Convex additively slowly varying functions

We study the problem of subtraction of slowly varying functions. It is well-known that the difference of two slowly varying functions need not be slowly varying and we look for some additional conditions which guarantee the slow variation of the difference. To this end we consider all possible decompositions L=F+G of a given increasing convex additively slowly varying function L into a sum of two increasing convex functions F and G. We characterize the class of functions L for which in every such decomposition the summands are necessarily additively slowly varying. The class OΠ₂⁺ we obtain is related to the well-known class OΠ_g where, instead of first order differences as in OΠ_g, we have second order differences.     

Sharp Hardy inequalities of fractional order involving slowly varying functions

We establish necessary and sufficient conditions for the validity of Hardy inequalities of fractional orders involving weights which are products of power-type functions and slowly varying functions. Consequently, for such weights, we solve Open Problems 1 and 2 mentioned in the book of Kufner and Persson, Weighted Inequalities of Hardy Type (World Scientific Publishing, 2003).     

Compact embeddings of Sobolev spaces with power weights perturbed by slowly varying functions

We study compact embeddings of weighted Sobolev spaces into Lebesgue spaces on the unit ball in . The weight is of slowly varyingly disturbed polynomial growth with a singularity at the origin. It extends 21 , 27 to a wider class of weights. Special attention is paid to the influence of the growth rate of the weight on the quality of compactness, measured in terms of entropy and approximation numbers. In case of Hilbert spaces, the results are related to the distribution of eigenvalues of some degenerate elliptic operators.     

Extending satisficing control strategy to slowly varying nonlinear systems

Based on the satisficing control strategy, a novel approach to design a stabilizing control law for nonlinear time varying systems with slowly varying parameters (slowly varying systems) is presented. The satisficing control strategy has been originally introduced for time-invariant systems; however, this technique does not have any stability proof for time varying systems. In this paper, first, a parametric version of the satisficing control strategy is developed. Then, by considering the time as a frozen parameter, the parametric satisficing control strategy is utilized. Finally, a theorem is presented which suggested a stabilizing satisficing control law for the slowly varying control systems. Moreover, in this theorem, the maximum admissible rate of change of the system dynamics is evaluated. The efficiency of the proposed approach is demonstrated by a computer simulation.     

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.     

Harvesting a logistic population in a slowly varying environment

The classic problem for a logistically evolving single species population being harvested involves three parameters: rate constant, carrying capacity and harvesting rate, which are taken to be positive constants. However, in real world situations, these parameters may vary with time. This paper considers the situation where these vary on a time scale much longer than that intrinsic to the population evolution itself. Application of a multiple time scale approach gives approximate explicit closed form expressions for the changing population, that compare favorably with those generated from numerical solutions.     

Localized vibration in elastic structures with slowly varying thickness

High frequency localized vibration modes for linear isotropicelastic plates and rods of slowly varying thickness are shownto occur within the vicinity of maximal or minimal cross-sections.Long wave asymptotic integration close to the eigenmodes ofsuch cross-sections is provided. The cases of a one integerparameter of thickness resonances (for a plate in plane stress)and two integer parameters (for a rod of circular cross-section)are fully investigated. One-dimensional Hermite-type equationsfor the eigenmodes are derived and associated frequency shiftsfrom those of the thickness resonances of the maximal or minimalcross-section obtained. It is established that the existenceof this type of localization phenomena is strongly dependenton the sign of the group velocity of the associated uniformthickness structure this being the same as that of the localgroup velocity calculated at the maximal or minimal cross-section.It is confirmed that positive group velocity allows localizationfor elevated structures with negative group velocity allowinglocalization only for ligaments. In each case intervals of thePoisson's ratio providing localization are given for the firstfew modes.     

Multiscaling analysis of a slowly varying anaerobic digestion model

We construct the slowly varying limiting state solutions to a nonlinear dynamical system for anaerobic digestion with Monod-based kinetics involving slowly varying model parameters arising from slow environmental variation. The advantage of these approximate solutions over numerical solutions is their applicability to a wide range of parameter values. We use these limiting state solutions to develop analytic approximations to the full nonlinear system by applying a multiscaling technique. The approximate solutions are shown to compare favorably with numerical solutions.     

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.     

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).     

Very badly approximable matrix functions

We study in this paper very badly approximable matrix functions on the unit circle i.e., matrix functions Φ such that the zero function is a superoptimal approximation of Φ. The purpose of this paper is to obtain a characterization of the continuous very badly approximable functions. Our characterization is more geometric than algebraic characterizations earlier obtained in [PY1] and [AP]. It involves analyticity of certain families of subspaces defined in terms of Schmidt vectors of the matrices This characterization can be extended to the wider class of admissible functions, i.e., the class of matrix functions Φ such that the essential norm ||H_Φ||_e of the Hankel operator H_Φ is less than the smallest nonzero superoptimal singular value of Φ. In the final section we obtain a similar characterization of badly approximable matrix functions.     

On a class of slowly changing functions

We study the slowly changing functions and use them in the study of entire functions. An erratum to this article is available at .     

Shoaling of nonlinear wave-groups on water of slowly varying depth

An approximate analytical solution describing the shoaling of modulated wave-trains is presented. This solution provides new information about the wave field evolution as well as about the wave-induced mean current and set down.

---

Zusammenfassung Eine angenäherte analytische Lösung wird gegeben, die das Brechen von modulierten Wellenzügen beschreibt. Die Lösung gibt neue Informationen über die Entwicklung des Wellenfeldes wie auch über die mittlere Strömung und mittlere Höhenänderung, die von den Wellen induziert wird.

     

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.