Asymptotic Analysis Research Articles

Prey-Taxis vs. An External Signal: Short-Wave Asymptotic and Stability Analysis

We consider two models of the predator–prey community with prey-taxis. Both models take into account the capability of the predators to respond to prey density gradients and also to one more signal, the production of which occurs independently of the community state (such a signal can be due to the spatiotemporal inhomogeneity of the environment arising for natural or artificial reasons). We call such a signal external. The models differ to one another through the description of their responses: the first one employs the Patlak–Keller–Segel law for both responses, and the second one employs Cattaneo’s model of heat transfer for both responses following to Dolak and Hillen. Assuming a short-wave external signal, we construct the complete asymptotic expansions of the short-wave solutions to both models. We use them to examine the effect of the short-wave signal on the formation of spatiotemporal patterns. We do so by comparing the stability of equilibria with no signal to that of the quasi-equilibria forced by the external signal. Such an approach refers back to Kapitza’s theory for an upside-down pendulum. The overall conclusion is that the external signal is likely not capable of creating the instability domain in the parametric space from nothing but it can substantially widen the one that is non-empty with no signal.

Cauchy–Logistic Unit Distribution: Properties and Application in Modeling Data Extremes

This manuscript deals with a novel two-parameter stochastic distribution, obtained by transforming the Cauchy distribution, using generalized logistic mapping, into a unit interval. In this way, according to the well-known properties of the Cauchy distribution, a unit random variable with significantly accentuated values at the ends of the unit interval is obtained. Therefore, the proposed stochastic distribution, named the Cauchy–logistic unit distribution, represents a stochastic model that may be suitable for modeling phenomena and processes with emphasized extreme values. Key stochastic properties of the CLU distribution are examined, such as moments, entropy, modality, and symmetry conditions. In addition, a quantile-based parameter estimation procedure, an asymptotic analysis of the thus obtained estimators, and their Monte Carlo simulation study are conducted. Finally, the application of the proposed distribution in stochastic modeling of some real-world data with emphasized extreme values is provided.

Asymptotic analysis for time fractional FitzHugh-Nagumo equations

Asymptotic analysis for time fractional FitzHugh-Nagumo equations

Internal Structure of Decaying Solitary Waves: Comparison of Analytic and Numerical Results

Abstract The evolution of solitary waves governed by perturbations of the Korteweg-de Vries (KdV) equation is considered, focussing in particular on the Burgers-Korteweg-de Vries (BKdV) equation. Using matched asymptotic expansions the structure of the wave is determined for all timescales. A tail appears behind the main waveform, the structure of which is determined in the form of a convolution integral. Numerical results are presented using a pseudospectral scheme but modified so that linear terms are incorporated into an integrating factor. All details of the asymptotic structure of the waveform are validated by numerical results. Comparisons are made with earlier asymptotic analyses of decaying solitary waves.

Asymptotic analyses on field-induced bending deformations of multi-layered hard-magnetic soft material plates

Asymptotic analyses on field-induced bending deformations of multi-layered hard-magnetic soft material plates

Modeling of spatial extremes in environmental data science: time to move away from max-stable processes

Abstract Environmental data science for spatial extremes has traditionally relied heavily on max-stable processes. Even though the popularity of these models has perhaps peaked with statisticians, they are still perceived and considered as the “state of the art” in many applied fields. However, while the asymptotic theory supporting the use of max-stable processes is mathematically rigorous and comprehensive, we think that it has also been overused, if not misused, in environmental applications, to the detriment of more purposeful and meticulously validated models. In this article, we review the main limitations of max-stable process models, and strongly argue against their systematic use in environmental studies. Alternative solutions based on more flexible frameworks using the exceedances of variables above appropriately chosen high thresholds are discussed, and an outlook on future research is given. We consider the opportunities offered by hybridizing machine learning with extreme-value statistics, highlighting seven key recommendations moving forward.

Asymptotic Analysis of Mis-Classified Linear Mixed Models

Asymptotic Analysis of Mis-Classified Linear Mixed Models

Splash on a liquid pool: coupled cavity–sheet unsteady dynamics

Splashes from impacts of drops on liquid pools are ubiquitous and generate secondary droplets important for a range of applications in healthcare, agriculture and industry. The physics of splash continues to comprise central unresolved questions. Combining experiments and theory, here we study the sequence of topological changes from drop impact on a deep, inviscid liquid pool, with a focus on the regime of crown splash with developing air cavity below the interface and crown sheet above it. We develop coupled evolution equations for the cavity–crown system, leveraging asymptotic theory for the cavity and conservation laws for the crown. Using the key coupling of sheet and cavity, we derive similarity solutions for the sheet velocity and thickness profiles, and asymptotic prediction of the crown height evolution. Unlike the cavity whose expansion is opposed by gravitational effects, the axial crown rise is mostly opposed by surface tension effects. Moreover, both the maximum crown height and the time of its occurrence scale as ${\textit {We}}^{5/7}$ . We find our analytical results to be in good agreement with our experimental measurements. The cavity–crown coupling achieved enables us to obtain explicit estimates of the crown splash spatio-temporal unsteady dynamics, paving the way to deciphering ultimate splash fragmentation.

Fragmentation from inertial detachment of a sessile droplet: implications for pathogen transport

Fragmentation of a fluid body into droplets underlies many contamination and disease transmission processes where pathogens are transported in a liquid phase. An important class of such processes involves formation of a fluid ligament and its destabilization into droplets. Inertial detachment (Gilet & Bourouiba, J. R. Soc. Interface, vol. 12, 2015, 20141092) is one of these modes: upon impact on a sufficiently compliant substrate, the substrate's motion can transfer its impulse to a contaminated sessile drop residing on it. The fragmentation of the sessile drop is efficient at producing contaminated ejected droplets with little dilution. Inertial detachment, particularly from substrates of intermediate wetting, is also interesting as a fundamental fragmentation process on its own merit, involving the asymmetric stretching of the sessile drop under impulsive axial forcing with one-sided pinning due to the substrate's intermediate wetting. Our experiments show that the radius, $R_{tip}$ , of the tip drop ejected become insensitive to the Bond number value for $Bo>1$ . Here, $Bo$ quantifies the inertial effects via the relative axial impulsive acceleration compared with capillarity. The time, $t_{tip}$ , of tip-drop breakup is also insensitive to $Bo$ . Combining experiments, theory and validated numerics, we decipher the selection of $R_{tip}$ and its sensitivity to the surface-wetting and substrate foot dynamics. Using asymptotic theory in the large $Bo$ limit for which the thin-film/slender-jet approximations hold, we derive a reduced physical model that predicts $R_{tip}$ consistent with our experiments. Finally, we discuss how pathogen physical properties (e.g. wetting and buoyancy) within the sessile drop determine their distribution in the tip and secondary fragmentation droplets.

Multiscale modelling and simulation of coexisting turbulent and rarefied gas flows

Simulating complex gas flows from turbulent to rarefied regimes is a long-standing challenge, since turbulence and rarefied flow represent contrasting extremes of computational aerodynamics. We propose a multiscale method to bridge this gap. Our method builds upon the general synthetic iterative scheme for the mesoscopic Boltzmann equation, and integrates the $k$ – $\omega$ model in the macroscopic synthetic equation to address turbulent effects. Asymptotic analysis and numerical simulations show that the macroscopic–mesoscopic coupling adaptively selects the turbulence model and the laminar Boltzmann equation. The multiscale method is then applied to opposing jet problems in hypersonic flight surrounding by rarefied gas flows, showing that the turbulence could cause significant effects on the surface heat flux, which cannot be captured by the turbulent model nor the laminar Boltzmann solution alone. This study provides a viable framework for advancing understanding of the interaction between turbulent and rarefied gas flows.

Performance Analysis of Artificial-Noise-Based Secure Transmission in Wiretap Channel

Artificial noise (AN)-aided techniques have been considered to be promising and practical candidates for enhancing physical layer security. However, there has been a lack of analysis regarding the AN effect on the eavesdropper (EV) from the perspective of the signal-to-interference plus noise ratio (SINR) regarding the existence of the EV’s channel state information (CSI) at the legitimate transmitter. In this paper, we analyze the performance of AN-aided secure transmission from the SINR perspective when a legitimate transmitter has and does not have the EV’s CSI. Based on the analyzed EV’s SINRs for the above two cases, the secrecy gap, which is the difference between the two secrecy capacities, is defined and analyzed. Based on the derived secrecy gap, we have analyzed the asymptotic performances of the secrecy capacity and gap when the number of antennas of the legitimate transmitter and the number of antennas of the EV have large values. Through asymptotic analysis, it is demonstrated that the AN-aided secure transmission under the practical environment (i.e., the case that the EV’s CSI is not available at the legitimate transmitter) can nearly achieve an ideal performance (i.e., the performance when the EV’s CSI is available at the legitimate transmitter) in a massive antenna system.

On affine multicolor urns grown under multiple drawing

Abstract Early investigation of Pólya urns considered drawing balls one at a time. In the last two decades, several authors have considered multiple drawing in each step, but mostly for schemes involving two colors. In this manuscript, we consider multiple drawing from urns of balls of multiple colors, formulating asymptotic theory for specific urn classes and addressing more applications. The class we consider is affine and tenable, built around a ‘core’ square matrix. We examine cases where the urn is irreducible and demonstrate its relationship to matrix irreducibility for its core matrix, with examples provided. An index for the drawing schema is derived from the eigenvalues of the core. We identify three regimes: small, critical, and large index. In the small-index regime, we find an asymptotic Gaussian law. In the critical-index regime, we also find an asymptotic Gaussian law, albeit with a difference in the scale factor, which involves logarithmic terms. In both of these regimes, we have explicit forms for the structure of the mean and the covariance matrix of the composition vector (both exact and asymptotic). In all three regimes we have strong laws.

Inference for two Weibull populations under joint generalized progressive type-I hybrid censoring with a simulation study and applications

The generalized progressive censoring scheme has been considered one of the most general cases of censoring schemes. In this study, we consider two Weibull populations under a jointly generalized progressive hybrid censoring scheme as a more flexible extension of the exponential distribution. The methods presented in this paper let experimenters evaluate life testing studies in the case of the most generalized censoring scheme based on a flexible distribution that has increasing, constant, and decreasing failure rates. The maximum likelihood method is used to obtain point estimates of the unknown parameters and the corresponding approximate confidence intervals by using asymptotic theory and bootstrap sampling. The Bayesian inferences are handled under informative and non-informative priors. The highest posterior density credible intervals are also obtained for the Bayesian estimations. We further obtained results with a challenging task an optimal censoring scheme using the A-optimality, D-optimality, and F-optimality criterion to let researchers determine the optimal censoring plan before conducting experiments or collecting data. Following the numerical results within this paper, A-optimality and D-optimality proposed the same scheme, while F-optimality proposed a scheme similar to them. In the last part of the study, we provide simulation studies under different censoring plans and use a numerical example to exemplify the theoretical outcomes. It is observed that the best estimation performances are obtained by informative Bayesian methods.

Theories without models: uncontrolled idealizations in particle physics

The perturbative treatment of realistic quantum field theories, such as quantum electrodynamics, requires the use of mathematical idealizations in the approximation series for scattering amplitudes. Such mathematical idealizations are necessary to derive empirically relevant models from the theory. Mathematical idealizations can be either controlled or uncontrolled, depending on whether current scientific knowledge can explain whether the effects of the idealization are negligible or not. Drawing upon negative mathematical results in asymptotic analysis (failure of Borel summability) and renormalization group theory (failure of asymptotic safety), we argue that the mathematical idealizations applied in perturbative quantum electrodynamics should be understood as uncontrolled. This, in turn, leads to the problematic conclusion that such theories do not have theoretical models in the natural understanding of this term. The existence of unquestionable empirically successful theories without theoretical models has significant implications both for our understanding of the theory-model relationship in physics and the concept of empirical adequacy.

Enhancing secrecy rates in visible-light communications with signal-dependent noise: exploring optical jamming technique

Abstract This research investigates the security of a visible light communication (VLC) system in an indoor setting. The system consists of multiple transmitters (referred to as Alice), a legitimate receiver (referred to as Bob), and a passive adversary (referred to as Eve). A scheme is proposed to enhance secrecy by using optical jamming signals alongside the confidential signal, under constraints of nonnegativity and average optical intensity. The optical jamming signals follow a truncated Gaussian distribution to adhere to predefined constraints. The analysis takes into account both noise independent of the signal and noise dependent on the signal. The study obtains lower and upper bounds on secrecy rate through the application of the variational method, the dual expression of secrecy rate, and the notion of ‘the optimal input distribution that tends towards infinity’. An asymptotic analysis reveals a minimal performance difference between the upper and lower bounds under large optical intensity. Introducing a peak optical intensity constraint enables further scrutiny of both the exact and asymptotic secrecy-rate bounds. Numerical results are provided to validate the obtained secrecy-rate bounds.

Outlier-robust unit root tests for score-driven models: critical values and applications

ABSTRACT Score-driven models for the t distribution are robust to outliers because outliers are discounted in the updating terms of the filters. We suggest outlier-robust unit root tests using two important score-driven models for the t distribution, the quasi-autoregressive (QAR) model and the Beta- t -EGARCH (exponential generalized autoregressive conditional heteroskedasticity) model. QAR and Beta- t -EGARCH have covariance stationary and unit root versions. The updating terms of these models are information-theoretically optimal extensions of the updating terms of the AR and GARCH models, respectively. For the asymptotic theory of the maximum likelihood (ML) estimation of the QAR and Beta- t -EGARCH models, the correct choice of the order of integration is important. QAR and Beta- t -EGARCH discount outliers in the filters, partly due to this, the covariance stationarity statistics are almost 1 in many applications. Motivated by these points, our contribution is to report unit root test critical value tables for QAR and Beta- t -EGARCH using extensive Monte Carlo simulation experiments. We present an application of the QAR unit root tests for the United States (US) inflation rate from January 1961 to June 2023. We present applications of the Beta- t -EGARCH unit root test for the NASDAQ Composite, S&P 500, and Bitcoin volatility from August 2018 to August 2023.

N-soliton solutions and asymptotic analysis for the massive Thirring model in the laboratory coordinates via the Riemann-Hilbert approach

Abstract In this paper, the N-soliton solutions for massive Thirring model (MTM) in the laboratory coordinates are analyzed via the Riemann-Hilbert approach. The direct scattering including the analyticity, symmetries and asymptotic behaviors of the Jost solutions as |λ| → ∞ and λ → 0 are given. Considering that the scattering coefficients have simple zeros, the matrix Riemann-Hilbert problem, reconstruction formulas and corresponding trace formulas are also derived. Further, the N-soliton solutions in the reflectionless case are obtained explicitly in the form of determinants. The propagation characteristics of one-soliton solutions and interaction properties of two-soliton solutions are discussed. In particular, the asymptotic expressions of two-soliton solutions as |t| → ∞ are obtained, which show that the velocities and amplitudes of the asymptotic solitons don’t change before and after interaction except the position shifts. In addition, three types of bounded states for two-soliton solutions are presented with certain parametric conditions.

Dynamics of a thin film of fluid spreading over a lubricated substrate

We examine the gravity-driven flow of a thin film of viscous fluid spreading over a rigid plate that is lubricated by another viscous fluid. We model the flow over such a ‘soft’ substrate by applying the principles of lubrication theory, assuming that vertical shear provides the dominant resistance to the flow. We do so in axisymmetric and two-dimensional geometries in settings in which the flow is self-similar. Different flow regimes arise, depending on the values of four key dimensionless parameters. As the viscosity ratio varies, the behaviour of the intruding layer ranges from that of a thin coating film, which exerts negligible traction on the underlying layer, to a very viscous gravity current spreading over a low-viscosity, near-rigid layer. As the density difference between the two layers approaches zero, the nose of the intruding layer steepens, approaching a shock front in the equal-density limit. We characterise a frontal stress singularity, which forms near the nose of the intruding layer, by performing an asymptotic analysis in a small neighbourhood of the front. We find from our asymptotic analysis that unlike single-layer viscous gravity currents, which exhibit a cube-root frontal singularity, the nose of a viscous gravity current propagating over another viscous fluid instead exhibits a square-root singularity, to leading order. We also find that large differences in the densities between the two fluids give rise to flows similar to that of thin films of a single viscous fluid spreading over a rigid, yet mobile, substrate.

Trees with flowers: a catalog of integer partition and integer composition trees with their asymptotic analysis

Trees with flowers: a catalog of integer partition and integer composition trees with their asymptotic analysis

Unsteady motion of nearly spherical particles in viscous fluids: a second-order asymptotic theory

The motion of small non-spherical particles is often studied using the unsteady Stokes equations. Zhang & Stone (J. Fluid Mech., vol. 367, 1998, pp. 329–358) reported an asymptotic treatment for nearly spherical particles, to first order in particle non-sphericity, i.e. $O(\epsilon )$ , where $\epsilon$ quantifies the shape deviation from a sphere. Importantly, key physical phenomena are absent at $O(\epsilon )$ , including (1) coupling between the torque experienced by the particle and its linear translation, (2) coupling between the force the particle experiences and its rotation and (3) the effect of non-sphericity on the orientation averages of these forces and torques. We present an explicit asymptotic theory to second order in particle non-sphericity, i.e. $O(\epsilon ^2)$ , for the force and torque acting on a particle in a general unsteady Stokes flow. The derived analytical formulae apply to particles of arbitrary shape, providing the leading-order asymptotic theory for the three above-mentioned phenomena. The theory is demonstrated for several example nearly spherical particles including a spheroid, a ‘pear-shaped’ particle and a simple model for a SARS-CoV-2 virion. This includes formulae for force and torque as a function of particle orientation and their corresponding orientation averages. Our study reveals that the orientation-averaged forces and torques experienced by a nearly spherical particle cannot be generally represented by a perfect sphere. The reported formulae are validated using finite-amplitude three-dimensional direct numerical simulations of the Navier–Stokes equations. A Mathematica notebook is also provided, facilitating implementation of the theory for particle shapes of the user's choosing.