Quadratic perturbations of quadratic codimension-four centers

We study the stratum in the set of all quadratic differential systems $\dot{x}=P_2(x,y), \dot{y}=Q_2(x,y)$ with a center, known as the codimension-four case $Q_4$. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincar\'e-Pontryagin-Melnikov integral $I$. We show that the orbits of the unperturbed system are elliptic curves, and $I$ is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center

Perturbations of quadratic centers of genus one

We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincare-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems.Comment: 28 page

Stability of periodic traveling waves for complex modified Korteweg-de Vries equation

We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg-de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.Comment: Latex, 20 page

In vitro evaluation of combination effects of doxorubicin with methylxanthine fractions isolated from Bancha and Pu-erh teas against breast cancer cells

Background: In the present study we investigated the combination effects of anthracycline antibiotic, doxorubicin, with methylxanthine fractions isolated from Bancha and Pu-erh tea leaves, against MCF-7 and MDA-MB-231 breast cancer cell lines.Methods: Neutral red uptake assay was used for assessment of cytotoxicity effects and fractional effect analysis and combination index for evaluation of the combination effects.Results: Doxorubicin was used in varying concentrations by a double dilution method, whereas the methylxanthine fractions were in fixed concentrations – 100, 200, 400 or 600 μg/ml. Results have shown that methylxanthine fraction isolated from Bancha has synergic effects with doxorubicin, while methylxanthines from Pu-erh displayed antagonistic effects.Conclusions: Тhe obtained results lead us to suspect, that even minor differences in the composition of natural products can lead to significant differences in the biological activity of the product