On some resonance cases in quasilinear differential systems with slowly varying parameters

For a quasilinear differential system of the second order whose coefficients have the form of Fourier series with slowly varying coefficients and frequency, we find features of existence of a partial solution of analogous structure with slowly varying coefficients and frequency in the case of purely imaginary eigenvalues of the matrix of the linear part that satisfy a certain resonance relation.     

A nonlinear theory for shells with slowly varying thickness

We study the Γ-limit of 3d nonlinear elasticity for shells of small, variable thickness, around an arbitrary smooth 2d surface. To cite this article: M. Lewicka et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Asymptotic splitting of systems of differential equations of second order with slowly varying and oscillating coefficients

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 40, No. 5, pp. 642–644, September–October, 1988.     

Fast-slow dynamics in Logistic models with slowly varying parameters

Fast-slow behaviors in the Logistic models with slowly varying parameters are revealed by using singular perturbation method. We first rewrite the Logistic models with slowly varying parameters in the form of singularly perturbed systems and separate their fast and slow limits. Then we apply matching to obtain the approximate solutions, which are explicit and analytical and compare very well with the numerically integrated ones. More importantly, we prove the uniform validity of the approximate solutions rigorously and give the error estimate between the approximate solutions and the exact solutions via the way of upper and lower solutions.     

Oscillations in multi-stable monotone systems with slowly varying feedback

The study of dynamics of gene regulatory networks is of increasing interest in systems biology. A useful approach to the study of these complex systems is to view them as decomposed into feedback loops around open loop monotone systems. Key features of the dynamics of the original system are then deduced from the input-output characteristics of the open loop system and the sign of the feedback. This paper extends these results, showing how to use the same framework of input-output systems in order to prove existence of oscillations, if the slowly varying strength of the feedback depends on the state of the system.     

Asymptotic theory for varying coefficient regression models with dependent data

The varying coefficient models (VCMs) are extremely important tools in the statistical literature and are widely used in many subject areas for data modeling and exploration. In linear VCMs, typically the errors are assumed to be independent. However, in many situations, especially in spatial or spatiotemporal settings, this is not a viable assumption. In this article, we consider nonparametric VCMs with a general dependent error structure which allows for both spatially autoregressive and spatial moving average models as special cases. We investigate asymptotic properties of local polynomial estimators of the model components. Specifically, we show that the estimates of the unknown functions and their derivatives are consistent and asymptotically normally distributed. We show that the rate of convergence and the asymptotic covariance matrix depend on the error dependence structure and we derive the explicit formula for the convergence results.     

Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces

This paper considers the existence and large time behavior of solutions to the convection-diffusion equation u _t −Δu+b(x)·∇(u|u| ^{q −1})=f(x, t) in ℝ ⁿ ×[0,∞), where f(x, t) is slowly decaying and q≥1+1/n (or in some particular cases q≥1). The initial condition u ₀ is supposed to be in an appropriate L ^p space. Uniform and nonuniform decay of the solutions will be established depending on the data and the forcing term.This work is partially supported by an AMO Grant     

On integral manifolds of oscillating systems with slowly varying frequencies

By using averages of functions, we construct the integral manifold of an oscillating system that passes through resonances in the course of its evolution. We investigate the smoothness of the integral manifold and obtain estimates for its partial derivatives. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 87–93, January, 1998.     

Peak congestion in multi-server service systems with slowly varying arrival rates

In this paper we consider the M _t queueing model with infinitely many servers and a nonhomogeneous Poisson arrival process. Our goal is to obtain useful insights and formulas for nonstationary finite-server systems that commonly arise in practice. Here we are primarily concerned with the peak congestion. For the infinite-server model, we focus on the maximum value of the mean number of busy servers and the time lag between when this maximum occurs and the time that the maximum arrival rate occurs. We describe the asymptotic behavior of these quantities as the arrival changes more slowly, obtaining refinements of previous simple approximations. In addition to providing improved approximations, these refinements indicate when the simple approximations should perform well. We obtain an approximate time-dependent distribution for the number of customers in service in associated finite-server models by using the modified-offered-load (MOL) approximation, which is the finite-server steady-state distribution with the infinite-server mean serving as the offered load. We compare the value and lag in peak congestion predicted by the MOL approximation with exact values for M _t/M/s delay models with sinusoidal arrival-rate functions obtained by numerically solving the Chapman–Kolmogorov forward equations. The MOL approximation is remarkably accurate when the delay probability is suitably small. To treat systems with slowly varying arrival rates, we suggest focusing on the form of the arrival-rate function near its peak, in particular, on its second and third derivatives at the peak. We suggest estimating these derivatives from data by fitting a quadratic or cubic polynomial in a suitable interval about the peak. This revised version was published online in June 2006 with corrections to the Cover Date.     

On the stability of oscillations representable by Fourier series with slowly varying parameters

For a quasilinear almost triangular differential system whose coefficients are representable by trigonometric series with slowly varying coefficients and frequency, we obtain conditions for the existence of a particular solution of similar structure and study the stability of this solution.     

On the equations of the theory of shells of slowly varying curvatures

Zusammenfassung Es werden für die Schalen mit langsam veränderlichen Krümmungen die Gleichungen angegeben. Die Lösung wird in Form von zwei Gleichungen für die Spannungsfunktion und die Normalverschiebung erhalten.