Exactly Solvable Anharmonic Oscillator, Degenerate Orthogonal Polynomials and Painlevé II.

Using WKB analysis, the paper addresses a conjecture of Shapiro and Tater on the similarity between two sets of points in the complex plane; on one side is the set the values of for which the spectrum of the quartic anharmonic oscillator in the complex plane with certain boundary conditions, has repeated eigenvalues. On the other side is the set of zeroes of the Vorob'ev-Yablonskii polynomials, i.e.

Soliton Shielding of the Focusing Nonlinear Schrödinger Equation.

We first consider a deterministic gas of N solitons for the focusing nonlinear Schrödinger (FNLS) equation in the limit N→∞ with a point spectrum chosen to interpolate a given spectral soliton density over a bounded domain of the complex spectral plane. We show that when the domain is a disk and the soliton density is an analytic function, then the corresponding deterministic soliton gas surprisingly yields the one-soliton solution with the point spectrum the center of the disk. We call this effect soliton shielding.

Entanglement of Two Disjoint Intervals in Conformal Field Theory and the 2D Coulomb Gas on a Lattice.

In the conformal field theories given by the Ising and Dirac models, when the system is in the ground state, the moments of the reduced density matrix of two disjoint intervals and of its partial transpose have been written as partition functions on higher genus Riemann surfaces with Z_{n} symmetry. We show that these partition functions can be expressed as the grand canonical partition functions of the two-dimensional two component classical Coulomb gas on certain circular lattices at specific values of the coupling constant.

On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations.

We study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (P[Formula: see text]) equation or its fourth-order analogue P[Formula: see text]. As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit.

Habitat quality affects early physiology and subsequent neuromotor development of juvenile black-capped chickadees.

In songbirds, the ability to learn and render the species-specific song is influenced by the development of both the song nuclei in the brain and the syrinx (bird's vocal apparatus) early in the bird's life. In black-capped chickadees (Poecille atricapillus), habitat quality is known to affect song structure, with birds in high-quality habitat (mature forest) having a higher song consistency than birds in low-quality habitat (young forest). Although this difference is suspected to stem from differences in development, the developmental status of juvenile birds in either habitat remains unexplored.