Green's formula and theorems on complete collection of isomorphisms for general elliptic boundary value problems for Douglis-Nirenberg systems

This work has been partially supported by Grant INTAS-94-2187

General-type discrete self-adjoint Dirac systems: explicit solutions of direct and inverse problems, asymptotics of Verblunsky-type coefficients and stability of solving inverse problem

We consider discrete self-adjoint Dirac systems determined by the potentials (sequences) $\{C_k\}$ such that the matrices $C_k$ are positive definite and $j$-unitary, where $j$ is a diagonal $m\times m$ matrix and has $m_1$ entries $1$ and $m_2$ entries $-1$ ($m_1+m_2=m$) on the main diagonal. We construct systems with rational Weyl functions and explicitly solve inverse problem to recover systems from the contractive rational Weyl functions. Moreover, we study the stability of this procedure. The matrices $C_k$ (in the potentials) are so called Halmos extensions of the Verblunsky-type coefficients $\rho_k$. We show that in the case of the contractive rational Weyl functions the coefficients $\rho_k$ tend to zero and the matrices $C_k$ tend to the indentity matrix $I_m$.Comment: This paper is a generalization and further development of the topics discussed in arXiv:math/0703369, arXiv:1206.2915, arXiv:1508.07954, arXiv:1510.0079

Discrete Dirac system: rectangular Weyl functions, direct and inverse problems

A transfer matrix function representation of the fundamental solution of the general-type discrete Dirac system, corresponding to rectangular Schur coefficients and Weyl functions, is obtained. Connections with Szeg\"o recurrence, Schur coefficients and structured matrices are treated. Borg-Marchenko-type uniqueness theorem is derived. Inverse problems on the interval and semiaxis are solved.Comment: Section 2 is improved in the second version: some new results on Halmos extension are added and arguments are simplifie

Geographic variation of life-history traits in the sand lizard, Lacerta agilis: testing Darwin's facundity-advantage hypothesis

The fecundity-advantage-hypothesis (FAH) explains larger female size relative to male size as a correlated response to fecundity selection. We explored FAH by investigating geographic variation in female reproductive output and its relation to sexual size dimorphism (SSD) in Lacerta agilis, an oviparous lizard occupying a major part of temperate Eurasia. We analysed how sex-specific body size and SSD are associated with two putative indicators of fecundity selection intensity (clutch size and the slope of the clutch size-female size relationship), and with two climatic variables throughout the species range and across two widespread evolutionary lineages. Variation within the lineages provides no support for FAH. In contrast, the divergence between the lineages is in line with FAH: the lineage with consistently female-biased SSD (L. a. agilis) exhibits higher clutch size and steeper fecundity slope than the lineage with an inconsistent and variable SSD (L. a. exigua). L. a. agilis shows lower offspring size (egg mass, hatchling mass) and higher clutch mass relative to female mass than L. a. exigua, i.e. both possible ways to enhance offspring number are exerted. As the SSD difference is due to male size (smaller males in L. a. agilis), fecundity selection favouring larger females, together with viability selection for smaller size in both sexes, would explain the female-biased SSD and reproductive characteristics of L. a. agilis. The pattern of intraspecific life-history divergence in L.agilis is strikingly similar to that between oviparous and viviparous populations of a related species Zootoca vivipara. Evolutionary implications of this parallelism are discussed