Remarks on oscillation of second-order linear difference equation

In this remark, we shall show that the Lemma 1 and parts of the proof of Theorem 1 in the paper “Oscillation criteria for second-order linear difference equations” (Applied Mathematics and Computation 145 (2003) 591-601) are incorrect.     

High-capacity purification of hen egg-white proteins by ion-exchange electrochromatography with an oscillatory transverse electric field

The ion-exchange electrochromatography with an oscillatory electric field perpendicular to the mobile-phase flow driven by pressure (pIEEC) was used to separate hen egg-white (HEW) proteins. The results were compared with those of normal ion-exchange chromatography (IEC). The column was designed as three-compartment rectangular column of 2-mL with dimensions (length x width x depth) of 40 x 10 x 5 mm(3) and the electric field was applied across the direction of column width. Q Sepharose FF was packed into the central compartment as the chromatographic bed. It was confirmed that the dynamic binding capacity (DBC) of different proteins (ovotransferrin and ovalbumin) in the HEW solution increased 2.3 times when an oscillatory electric current of 30 mA at 1/20 Hz was applied in the transverse column direction. Then, the HEW proteins were separated by the pIEEC at loading amounts 2.3-fold higher than those by the IEC. When the feedstock of about one-third of the DBC was applied to the columns (i.e., 7 mL for the pIEEC and 3 mL for the IEC), similar separation efficiencies of the two chromatographic modes were achieved. Both the recovery yield and purity reached 73% to over 90%. The results indicate that the pIEEC is promising for high-capacity purification of proteins.     

Existence of oscillatory solutions along the path of longitudinal flight equilibriums of an unmanned aircraft, when the automatic flight control system fails

The motion around the center of mass of a rigid unmanned aircraft, whose flight control system fails, in an “Aero Data Model In a Research Environment” is described, by a set of nine nonlinear ordinary differential equations. The longitudinal flight with constant forward velocity is described by a subset of three nonlinear differential equations, obtained from the general system. In this paper, the existence of oscillatory solutions of this system of three differential equations is proved by means of coincidence degree theory and Mawhin's continuation theorem.     

Fast directional algorithms for the Helmholtz kernel

This paper presents a new directional multilevel algorithm for solving N-body or N-point problems with highly oscillatory kernels. We address the problem by first proving that the interaction between a ball of radius r and a well-separated region has an approximate low rank representation, as long as the well-separated region belongs to a cone with a spanning angle of O(1/r) and is at a distance which is at least O(r²) away from the ball. Based on this representation, our algorithm organizes the high frequency computation using a multidirectional and multiscale strategy. Our algorithm is proved to have an optimal O(NlogN) computational complexity for any given accuracy when the points are sampled from a two-dimensional surface.     

A sixth order diagonally implicit symmetric and symplectic Runge-Kutta method for solving Hamiltonian systems

The paper is concerned with construction of symmetric and symplectic Runge-Kutta methods for Hamiltonian systems. Based on the symplectic and symmetrical properties, a sixth-order diagonally implicit symmetric and symplectic Runge-Kutta method with seven stages is presented, the proposed method proved to be P-stable. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing Runge-Kutta methods in scientic literature.     

Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations

In this paper, we study time‐asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space‐time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time‐asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support.     

On the calculation of highly oscillatory integrals with an algebraic singularity

In this paper we explore the evaluation of highly oscillatory integrals containing an algebraic singularity based on the change of variable t = x^−r, where r is a positive real number, and the analytical continuation. Robust numerical results demonstrate the accuracy and effectiveness of the proposed approach.     

Autocatalator as the source of instability in the complex non-linear neuroendocrine model

The mathematical analogy between properties of simple and complex models of non-linear reactions was used to determine reaction steps in the complex model, necessary to generate instability region and appropriate type of bifurcations on the border between stable and unstable non-equilibrium stationary state. The autocatalator was recognized as the simple prototype two-variable non-linear model practical for examination of the complex four-variable non-linear neuroendocrine system known as the Hypothalamic-Pituitary-Adrenal (HPA) axis. In both cases, we derived the instability criteria by stoichiometric network analysis (SNA), determine conditions under which dynamic transitions, i.e. bifurcations occur, and identify the type of bifurcation. The supercritical Andronov–Hopf bifurcation was found in both cases whereas saddle-node bifurcation was detected only in the model for HPA axis.Thus, by stoichiometric network analysis, the mathematical analogy is found between two different models with same autocatalytic steps, that is, between two models with easily recognizable qualitative analogy.     

Traveling waves of some integral-differential equations arising from neuronal networks with oscillatory kernels

In this paper, we study the existence, uniqueness and profiles of traveling waves to some integral-differential equations arising from nonlinear nonlocal neuronal networks with oscillatory kernels. Our approach is fundamental ideas in differential equation and function analysis.     

Global attractivity in delay “food-limited” models with exponential impulses

In this paper, we are concerned with the global attractivity of the positive equilibrium of the following delay “food-limited” model with exponential impulses: