Newtonian limit of conformal gravity and the lack of necessity of the second order Poisson equation

We study the interior structure of a locally conformal invariant fourth order theory of gravity in the presence of a static, spherically symmetric gravitational source. We find, quite remarkably, that the associated dynamics is determined exactly and without any approximation at all by a simple fourth order Poisson equation which thus describes both the strong and weak field limits of the theory in this static case. We present the solutions to this fourth order equation and find that we are able to recover all of the standard Newton-Euler gravitational phenomenology in the weak gravity limit, to thus establish the observational viability of the weak field limit of the fourth order theory. Additionally, we make a critical analysis of the second order Poisson equation, and find that the currently available experimental evidence for its validity is not as clearcut and definitive as is commonly believed, with there not apparently being any conclusive observational support for it at all either on the very largest distance scales far outside of fundamental sources, or on the very smallest ones within their interiors. Our study enables us to deduce that even though the familiar second order Poisson gravitational equation may be sufficient to yield Newton's Law of Gravity it is not in fact necessary.     

On the scattering amplitudes for elastic waves

Reciprocity and scattering theorems for the normalized spherical scattering amplitude for elastic waves are obtained for the case of a rigid scatterer, a cavity and a penetrable scattering region. Depending on the polarization of the two incident waves reciprocity relations of the radial-radial, radial-angular, and angular-angular type are established. Radial and angular scattering theorems, expressing the corresponding scattering amplitudes via integrals of the amplitudes over all directions of observation, as well as their special forms for scatterers with inversion symmetry are also provided. As a consequence of the stated scattering theorems the scattering cross-section for either a longitudinal, or a transverse incident wave is expressed through the forward value of the radial, or the angular amplitude, correspondingly. All the known relative theorems for acoustic scattering are trivially recovered from their elastic counterparts.Part of this work was done during the time that the first author was visiting the Department of Mathematics of The University of Tennessee at Knoxville. The authors want to thank the Greek Ministry of Research and Technology for partially supporting the present work.     

The Inverse Spectral Method for Colliding Gravitational Waves

The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular region 1 x < y 1 for a certain 2 × 2 matrix valued nonlinear equation. This equation, which is a particular exact reduction of the vacuum Einstein equations, is integrable, i.e. it possesses a Lax pair formulation. Using the simultaneous spectral analysis of this Lax pair we study the above Goursat problem as well as its linearized version. It is shown that the linear problem reduces to a scalar Riemann–Hilbert problem, which can be solved in closed form, while the nonlinear problem reduces to a 2 × 2 matrix Riemann–Hilbert problem, which under certain conditions is solvable.     

Age-dependent dichotomous effect of superoxide dismutase Ala16Val polymorphism on oxidized LDL levels

We investigated the association between superoxide dismutase (SOD) Ala16Val polymorphism and the levels of oxidized LDL lipoprotein-C (ox-LDL-C) in two age-different Greek cohorts. Four hundred fifteen middle-aged (n=147 females: 43.2+/-13 years, n=268 males: 43.3+/-14 years) Caucasian Greek subjects consisted the middle aged cohort. One hundred seventy five elderly (n=88 females: 79.9+/-4 years; n=87 males: 80.6+/-4 years) were selected from the elderly cohort. Genotype data were obtained for all of them. Multiple linear regression analysis, stratified by gender and adjusted for age, smoking habits and body mass index as covariates, showed higher ox-LDL-C levels for the middle aged men with the Val/Val genotype, compared to the other allele (Ala/Ala and Ala/Val) carriers (65.9+/-25.7 vs. 55.7+/-20.5 mg/dl; standardized beta coefficient=0.192, P=0.012). On the contrary, elderly women with the Val/Val genotype occurred with lower ox-LDL-C levels compared to the Ala/Ala or Ala/Val genotype (74.2+/-22.1 vs. 86.5+/-26.6 mg/dl; standardized beta coefficient= -0.269, P=0.015). The same trend was also recorded in elderly men, however without reaching statistical significance (standardized beta coefficient= -0.187, P=0.077). Moreover, elderly men and women with the Ala/Ala or Ala/Val genotype presented higher triglycerides levels compared to Val/Val (women: 145.2+/-68.7 vs. 114.3+/- 34.3 mg/dl, P= 0.027; men: 147.8+/-72.4 vs. 103.7 +/-38.0 mg/dl, P=0.002). Additionally, middle aged men with the Val/Val genotype had higher HDL-C levels compared to the Ala allele carriers. The results suggest that SOD Ala16Val polymorphism is an age-dependent modulator of ox-LDL-C levels in middle-aged men and elderly women.     

Interaction of solid particles with a tangle of vortex filaments in a viscous fluid

Homogeneous isotropic turbulence consists of coherent filamentary vortex structures superimposed to a more incoherent background. The question which we address is the effect of these structures on the dynamics of small, neutrally buoyant solid particles. Rather than generating the turbulence by direct numerical simulation (DNS) of the Navier-Stokes equations, we use a model of turbulence based entirely on viscous vortex filaments which interact via inertial forces and reconnect with each other. Using this model, we show that solid particles can become trapped around vortex filaments, something difficult to achieve with DNS. Unlike most studies, we have not neglected inviscid inertial effects. By comparing the Stokes, local, and convective components of the particle's acceleration, we also show that the convective part clearly identifies the trapping.     

Integrability and beyond

An algebraic approach to solving nonlinear functional equations in the Riemann theta functions is stated. By the inverse scattering method and some general methods of the theory of partial differential equations, the solution of the initial boundary value problem for the nonlinear Schrödinger equation is presented. Bibliography:17 titles.