MC algorithm converges to equilibrium for all choices of jump parameters, the localization transition changes drastically the asymptotic shape of the difference between the probability distribution reached after a finite number of steps of the algorithm and the target equilibrium distribution. We argue that the relaxation before and after the localization transition is respectively limited by diffusion and rejection rates.

Path Counting on Tree-like Graphs with a Single Entropic Trap: Critical Behavior and Finite Size Effects.

It is known that maximal entropy random walks and partition functions that count long paths on graphs tend to become localized near nodes with a high degree. Here, we revisit the simplest toy model of such a localization: a regular tree of degree p with one special node ("root") that has a degree different from all the others. We present an in-depth study of the path-counting problem precisely at the localization transition. We study paths that start from the root in both infinite trees and finite, locally tree-like regular random graphs (RRGs). For the infinite tree, we prove that the probability distribution function of the endpoints of the path is a step function. The position of the step moves away from the root at a constant velocity v=(p-2)/p. We find the width and asymptotic shape of the distribution in the vicinity of the shock. For a finite RRG, we show that a critical slowdown takes place, and the trajectory length needed to reach the equilibrium distribution is on the order of N instead of logp-1N away from the transition. We calculate the exact values of the equilibrium distribution and relaxation length, as well as the shapes of slowly relaxing modes.

On the Fluctuations of Internal DLA on the Sierpinski Gasket Graph

Internal diffusion limited aggregation (IDLA) is a random aggregation model on a graph G, whose clusters are formed by random walks started in the origin (some fixed vertex) and stopped upon visiting a previously unvisited site. On the Sierpinski gasket graph, the asymptotic shape is known to be a ball in the graph metric. In this paper, we improve the sublinear bounds for the fluctuations known from its known asymptotic shape result by establishing bounds for the odometer function for a divisible sandpile model.

Limiting shape for first-passage percolation models on random geometric graphs

AbstractLet a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed random variables on the random infinite connected component. We provide sufficient conditions for the existence of the asymptotic shape, and we show that the shape is a Euclidean ball. We give some examples exhibiting the result for Bernoulli percolation and the Richardson model. In the latter case we further show that it converges weakly to a nonstandard branching process in the joint limit of large intensities and slow passage times.

Asymptotic shapes for stationary first passage percolation on virtually nilpotent groups

We study first passage percolation (FPP) with stationary edge weights on Cayley graphs of finitely generated virtually nilpotent groups. Previous works of Benjamini and Tessera (Electron J Probab 20:1–20, 2015) and Cantrell and Furman (Groups Geom Dyn 11(4):1307–1345, 2017) show that scaling limits of such FPP are given by Carnot-Carathéodory metrics on the associated graded nilpotent Lie group. We show a converse, i.e. that for any Cayley graph of a finitely generated nilpotent group, any Carnot-Carathéodory metric on the associated graded nilpotent Lie group is the scaling limit of some FPP with stationary edge weights on that graph. Moreover, for any Cayley graph of any finitely generated virtually nilpotent group, any “conjugation-invariant” metric is the scaling limit of some FPP with stationary edge weights on that graph. We also show that the “conjugation-invariant” condition is also a necessary condition in all cases where scaling limits are known to exist.

Axisymmetric plumes due to fluid injection through a small source in a wet porous medium

A small spherical source discharges a fluid into a porous medium that is already fully saturated with another fluid. The injected fluid has higher density than the ambient fluid, and so it forms a plume that moves downward under the effects of gravity. We present a simple asymptotic analysis assuming the two fluids do not mix that gives the width of the plume far from the source as a function of the injected volume flux. A spectral method is then developed for solving the full nonlinear problem in Boussinesq theory. Accurate numerical solutions are presented, which show in detail the evolution of the plume of heavier injected fluid over time. Close agreement with the asymptotic plume shape far from the source is demonstrated at later times.

Slow melting of a disordered quantum crystal

The melting of the corner of a crystal is a classical, real-world, non-equilibrium statistical mechanics problem which has shown several connections with other branches of physics and mathematics. For a perfect, classical crystal in two and three dimensions the solution is known: the crystal melts reaching a certain asymptotic shape, which keeps expanding ballistically. In this paper, we move onto the quantum realm and show that the presence of quenched disorder slows down severely the melting process. Nevertheless, we show that there is no many-body localization transition, which could impede the crystal to be completely eroded. We prove such claim both by a perturbative argument, using the forward approximation, and via numerical simulations. At the same time we show how, despite the lack of localization, the erosion dynamics is slowed from ballistic to logarithmic, therefore pushing the complete melting of the crystal to extremely long timescales.

A dynamics-based density profile for dark haloes – II. Fitting function

ABSTRACT The density profiles of dark matter haloes are commonly described by fitting functions such as the NFW or Einasto models, but these approximations break down in the transition region where haloes become dominated by newly accreting matter. Here, we present a simple accurate new fitting function that is inspired by the asymptotic shapes of the separate orbiting and infalling halo components. The orbiting term is described as a truncated Einasto profile, ρorb ∝ exp [ − 2/α (r/rs)α − 1/β (r/rt)β], with a five-parameter space of normalization, physically distinct scale and truncation radii, and α and β, which control how rapidly the profiles steepen. The infalling profile is modelled as a power law in overdensity that smoothly transitions to a constant at the halo centre. We show that these formulae fit the averaged total profiles in simulations to about 5 per cent accuracy across almost all of an expansive parameter space in halo mass, redshift, cosmology, and accretion rate. When fixing α = 0.18 and β = 3, the formula becomes a three-parameter model that fits individual haloes better than the Einasto profile on average. By analogy with King profiles, we show that the sharp truncation resembles a cut-off in binding energy.

Asymptotic shape of the concave majorant of a Lévy process

We establish distributional limit theorems for the shape statistics of a concave majorant (i.e. the fluctuations of its length, its supremum, the time it is attained and its value at $T$) of any Levy process on $[0,T]$ as $T\to\infty$. The scale of the fluctuations of the length and other statistics, as well as their asymptotic dependence, vary significantly with the tail behaviour of the Levy measure. The key tool in the proofs is the recent representation of the concave majorant for all Levy processes using a stick-breaking representation.

Asymptotic shape of isolated magnetic domains

We investigate the energy of an isolated magnetized domainΩ⊂Rnforn=2,3. In non-dimensionalized variables, the energy given byE(Ω)=∫Rn|∇χΩ| dx+∫Rn|∇hΩ|2 dxpenalizes the interfacial area of the domain as well as the energy of the corresponding magnetostatic field. Here, the magnetostatic potentialhΩis determined byΔhΩ=∂1χΩ, corresponding to uniform magnetization within the domain. We consider the macroscopic regime|Ω|→∞, in which we derive compactness andΓ-limit which is formulated in terms of the cross-sectional area of the anisotropically rescaled configuration. We then give the solutions for the limit problems.

Asymptotic of the smallest eigenvalues of the continuous Anderson Hamiltonian in $$d\le 3$$

We consider the continuous Anderson Hamiltonian with white noise potential on $(-L/2,L/2)^d$ in dimension $d\le 3$, and derive the asymptotic of the smallest eigenvalues when $L$ goes to infinity. We show that these eigenvalues go to $-\infty$ at speed $(\log L)^{1/(2-d/2)}$ and identify the prefactor in terms of the optimal constant of the Gagliardo-Nirenberg inequality. This result was already known in dimensions $1$ and $2$, but appears to be new in dimension $3$. We present some conjectures on the fluctuations of the eigenvalues and on the asymptotic shape of the corresponding eigenfunctions near their localisation centers.

Contributed Session III: The benefits of naturally moving over stabilized stimuli for acuity increase with longer presentation times

Longer stimulus durations aid in resolving fine spatial detail (Graham & Cook, 1937) - an effect partially attributed to fixational eye movements (FEM; Rucci et al., 2007, Anderson et al., 2020). While FEM do not appear to impact acuity at short presentations (6-300 ms; Tulunay-Keesey & Jones, 1976), recent work has shown that stabilizing stimuli on the retina, eliminating the effects of FEM, impedes acuity at longer durations (750 ms; Ratnam et al., 2017). At intermediate durations, the interplay between acuity and stimulus motion on the retina is not well understood. We used an adaptive optics scanning laser ophthalmoscope to stabilize stimuli on the retina at 1 degree from fixation. A 4 AFC tumbling E staircase was used to find the minimum angle of resolution (MAR) for 100, 375 and 750 ms presentations of unstabilized stimuli. The MAR-duration relationship followed an asymptotic shape, improving from 100 ms (mean MAR ± SEM: 1.84 ± 0.17 arcmin) to 375 ms (1.45 ± 0.09) but not to 750 ms (1.48 ± 0.08). Next, performance (% correct) on stabilized and unstabilized stimuli was measured at each duration's MAR. For stabilized stimuli, we measured performance as a ratio relative to unstabilized performance and found that it degraded with increasing stimulus duration (1.1 ± 0.01, 0.91 ± 0.07, 0.65 ± 0.11 for 100, 375, and 750 ms durations, respectively). These results suggest that retinal stabilization may disrupt the mechanisms by which integration over time enhances visual acuity.

A branching process approach to level‐k phylogenetic networks

AbstractThe mathematical analysis of random phylogenetic networks via analytic and algorithmic methods has received increasing attention in the past years. In the present work we introduce branching process methods to their study. This approach appears to be new in this context. Our main results focus on random level‐k networks with n labeled leaves. Although the number of reticulation vertices in such networks is typically linear in n, we prove that their asymptotic global and local shape is tree‐like in a well‐defined sense. We show that the depth process of vertices in a large network converges towards a Brownian excursion after rescaling by . We also establish Benjamini–Schramm convergence of large random level‐k networks towards a novel random infinite network.

Asymptotic Properties of Random Unlabelled Block-Weighted Graphs

We study the asymptotic shape of random unlabelled graphs subject to certain subcriticality conditions. The graphs are sampled with probability proportional to a product of Boltzmann weights assigned to their $2$-connected components. As their number of vertices tends to infinity, we show that they admit the Brownian tree as Gromov–Hausdorff–Prokhorov scaling limit, and converge in a strengthened Benjamini–Schramm sense toward an infinite random graph. We also consider models of random graphs that are allowed to be disconnected. Here a giant connected component emerges and the small fragments converge without any rescaling towards a finite random limit graph. Our main application of these general results treats subcritical classes of unlabelled graphs. We study the special case of unlabelled outerplanar graphs in depth and calculate its scaling constant.

Local convergence of random planar graphs

The present work describes the asymptotic local shape of a graph drawn uniformly at random from all connected simple planar graphs with n labelled vertices. We establish a novel uniform infinite planar graph (UIPG) as quenched limit in the local topology as n\to\infty . We also establish such limits for random 2 -connected planar graphs and maps as their number of edges tends to infinity. Our approach encompasses a new probabilistic view on the Tutte decomposition. This allows us to follow the path along the decomposition of connectivity from planar maps to planar graphs in a uniform way, basing each step on condensation phenomena for random walks under subexponentiality and Gibbs partitions. Using large deviation results, we recover the asymptotic formula by Giménez and Noy (2009) for the number of planar graphs.

Erratum to “Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domains”

We correct two errors in [J. Spectr. Theory 9 (2019), no. 3, 857–895]. All the main results of the paper remain unchanged, but two of the proofs need to be altered.

On spectral asymptotic of quasi-exactly solvable quartic potential

Motivated by the earlier results of Masoero and De Benedetti (Nonlinearity 23:2501, 2010) and Shapiro et al. (Commun Math Phys 311(2):277–300, 2012), we discuss below the asymptotic of the solvable part of the spectrum for the quasi-exactly solvable quartic oscillator. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the level crossings of the latter oscillator with the asymptotic shape of zeros of the Yablonskii–Vorob’ev polynomials. Further we present a numerical study of the spectral monodromy for the oscillator in question.