New Ther1-derived SINE Squam3 in Scaled Reptiles

Background SINEs compose a significant part of animal genomes and are used to study the evolution of taxa. Despite significant advances in SINE studies in vertebrates and higher eukaryotes in general, their own evolution is poorly understood. Results We have found and described in detail a new SINE family Squam3 specific for scaled reptiles (Squamata). The subfamilies of this SINE demonstrate different distribution in the genomes of squamates, which together with the data on similar SINEs in the tuatara allowed us to propose a scenario of their evolution in the context of reptilian evolution. Conclusions Ancestral SINEs preserved in small numbers in most genomes can give rise to taxon-specific SINE families. Analysis of this aspect of SINEs can shed light on the history and mechanisms of SINE variation in reptilian genomes.

On PZ type Siegel disks of the sine family

We prove that for typical rotation numbers $0<{\it\theta}<1$, the boundary of the Siegel disk of $f_{{\it\theta}}(z)=e^{2{\it\pi}i{\it\theta}}\sin (z)$ centered at the origin is a Jordan curve which passes through exactly two critical points ${\it\pi}/2$ and $-{\it\pi}/2$.

Exponentially Convergent Approximation to the Elliptic Solution Operator

We develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].