Genome Assembly Improvement and Mapping Convergently Evolved Skeletal Traits in Sticklebacks with Genotyping-by-Sequencing.

Marine populations of the threespine stickleback (Gasterosteus aculeatus) have repeatedly colonized and rapidly adapted to freshwater habitats, providing a powerful system to map the genetic architecture of evolved traits. Here, we developed and applied a binned genotyping-by-sequencing (GBS) method to build dense genome-wide linkage maps of sticklebacks using two large marine by freshwater F2 crosses of more than 350 fish each. The resulting linkage maps significantly improve the genome assembly by anchoring 78 new scaffolds to chromosomes, reorienting 40 scaffolds, and rearranging scaffolds in 4 locations. In the revised genome assembly, 94.6% of the assembly was anchored to a chromosome. To assess linkage map quality, we mapped quantitative trait loci (QTL) controlling lateral plate number, which mapped as expected to a 200-kb genomic region containing Ectodysplasin, as well as a chromosome 7 QTL overlapping a previously identified modifier QTL. Finally, we mapped eight QTL controlling convergently evolved reductions in gill raker length in the two crosses, which revealed that this classic adaptive trait has a surprisingly modular and nonparallel genetic basis

The optimized Rayleigh-Ritz scheme for determining the quantum-mechanical spectrum

The convergence of the Rayleigh-Ritz method with nonlinear parameters optimized through minimization of the trace of the truncated matrix is demonstrated by a comparison with analytically known eigenstates of various quasi-solvable systems. We show that the basis of the harmonic oscillator eigenfunctions with optimized frequency ? enables determination of boundstate energies of one-dimensional oscillators to an arbitrary accuracy, even in the case of highly anharmonic multi-well potentials. The same is true in the spherically symmetric case of V (r) = {\omega}2r2 2 + {\lambda}rk, if k > 0. For spiked oscillators with k < -1, the basis of the pseudoharmonic oscillator eigenfunctions with two parameters ? and {\gamma} is more suitable, and optimization of the latter appears crucial for a precise determination of the spectrum.Comment: 22 pages,8 figure

A basis for variational calculations in d dimensions

In this paper we derive expressions for matrix elements (\phi_i,H\phi_j) for the Hamiltonian H=-\Delta+\sum_q a(q)r^q in d > 1 dimensions. The basis functions in each angular momentum subspace are of the form phi_i(r)=r^{i+1+(t-d)/2}e^{-r^p/2}, i >= 0, p > 0, t > 0. The matrix elements are given in terms of the Gamma function for all d. The significance of the parameters t and p and scale s are discussed. Applications to a variety of potentials are presented, including potentials with singular repulsive terms of the form b/r^a, a,b > 0, perturbed Coulomb potentials -D/r + B r + Ar^2, and potentials with weak repulsive terms, such as -g r^2 + r^4, g > 0.Comment: 22 page

Variational collocation for systems of coupled anharmonic oscillators

We have applied a collocation approach to obtain the numerical solution to the stationary Schr\"odinger equation for systems of coupled oscillators. The dependence of the discretized Hamiltonian on scale and angle parameters is exploited to obtain optimal convergence to the exact results. A careful comparison with results taken from the literature is performed, showing the advantages of the present approach.Comment: 14 pages, 10 table

The Fokker-Planck equation for bistable potential in the optimized expansion

The optimized expansion is used to formulate a systematic approximation scheme to the probability distribution of a stochastic system. The first order approximation for the one-dimensional system driven by noise in an anharmonic potential is shown to agree well with the exact solution of the Fokker-Planck equation. Even for a bistable system the whole period of evolution to equilibrium is correctly described at various noise intensities.Comment: 12 pages, LATEX, 3 Postscript figures compressed an

Effective-mass Klein-Gordon Equation for non-PT/non-Hermitian Generalized Morse Potential

The one-dimensional effective-mass Klein-Gordon equation for the real, and non-\textrm{PT}-symmetric/non-Hermitian generalized Morse potential is solved by taking a series expansion for the wave function. The energy eigenvalues, and the corresponding eigenfunctions are obtained. They are also calculated for the constant mass case.Comment: 14 page

Convergence of the Gaussian Expansion Method in Dimensionally Reduced Yang-Mills Integrals

We advocate a method to improve systematically the self-consistent harmonic approximation (or the Gaussian approximation), which has been employed extensively in condensed matter physics and statistical mechanics. We demonstrate the {\em convergence} of the method in a model obtained from dimensional reduction of SU($N$) Yang-Mills theory in $D$ dimensions. Explicit calculations have been carried out up to the 7th order in the large-N limit, and we do observe a clear convergence to Monte Carlo results. For $D \gtrsim 10$ the convergence is already achieved at the 3rd order, which suggests that the method is particularly useful for studying the IIB matrix model, a conjectured nonperturbative definition of type IIB superstring theory.Comment: LaTeX, 4 pages, 5 figures; title slightly changed, explanations added (16 pages, 14 figures), final version published in JHE

Spiked oscillators: exact solution

A procedure to obtain the eigenenergies and eigenfunctions of a quantum spiked oscillator is presented. The originality of the method lies in an adequate use of asymptotic expansions of Wronskians of algebraic solutions of the Schroedinger equation. The procedure is applied to three familiar examples of spiked oscillators

A class of nonlinear wave equations containing the continuous Toda case

We consider a nonlinear field equation which can be derived from a binomial lattice as a continuous limit. This equation, containing a perturbative friction-like term and a free parameter $\gamma$, reproduces the Toda case (in absence of the friction-like term) and other equations of physical interest, by choosing particular values of $\gamma$. We apply the symmetry and the approximate symmetry approach, and the prolongation technique. Our main purpose is to check the limits of validity of different analytical methods in the study of nonlinear field equations. We show that the equation under investigation with the friction-like term is characterized by a finite-dimensional Lie algebra admitting a realization in terms of boson annhilation and creation operators. In absence of the friction-like term, the equation is linearized and connected with equations of the Bessel type. Examples of exact solutions are displayed, and the algebraic structure of the equation is discussed.Comment: Latex file + [equations.sty], 22 p