On the algebraic invariant curves of plane polynomial differential systems

We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The asymptotic estimate of the degree of an arbitrary algebraic invariant curve is found. In the smooth case this estimate was already found by D. Cerveau and A. Lins Neto [Ann. Inst. Fourier Grenoble 41, 883-903] in a different way.Comment: 10 pages, Latex, to appear in J.Phys.A:Math.Ge

How to find discrete contact symmetries

This paper describes a new algorithm for determining all discrete contact symmetries of any differential equation whose Lie contact symmetries are known. The method is constructive and is easy to use. It is based upon the observation that the adjoint action of any contact symmetry is an automorphism of the Lie algebra of generators of Lie contact symmetries. Consequently, all contact symmetries satisfy various compatibility conditions. These conditions enable the discrete symmetries to be found systematically, with little effort

Gradient Optics of subwavelength nanofilms

Propagation and tunneling of light through subwavelength photonic barriers, formed by dielectric layers with continuous spatial variations of dielectric susceptibility across the film are considered. Effects of giant heterogeneity-induced non-local dispersion, both normal and anomalous, are examined by means of a series of exact analytical solutions of Maxwell equations for gradient media. Generalized Fresnel formulae, visualizing a profound influence of gradient and curvature of dielectric susceptibility profiles on reflectance/transmittance of periodical photonic heterostructures are presented. Depending on the cutoff frequency of the barrier, governed by technologically managed spatial profile of its refractive index, propagation or tunneling of light through these barriers are examined. Nonattenuative transfer of EM energy by evanescent waves, tunneling through dielectric gradient barriers, characterized by real values of refractive index, decreasing in the depth of medium, is shown. Scaling of the obtained results for different spectral ranges of visible, IR and THz waves is illustrated. Potential of gradient optical structures for design of miniaturized filters, polarizers and frequency-selective interfaces of subwavelength thickness is considered

Critical behavior of magnetic systems with extended impurities in general dimensions

We investigate the critical properties of d-dimensional magnetic systems with quenched extended defects, correlated in $\epsilon_d$ dimensions (which can be considered as the dimensionality of the defects) and randomly distributed in the remaining $d-\epsilon_d$ dimensions; both in the case of fixed dimension d=3 and when the space dimension continuously changes from the lower critical dimension to the upper one. The renormalization group calculations are performed in the minimal subtraction scheme. We analyze the two-loop renormalization group functions for different fixed values of the parameters $d, \epsilon_d$. To this end, we apply the Chisholm-Borel resummation technique and report the numerical values of the critical exponents for the universality class of this system.Comment: 8 figures. To appear in Phys. Rev.

Mitochondria and Energetic Depression in Cell Pathophysiology

Mitochondrial dysfunction is a hallmark of almost all diseases. Acquired or inherited mutations of the mitochondrial genome DNA may give rise to mitochondrial diseases. Another class of disorders, in which mitochondrial impairments are initiated by extramitochondrial factors, includes neurodegenerative diseases and syndromes resulting from typical pathological processes, such as hypoxia/ischemia, inflammation, intoxications, and carcinogenesis. Both classes of diseases lead to cellular energetic depression (CED), which is characterized by decreased cytosolic phosphorylation potential that suppresses the cell’s ability to do work and control the intracellular Ca2+ homeostasis and its redox state. If progressing, CED leads to cell death, whose type is linked to the functional status of the mitochondria. In the case of limited deterioration, when some amounts of ATP can still be generated due to oxidative phosphorylation (OXPHOS), mitochondria launch the apoptotic cell death program by release of cytochrome c. Following pronounced CED, cytoplasmic ATP levels fall below the thresholds required for processing the ATP-dependent apoptotic cascade and the cell dies from necrosis. Both types of death can be grouped together as a mitochondrial cell death (MCD). However, there exist multiple adaptive reactions aimed at protecting cells against CED. In this context, a metabolic shift characterized by suppression of OXPHOS combined with activation of aerobic glycolysis as the main pathway for ATP synthesis (Warburg effect) is of central importance. Whereas this type of adaptation is sufficiently effective to avoid CED and to control the cellular redox state, thereby ensuring the cell survival, it also favors the avoidance of apoptotic cell death. This scenario may underlie uncontrolled cellular proliferation and growth, eventually resulting in carcinogenesis