Asymptotic Expansion Research Articles

Understanding how chronic environmental stressors shape animal development is essential for predicting ecological responses and optimizing rearing systems. This perspective complements the use of short-term tolerance assays, which overlook the cumulative effects of sustained stress. Temperature and nutrition affect key life-history traits such as growth, development rate, and survival. While both factors have been widely studied, their relative impacts aren't clearly defined. We investigated how constant temperature (26-41°C) and dietary protein-to-carbohydrate ratio (P:C; 0.15-2.18) influence development in two cricket species, Acheta domesticus and Gryllodes sigillatus. Growth trajectories were modelled using a unified-logistic equation to estimate asymptotic mass and maximum growth rate, thereby capturing the growth trajectory in a simplified and interpretable way, enabling comparisons across treatments. Asymptotic mass was combined with developmental rate and survival to calculate a composite metric of developmental performance. Developmental performance peaked at 35°C but fell at thermal extremes due to delayed development (in cold) or reduced mass and survival (in heat). Diet had more modest effects, as performance was stable across most P:C ratios, and only declined at extreme imbalances. Notably, the performance cost of the most unbalanced diets was comparable to a 4-5°C shift from thermal optimum. Our results demonstrate that temperature, more than diet, drives variation in developmental performance during ad libitum feeding. This integrative framework provides a robust approach to quantify environmental sensitivity, define performance limits, and guide us toward the mechanisms underlying those limits and/or performance trade-offs.

Abstract The Cahn–Hilliard model with reaction terms can lead to situations in which no coarsening is taking place and, in contrast, growth and division of droplets occur which all do not grow larger than a certain size. This phenomenon has been suggested as a model for protocells, and a model based on the modified Cahn–Hilliard equation has been formulated. We introduce this equation and show the existence and uniqueness of solutions. Then, formally matched asymptotic expansions are used to identify a sharp interface limit using a scaling of the reaction term, which becomes singular when the interfacial thickness tends to zero. We compute planar solutions and study their stability under non-planar perturbations. Numerical computations for the suggested model are used to validate the sharp interface asymptotics. In addition, the numerical simulations show that the reaction terms lead to diverse phenomena such as growth and division of droplets in the obtained solutions, as well as the formation of shell-like structures.

Abstract An integrable polygon is one whose interior angles are fractions of $$\pi $$ π ; that is to say of the form $$\frac{\pi }{n}$$ π n for positive integers n . We consider the Laplace spectrum on these polygons with the Dirichlet and Neumann boundary conditions, and we obtain new spectral invariants for these polygons. This includes new expressions for the spectral zeta function and zeta-regularized determinant as well as a new spectral invariant contained in the short-time asymptotic expansion of the heat trace. Moreover, we demonstrate relationships between the short-time heat trace invariants of general polygonal domains (not necessarily integrable) and smoothly bounded domains and pose conjectures and further related directions of investigation.

The paper considers the Cauchy problem for a singularly perturbed integro-differential partial differential equation with a rapidly oscillating right-hand side. When considering such problems, it turned out that the existing technique for regularizing singularly perturbed equations is not effective and requires significant rethinking. The development of a new technique for constructing regularized asymptotic solutions for integro-differential equations with partial derivatives and exponential inhomogeneity constitutes the main content of this work. The problem was regularized and the normal and unique solvability of general iterative problems was proved. The asymptotic convergence of formal solutions is proved and a solution to the first iterative problem is constructed.

Funnel control is remarkably a low-complexity control technique which ensures a model-free tracking of reference signal for nonlinear systems. It also ensures a prescribed transient behaviour of error signal; however, it only guarantees the error signal to remain within or close to the funnel boundary rather than to converge to zero. Therefore, for ensuring the asymptotic convergence of the error to zero, in this paper, we introduce the concept of conditional error in funnel control without any complex condition or requirement(s). Thus, we first introduce the idea of conditional funnel control for relative degree one systems and then, we extend this idea for arbitrary relative degree nonlinear systems. Mathematical analyses are provided to ensure the boundedness of the tracking error within the funnel boundary as well as the occurrence of asymptotic convergence. For validating the efficacy of the proposed scheme, we provide numerical examples to compare the numerical results with the recently proposed schemes. Finally, we demonstrate the industrial applications (through simulation case-studies) of the proposed method for a Continuous Stirred Tank Reactor (CSTR), Bioreactor and Flexible Joint Manipulator.

Asymptotic expansions in the central limit theorem for a super-Brownian motion

Double-layer structure and interfacial tension at an ionic surfactant-laden interface.

Global rational approximations of functions with factorially divergent asymptotic series

Sclerochronological age synchrony corroborates remarkable lifespan and protracted asymptotic growth for a deepwater snapper (Pristipomoides zonatus) in the Indian and Pacific Oceans

Simple asymptotic expansions for the Jacobi functions P ν ( α , β ) ( z ) P_\nu ^{(\alpha , \beta )}(z) and Q ν ( α , β ) ( z ) Q_\nu ^{(\alpha , \beta )}(z) for large degree ν \nu , with fixed parameters α \alpha and β \beta , are surprisingly rare in the literature, with only a few special cases covered. This paper addresses this notable gap by deriving simple (inverse) factorial expansions for these functions, complemented by explicit and computable error bounds. Additionally, we provide analogous results for the associated functions Q ν ( α , β ) ( x ) \mathrm {Q}_\nu ^{(\alpha , \beta )}(x) and Q ν ( α , β ) ( x ) \mathsf {Q}_\nu ^{(\alpha , \beta )}(x) .

The dilatancy behaviour of clays significantly influences the stability and performance of underground engineering. In this work, comprehensive laboratory tests on both natural and reconstituted Shanghai clays are conducted to investigate the influence of natural structure and stress history (normal consolidation and overconsolidation) on the dilatancy behaviour. Meanwhile, comparative analysis is performed with data from the literature on Shanghai clays. Results indicate that reconstituted specimens, despite being normally consolidated, exhibit distinct dilatancy behaviours (i.e. shear-induced contraction and shear-induced dilation). Dilatancy intensifies as the overconsolidation ratio increases. The natural specimens from different layers tend to exhibit full contraction behaviour, influenced by their natural structure. Further investigation demonstrates that the confining pressure (p')–void ratio (e) states of specimens fundamentally control the dilatancy behaviours. Clays with denser states exhibit a marked tendency for shear-induced dilation, while those with looser states often display contractive behaviour. Based on these findings, a novel concept of the bounding compression line (BCL) is proposed, which captures the asymptotic convergence of Shanghai clays’ compression behaviour. By integrating the BCL with critical state soil mechanics, this work provides a unified interpretation of the diverse dilatancy behaviours observed in Shanghai clays.

Asymptotic expansion of the solution of a boundary value problem for singularly perturbed integro-differential equations of higher orders

Let Γ \Gamma be a cocompact Fuchsian group, and l l a fixed closed geodesic. We study the counting of those images of l l under the action of Γ \Gamma that have a distance from l l less than or equal to R R . We prove an Ω \Omega -result for the error term in the asymptotic expansion of the counting function. More specifically, we prove that, for every δ > 0 \delta >0 , the error term is equal to Ω δ ( X 1 / 2 ( log ⁡ log ⁡ X ) 1 / 4 − δ ) \Omega _{\delta }\left (X^{1/2}\left (\log {\log {X}}\right )^{1/4-\delta } \right ) , where X = cosh ⁡ R X=\cosh {R} .

This paper presents a robust vision-based motion planning framework for dual-arm manipulators that introduces a novel three-way force equilibrium with velocity-dependent stabilization. The framework combines an improved Artificial Potential Field (iAPF) for linear velocity control with a Proportional-Derivative (PD) controller for angular velocity, creating a hybrid twist command for precise manipulation. A priority-based state machine enables human-like asymmetric dual-arm manipulation. Lyapunov stability analysis proves the asymptotic convergence to desired configurations. The method introduces a computationally efficient continuous distance calculation between links based on line segment configurations, enabling real-time collision monitoring. Experimental validation integrates a real-time vision system using YOLOv8 OBB that achieves 20 frames per second with 0.99/0.97 detection accuracy for bolts/nuts. Comparative tests against traditional APF methods demonstrate that the proposed approach provides stabilized motion planning with smoother trajectories and optimized spatial separation, effectively preventing inter-arm collisions during industrial component sorting.

ABSTRACT In recent years, time‐varying formations have emerged as a crucial control strategy due to their distinct advantages in adapting to environmental changes. However, achieving time‐varying formation tracking in heterogeneous multi‐agent systems via reinforcement learning (RL), especially when the leader's state is unavailable, remains a significant challenge. This paper investigates the problem of time‐varying formation tracking for heterogeneous multi‐agent systems (MASs) with unknown dynamics and inaccessible leader states. A fully distributed output‐feedback reinforcement learning framework is developed, which integrates adaptive observer design, optimal regulation, and data‐driven policy iteration into a unified control scheme. Specifically, a fully distributed adaptive observer is proposed to estimate the leader's state using only its output, without requiring Laplacian eigenvalues or global network knowledge. Based on this observer, an output‐feedback reinforcement learning controller is constructed to achieve asymptotic convergence of the formation tracking error, in contrast to existing state‐feedback‐based methods that only ensure bounded errors. Furthermore, a state reconstruction mechanism, originally used in synchronization problems, is extended to time‐varying formation tracking, enabling policy learning directly from input‐output data under unknown dynamics. Theoretical analysis and simulation studies demonstrate that the proposed framework achieves robust, scalable, and model‐free time‐varying formation tracking, offering clear advantages over existing approaches.

The explicit form of the sixth order radiative corrections to the lepton L ( L = e , μ , and τ ) anomalous magnetic moment from QED Feynman diagrams with insertion of the fourth order polarization operators consisting of either two closed lepton loops or one lepton loop crossed by a photon line is discussed in detail. The approach is based on the consistent application of dispersion relations for vacuum polarization operators and the Mellin-Barnes transform for massive photon propagators. Explicit analytical expressions for corrections to the lepton anomaly are obtained for the first time in the whole interval 0 < r < ∞ of the ratio r of lepton masses m ℓ / m L . Asymptotic expansions in the limit of both small r ≪ 1 and large r ≫ 1 computed from the exact expressions are found to be in perfect agreement with the ones earlier reported in the literature. We argue that in the region where the physical lepton mass ratios are located, the asymptotic expansions hold with an accuracy higher than the experimentally measured anomalies. The two loop diagrams with all three leptons different from each other are computed numerically and compared with the corresponding corrections from the pure two-bubble and one-bubble mixed diagrams. It is shown that there are regions of ratios r where all three types of the fourth order polarization operator contribute equally to the anomaly.

Abstract We develop a high-order asymptotic expansion for the mean first passage time (MFPT) of the capture of Brownian particles by a small elliptical trap in a bounded two-dimensional region. This new result describes the effect that trap orientation plays on the capture rate and extends existing results that give information only on the role of trap position on the capture rate. Our results are validated against numerical simulations that confirm the accuracy of the asymptotic approximation. In the case of the unit disk domain, we identify a bifurcation such that the high-order correction to the global MFPT (GMFPT) is minimized when the trap is orientated in the radial direction for traps centred at $0\lt r\lt r_c :=\sqrt {2-\sqrt {2}}$ . When centred at position $r_c\lt r\lt 1$ , the GMFPT correction is minimized by orientating the trap in the angular direction. In the scenario of a general two-dimensional geometry, we identify the orientation that minimizes the GMFPT in terms of the regular part of the Neumann Green’s function. This theory is demonstrated on several regular domains such as disks, ellipses and rectangles.

The problem of forecasting methodology for special modes of dynamic processes the nonlinear effect that occurs in the marine environment, called “rogue waves”, is considered. Rogue waves are waves that occur in the ocean, as a rule, suddenly, exist for a short period of time and have a huge destructive potential. There are many directions in the study of this phenomenon based on the application of computer modeling and numerical methods. At the same time, there is a tendency to search for rogue waves not only in hydrodynamics, but also in other subject areas, in which, when constructing models of the phenomena and processes under study, the apparatus for solving the corresponding initial boundary value problems for systems of differential equations is used. As a rule, the authors try to find solutions to differential equations, based on which it is possible to demonstrate the occurrence of abnormally high waves. It should be noted that the search for analytical solutions for some differential equations is an extremely difficult task or even impossible to solve. An alternative approach is proposed that makes it possible to prove the existence of the possibility of an anomaly without the need to solve the corresponding system of differential equations, and a model of a dynamic system is constructed similar to the formalism of Koopman theory which takes into account the asymptotic growth rate of the image of a dynamic operator in the energy space, on the basis of which an ordered hierarchy of classes of dynamic operators arises. The definition of an anomaly in the formalism of the mathematical apparatus under consideration is proposed, while the phenomenon of a rogue wave is interpreted as a special case of the occurrence of an anomalous phenomenon in a hydrodynamic system with a sufficiently high average value of the wave background. Within the framework of the proposed approach, it is possible to formulate the necessary conditions for the occurrence of an abnormal phenomenon and sufficient conditions for the absence of anomalies. A time series processing method is proposed that considers the hypothesis of the frequency of occurrence of anomalous phenomena. The existence of anomalies in magnetohydrodynamic processes is demonstrated, which is proved by constructing a model of magnetic field inversion, and the solution of the corresponding dispersion equation is carried out using a modification of the numerical Ivanisov-Polishchuk method consisting in combining the Ivanisov-Polishchuk algorithm and the Adam optimization method. The results obtained may be in demand for further development of the study of the structure of dynamic systems and for identifying more interdisciplinary connections that allow constructive transfer of some of the results from one subject area to another.

This study introduces a novel control strategy aimed at achieving projective synchronization in incommensurate fractional-order chaotic systems (IFOCS). The approach integrates the mathematical framework of fractional calculus with the recursive structure of the backstepping control technique. A key feature of the proposed method is the systematic use of the Mittag–Leffler function to verify stability at every step of the control design. By carefully constructing the error dynamics and proving their asymptotic convergence, the method guarantees the overall stability of the coupled system. In particular, stabilization of the error signals around the origin ensures perfect projective synchronization between the master and slave systems, even when these systems exhibit fundamentally different fractional-order chaotic behaviors. To illustrate the applicability of the method, the proposed fractional order backstepping control (FOBC) is implemented for the synchronization of two representative systems: the fractional-order Van der Pol oscillator and the fractional-order Rayleigh oscillator. These examples were deliberately chosen due to their structural differences, highlighting the robustness and versatility of the proposed approach. Extensive simulations are carried out under diverse initial conditions, confirming that the synchronization errors converge rapidly and remain stable in the presence of parameter variations and external disturbances. The results clearly demonstrate that the proposed FOBC strategy not only ensures precise synchronization but also provides resilience against uncertainties that typically challenge nonlinear chaotic systems. Overall, the work validates the effectiveness of FOBC as a powerful tool for managing complex dynamical behaviors in chaotic systems, opening the way for broader applications in engineering and science.

This paper addresses the trajectory tracking control problem of underactuated autonomous underwater vehicles (AUVs) subject to model uncertainties, external disturbances, and asymmetric input saturation. A robust predefined-time controller is proposed to ensure rapid convergence of tracking errors within a predefined time, rather than the asymptotic convergence achieved by most existing methods. First, an output redefinition-based dynamic transformation (ORDT) is introduced to overcome the relative degree deficiency caused by underactuation, enabling direct control of sway and heave motions. Subsequently, a predefined-time stable integrated controller is designed, which employs a Gaussian error function to handle asymmetric input saturation and uses a neural network (NN) with minimal learning parameters for disturbance compensation. The convergence time of the proposed controller can be preset by a simple parameter, without dependence on initial conditions or complex tuning. The stability analysis based on the Lyapunov method proves that all closed-loop signals converge to a sufficiently small neighborhood of the origin within a predefined time. Finally, simulation results demonstrate the effectiveness and superiority of the proposed control scheme.