Asymptotic Expansion Research Articles

Small time asymptotics of the entropy of the heat kernel on a Riemannian manifold

We give an asymptotic expansion of the relative entropy between the heat kernel qZ(t,z,w) of a compact Riemannian manifold Z and the normalized Riemannian volume for small values of t and for a fixed element z∈Z. We prove that coefficients in the expansion can be expressed as universal polynomials in the components of the curvature tensor and its covariant derivatives at z, when they are expressed in terms of normal coordinates. We describe a method to compute the coefficients, and we use the method to compute the first three coefficients. The asymptotic expansion is necessary for an unsupervised machine-learning algorithm called the Diffusion Variational Autoencoder.

An asymptotic expansion of the solution of a semi-linear partial differential equation implied by a nonlinear Feynman–Kac formula

This paper introduces an asymptotic expansion for the smooth solution of a semi-linear partial differential equation. Our scheme is based on Itô’s formula, Taylor’s expansion, nonlinear Feynman–Kac formula and some algebras.

Application Of Multipoint Secant-Type Method ForFinding Roots 0f Nonlinear Equations

In this paper, we introduce a family of pk-order iterative schemes for finding the simple root of a nonlinear algebraic equation of the function fx=0 by using the divided difference approximation. The proposed method uses one evaluation of the function per iteration and can achieve convergence order pk. The error equation and asymptotic convergence constant are proved theoretically and numerically. Numerical examples are included to demonstrate the exceptional convergence speed of the proposed method and thus verify the theoretical results

Prescribed intelligent elliptical pursuing by UAVs: A reinforcement learning policy

This paper proposes a prescribed intelligent elliptical pursuing algorithm driven by deep reinforcement learning (DRL) for Unmanned Aerial Vehicles (UAVs), aiming to achieve an optimal performance without violating appointed-time constraints. Specifically, an unknown system dynamics estimator (USDE) with merely one filtering coefficient is employed to counteract uncertainties effectively. Subsequently, an elliptical enclosing controller is devised for UAVs to achieve asymptotic convergence towards the target ellipse orbit. Particularly, a proximal policy optimization (PPO)-based optimal learning term is contrived to achieve control optimality related to consumption and efficiency, where a refined reward function is established to characterize the appointed-time constraints. The salient merit is that an efficient learning paradigm is formulated to fulfill a near optimal appointed-time enclosing free from prescribed performance control. Finally, a plenty of simulations substantiate the feasibility of developed algorithm.

On matched asymptotic expansion solutions of a bucklewave propagation problem

AbstractIn the present paper, the method of matched asymptotic expansions is used to derive approximate analytical solutions to a model problem related to bucklewave propagation in undersea pipelines. The model problem, introduced by Chater, Hutchinson and Neale (in: Collapse – The Buckling of Structures in Theory and Practice (Thompson, J.M.T., Hunt, G.W., eds.) Cambridge Univ. Press, Cambridge, 1983), consists of a linear elastic beam resting on a nonlinear elastic foundation, with inclusion of axial tension and inertia, related to a propagating buckle. When tension dominates over inertia, it is found that the asymptotic solution describing a propagating buckle has the character of nonoscillatory motion. When tension and inertia forces are of equal magnitude, or when inertia forces dominate over tension, the solutions are found to be oscillatory, but still localized around the bucklewave front. Finally, it is shown that when the propagation speed of the buckle exceeds a critical value, a spatial instability sets in, with exponentially growing amplitude of the beam deflection.

Self‐avoiding walks and polygons on hyperbolic graphs

AbstractWe prove that for the ‐regular tessellations of the hyperbolic plane by ‐gons, there are exponentially more self‐avoiding walks of length than there are self‐avoiding polygons of length . We then prove that this property implies that the self‐avoiding walk is ballistic, even on an arbitrary vertex‐transitive graph. Moreover, for every fixed , we show that the connective constant for self‐avoiding walks satisfies the asymptotic expansion as ; on the other hand, the connective constant for self‐avoiding polygons remains bounded. Finally, we show for all but two tessellations that the number of self‐avoiding walks of length is comparable to the th power of their connective constant. Some of these results were previously obtained by Madras and Wu for all but finitely many regular tessellations of the hyperbolic plane.

Incompatible Deformations of Elastic Plates

This article considers the methods for mathematical modeling of incompatible finite deformations of elastic plates by using the principles of the differential geometry theory underlying continuously distributed defects. Equilibrium equations were derived by asymptotic expansions of the finite strain measures with respect to two small parameters. One parameter defines the order of smallness of displacements from the reference shape (self-stressed state), while the other specifies the thickness. Asymptotic orders were different for the deflections and displacements in the plate plane, as well as for their derivatives. They were selected in such a way that, with additional assumptions on the possibility of ignoring certain terms in the resulting expressions and the compatibility of deformations, the equations could be reduced to the system of F¨oppl–von Ka´rm´an equations.

Long-term recovery dynamics determined by the degree of the disturbance – Ten years tracking of aquatic macroinvertebrate recolonisation after an industrial disaster (Red Sludge Disaster, Hungary)

A ten-year-long examination of macroinvertebrate community recovery was conducted following a catastrophic spill of highly alkaline red sludge (pH >13) into lowland streams. Our primary objective was to compare recovery patterns after coarse- and fine-grain disturbances, focusing on two aspects: i) trend analysis to reveal long-term changes of six community parameters, and ii) variation analyses to assess parameter changes over time. We conducted statistical analysis on long-term data series of macroinvertebrates obtained from quantitative samples collected at four sections with varying degrees of disturbance along the impacted stream sections.We developed a comprehensive theoretical framework comprising a series of sequential phases: Ramp-up, Overshoot, and Oscillation Phases.i) A trend analysis revealed that disturbances show a gradual recovery pattern, while variance analyses showed an asymptotic convergence to an equilibrium.ii) Evaluating these trends across phases unveiled that the initial recovery phase exhibited a steep trajectory, lasting 4–9 months, irrespective of disturbance severity. Coarse-grain disturbances induced a remarkable Overshoot phenomenon across all community metrics. The more severe the disturbance, the greater the height and duration of the Overshoot. Our results suggest that the presence or absence of Overshoot can serve as an indicator for coarse-grain disturbances in the context of large and infrequent disturbances (LID). The entire recovery process lasts for 2.5–3 years irrespective of the severity of the LID.In conclusion, a minimum survey duration of two and half years is deemed imperative to capture the phases of recovery, and changes associated with LID are not expected to extend beyond the three-year threshold. The theoretical framework, including Overshoot parameters, may assist future studies in comparing recovery patterns of different LID types. Furthermore, our theoretical framework is likely to be applicable to other groups of organisms given a sufficiently long monitoring of recovery, influenced also by the length of reproductive cycles.

Echo State Property upon Noisy Driving Input

The echo state property (ESP) is a key concept for understanding the working principle of the most widely used reservoir computing model, the echo state network (ESN). The ESP is achieved most of the operation time under general conditions, yet the property is lost when a combination of driving input signals and intrinsic reservoir dynamics causes unfavorable conditions for forgetting the initial transient state. A widely used treatment, setting the spectral radius of the weight matrix below the unity, is not sufficient as it may not properly account for the nature of driving inputs. Here, we characterize how noisy driving inputs affect the dynamical properties of an ESN and the empirical evaluation of the ESP. The standard ESN with a hyperbolic tangent activation function is tested using the MNIST handwritten digit datasets at different additive white Gaussian noise levels. The correlations among the neurons, input mapping, and memory capacity of the reservoir nonlinearly decrease with the noise level. These trends agree with the deterioration of the MNIST classification accuracy against noise. In addition, the ESP index for noisy driving input is developed as a tool to help easily assess ESPs in practical applications. Bifurcation analysis explicates how the noise destroys an asymptotical convergence in an ESN and confirms that the proposed index successfully captures the ESP against noise. These results pave the way for developing noise-robust reservoir computing systems, which may promote the validity and utility of reservoir computing for real-world machine learning applications.

Thermoelastic analysis of variable thickness truncated conical shell subjected to thermomechanical load with internal heat generation using perturbation technique

In this article, the behavior of the truncated conical shell subjected to thermomechanical loading in their inner and outer layers with internal heat generation source is investigated. The displacement field obeys kinematic of the first order shear deformation theory and the first-order temperature theory is used. Two-dimensional temperature analysis has been performed along the thickness and axis of the shell, which can be defined under various loading and thermomechanical boundary conditions. The set of governing equations is a system of differential equations with variable coefficients which are solved by using the analytical matched asymptotic expansion of the perturbations technique. The mentioned solution has little computational cost, therefore can be used well in parametric studies for optimization. It was shown, axial displacement is somehow independent of the radial axis and its maximum value occurs near the upper boundary. The conical shell has expanded in the radial direction. Also the maximum temperature happens approximately in the middle of the length of the conical shell. The parametric study showed, with the increase heat flux in the outer layer thereby expanding the area of heat application which would let more heat in than out. Also it was found, controlling the lifetime of the structure is the result of directing the cone angle. The results obtained from the analytical solution were compared with the Finite Element Method (FEM) analysis and the results of similar related articles, only to show a good agreement.

Graded identities with involution for the algebra of upper triangular matrices

Let F be a field of characteristic zero and let m≥2 be an integer. In this paper, we prove that if a group grading on UTm(F) admits a graded involution then this grading is a coarsening of a Z⌊m2⌋-grading on UTm(F) and the graded involution is equivalent to the reflection or symplectic involution on UTm(F), this grading is called the finest grading on UTm(F). Furthermore, if m≤4 the algebra UTm(F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z⌊m2⌋,⁎)-identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z⌊m2⌋,⁎)-codimensions is determined. As a consequence, we prove that for any G-grading on UTm(F) and any graded involution the (G,⁎)-exponent is m. Finally, we study the algebra UT3(F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z-grading and the Z2-grading induced by (0,1,0). We determine a basis for the (Z2,⁎)-identities and we compute the codimension sequence for the (Z2,⁎)-graded identities for UT3(F).

Asymptotic growth patterns for class field towers

Let p be an odd prime number. We study growth patterns associated with finitely ramified Galois groups considered over the various number fields varying in a \mathbb{Z}_{p} -tower. These Galois groups can be considered as non-commutative analogues of ray class groups. For certain \mathbb{Z}_{p} -extensions in which a given prime above p is completely split, we prove precise asymptotic lower bounds. Our investigations are motivated by the classical results of Iwasawa, who showed that there are growth patterns for p -primary class numbers of the number fields in a \mathbb{Z}_{p} -tower.

Strain‐controlled dynamics of transverse domain walls in hybrid piezoelectric–magnetostrictive heterostructures: An asymptotic approach

AbstractThe present work deals with an analytical study of the statical and dynamical behavior of transverse magnetic domain walls in hybrid bilayer piezoelectric/magnetostrictive heterostructures under the combined influence of spin‐polarized electric currents, transverse, and axial applied magnetic fields. The magnetostrictive material is considered to be isotropic and linearly elastic. We perform the investigation within the framework of the one‐dimensional Landau–Lifshitz–Gilbert equation. By employing the regular perturbation asymptotic expansion approach, we derive the explicit analytical expression of the most significant dynamic features, namely, traveling wave profile, DW velocity, and displacement under the influence of small and moderate transverse magnetic fields. Finally, we illustrate numerically the analytical results and delineate the domain wall motion for metallic and semiconductor ferromagnets.

The asymptotic behavior of Bergman kernels

Let (X,d,p) be the pointed Gromov-Hausdorff limit of a sequence of pointed complete polarized Kähler manifolds (Ml,ωl,Ll,hl,pl) with Ric(hl)=2πωl, Ric(ωl)≥−Λωl and Vol(B1(pl))≥v, ∀l∈N, where Λ,v>0 are constants. Then X is a normal complex space [33].In this paper, we discuss the convergence of the Hermitian line bundles (Ll,hl) and the Bergman kernels. In particular, we show that the Kähler forms ωl converge to a unique closed positive current ωX on Xreg. By establishing a version of L2 estimate on the limit line bundle on X, we give a convergence result of Fubini-Study currents on X. Then we prove that the convergence of Bergman kernels implies a uniform Lp asymptotic expansion of Bergman kernel on the collection of n-dimensional polarized Kähler manifolds (M,ω,L,h) with Ricci lower bound −Λ and non-collapsing condition Vol(B1(x))≥v>0. Under the additional orthogonal bisectional curvature lower bound, we will also give a uniform C0 asymptotic estimate of Bergman kernel for all sufficiently large m, which improves a theorem of Jiang [28]. By calculating the Bergman kernels on orbifolds, we disprove a conjecture of Donaldson-Sun in [20].

Vision-based UAV adaptive tracking control for moving targets with velocity observation

An adaptive image-based visual servoing (IBVS) controller is designed for a quadrotor unmanned aerial vehicle (UAV) to achieve robust tracking for moving targets, under underactuation and tight coupling constraints of UAV kinematics. Specifically, image features are selected from perspective image moments of a planar target to obtain virtual feature dynamics regarding UAV kinematics and dynamics. By constructing an auxiliary variable, a translational velocity observer for moving target is constructed by using virtual image features. An IBVS tracking controller is designed without target geometric information by combining UAV and visual feature dynamics. Designed controller and observer make the UAV robustly reach desired height and track the moving target, despite uncertainty of target movement. The controller has asymptotical convergence performance, and the target velocity is observed according to Lyapunov stability analysis. Simulation and experimental results show that the proposed method has smoother and more accurate performance in motion tracking and target velocity prediction under system uncertainty.

The study of heavy quarks pair production at the LHeC and FCC-eh in the collinear generalized double asymptotic scaling (DAS) approach

The study of heavy quarks pair production at the LHeC and FCC-eh in the collinear generalized double asymptotic scaling (DAS) approach

Soliton confinement in the double sine-Gordon model

The double sine-Gordon field theory in the weak confinement regime is studied. It represents the small non-integrable deformation of the standard sine-Gordon model caused by the cosine perturbation with the frequency reduced by the factor of 2. This perturbation leads to the confinement of the sine-Gordon solitons, which become coupled into the ‘meson’ bound states. We classify the meson states in the weak confinement regime, and obtain three asymptotic expansions for their masses, which can be used in different regions of the model parameters. It is shown, that the sine-Gordon breathers, slightly deformed by the perturbation term, transform into the mesons upon increase of the sine-Gordon coupling constant.

Lattice walks confined to an octant in dimension 3: (non-)rationality of the second critical exponent

In the field of enumeration of walks in cones, it is known how to compute asymptotically the number of excursions (finite paths in the cone with fixed length, starting and ending points, using jumps from a given step set). As it turns out, the associated critical exponent is related to the eigenvalues of a certain Dirichlet problem on a spherical domain. An important underlying question is to decide whether this asymptotic exponent is a (non-)rational number, as this has important consequences on the algebraic nature of the associated generating function. In this paper, we ask whether such an excursion sequence might admit an asymptotic expansion with a first rational exponent and a second non-rational exponent. While the current state of the art does not give any access to such many-term expansions, we look at the associated continuous problem, involving Brownian motion in cones. Our main result is to prove that in dimension three there exists a cone such that the heat kernel (the continuous analog of the excursion sequence) has the desired rational/non-rational asymptotic property. Our techniques come from spectral theory and perturbation theory. More specifically, our main tool is a new Hadamard formula, which has an independent interest and allows us to compute the derivative of eigenvalues of spherical triangles along infinitesimal variations of the angles.

Population Dynamics of Gudusia chapra (Hamilton, 1822) from Lake Kaptai, Bangladesh

Aims: This study aimed to assess the population dynamics of Gudusia chapra in Lake Kaptai, Bangladesh, utilizing FiSAT II software for comprehensive analysis. The key objective was to determine growth parameters, mortality rates, exploitation status, and length-weight relationships to inform policy.
 Study Design: The study employed a quantitative approach, utilizing length frequency data collected from Lake Kaptai. FiSAT II software facilitated the estimation of population dynamics parameters, employing the Von Bertalanffy growth equation and other relevant analyses.
 Place and Duration of Study: The research was conducted in Lake Kaptai, Bangladesh, focusing on the intricate aquatic ecosystem from November 2003 to October 2004. Data collection and analysis spanned an extensive duration, providing a robust understanding of the population dynamics of G. chapra.
 Methodology: FiSAT II software was utilized for population dynamics estimation, with a particular focus on growth parameters, mortality rates, and exploitation status using Von Bertalanffy growth equation. Length-weight relationships were also determined. Recruitment patterns were observed, and key indices, including the growth performance index calculated.
 Results: The estimated asymptotic length (L∞) and growth coefficient (K) of G. chapra were found to be 19.95 cm and 0.89 year-1 respectively. Natural mortality (M), fishing mortality (F) and total mortality (Z) rates were 1.85, 2.21 and 4.06 year-1 respectively. The exploitation rate (E) indicated overfishing, with a value of 0.54, which exceeded the optimum E (E=0.5). The growth performance index (φ´) of 2.54 was recorded, and the recruitment pattern occurred between April and July.
 Conclusion: The study concludes that G. chapra in Lake Kaptai is experiencing overfishing, as indicated by the elevated exploitation rate. Understanding the growth parameters and mortality rates is vital for sustainable fisheries management in the region.
 Implication: The findings have significant implications for fisheries management, highlighting the urgent need for measures to mitigate overfishing of G. chapra in Lake Kaptai. Implementing conservation strategies and monitoring recruitment patterns are crucial for ensuring the long-term sustainability of this important fish species.

Space–time structure of weak magnetohydrodynamic turbulence

The two-time energy spectrum of weak magnetohydrodynamic turbulence is found by applying a wave-turbulence closure to the cumulant hierarchy constructed from the dynamical equations. Solutions are facilitated via asymptotic expansions in terms of the small parameter $\varepsilon$ , describing the ratio of time scales corresponding to Alfvénic propagation and nonlinear interactions between counter-propagating Alfvén waves. The strength of nonlinearity at a given spatial scale is further quantified by an integration over all possible delta-correlated modes compliant in a given set of three-wave interactions that are associated with energy flux through the said scale. The wave-turbulence closure for the two-time spectrum uncovers a secularity occurring on a time scale of order $\varepsilon ^{-2}$ , and the asymptotic expansion for the spectrum is reordered in a manner comparable to the one-time case. It is shown that for the regime of stationary turbulence, the two-time energy spectrum exponentially decays on a lagged time scale $(\varepsilon ^2 \gamma _k^s)^{-1}$ in proportion to the strength of the associated three-wave interactions, characterized by nonlinear decorrelation frequency $\gamma _k^s$ . The scaling of the form $k_{\perp } v_0 \chi _0$ exhibited by this frequency is reminiscent of random sweeping by the outer scale with characteristic fluctuation velocity $v_0$ that is modified due to competition with Alfvénic propagation (characterized by $\chi _0$ ) at the said scale. A brief calculation of frequency broadening of the power spectrum due to nonlinear interactions is also presented.