Universal description of three two-component fermions

A quantum mechanical three-body problem for two identical fermions of mass $m$ and a distinct particle of mass $m_1$ in the universal limit of zero-range two-body interaction is studied. For the unambiguous formulation of the problem in the interval $\mu_r < m/m_1 \le \mu_c$ ($\mu_r \approx 8.619$ and $\mu_c \approx 13.607$) an additional parameter $b$ determining the wave function near the triple-collision point is introduced; thus, a one-parameter family of self-adjoint Hamiltonians is defined. The dependence of the bound-state energies on $m/m_1$ and $b$ in the sector of angular momentum and parity $L^P = 1^-$ is calculated and analysed with the aid of a simple model

On a class of second-order PDEs admitting partner symmetries

Recently we have demonstrated how to use partner symmetries for obtaining noninvariant solutions of heavenly equations of Plebanski that govern heavenly gravitational metrics. In this paper, we present a class of scalar second-order PDEs with four variables, that possess partner symmetries and contain only second derivatives of the unknown. We present a general form of such a PDE together with recursion relations between partner symmetries. This general PDE is transformed to several simplest canonical forms containing the two heavenly equations of Plebanski among them and two other nonlinear equations which we call mixed heavenly equation and asymmetric heavenly equation. On an example of the mixed heavenly equation, we show how to use partner symmetries for obtaining noninvariant solutions of PDEs by a lift from invariant solutions. Finally, we present Ricci-flat self-dual metrics governed by solutions of the mixed heavenly equation and its Legendre transform.Comment: LaTeX2e, 26 pages. The contents change: Exact noninvariant solutions of the Legendre transformed mixed heavenly equation and Ricci-flat metrics governed by solutions of this equation are added. Eq. (6.10) on p. 14 is correcte

Genetic and Environmental Influences on Temperament in Adolescence

This study, which is a part of a Moscow longitudinal twin project, aims to explore genetic and environmental contributions to inter-individual variability of temperamental traits in adolescence on the basis of a Russian sample. 85 monozygotic (MZ) and 64 same-sex dizygotic (DZ) twin pairs aged 12 – 14 years completed the children version of Rusalov Structure of Temperament Questionnaire (C-STQ). The results of model-fitting analyses indicate considerable hereditary determination of individual differences in 3 out of the 8 C-STQ dimensions - social tempo, objectrelated emotional sensitivity, and social emotional sensitivity. Non-shared environmental effects explained the rest of the total variance in these dimensions. Individual differences in the other STQ dimensions were due to environmental factors

Low-energy three-body dynamics in binary quantum gases

The universal three-body dynamics in ultra-cold binary Fermi and Fermi-Bose mixtures is studied. Two identical fermions of the mass $m$ and a particle of the mass $m_1$ with the zero-range two-body interaction in the states of the total angular momentum L=1 are considered. Using the boundary condition model for the s-wave interaction of different particles, both eigenvalue and scattering problems are treated by solving hyper-radial equations, whose terms are derived analytically. The dependencies of the three-body binding energies on the mass ratio $m/m_1$ for the positive two-body scattering length are calculated; it is shown that the ground and excited states arise at $m/m_1 \ge \lambda_1 \approx 8.17260$ and $m/m_1 \ge \lambda_2 \approx 12.91743$, respectively. For m/m_1 \alt \lambda_1 and m/m_1 \alt \lambda_2, the relevant bound states turn to narrow resonances, whose positions and widths are calculated. The 2 + 1 elastic scattering and the three-body recombination near the three-body threshold are studied and it is shown that a two-hump structure in the mass-ratio dependencies of the cross sections is connected with arising of the bound states.Comment: 16 page

The interplay of chaos between the terrestrial and giant planets

We report on some simple experiments on the nature of chaos in our planetary system. We make the following interesting observations. First, we look at the system of Sun + four Jovian planets as an isolated five-body system interacting only via Newtonian gravity. We find that if we measure the Lyapunov time of this system across thousands of initial conditions all within observational uncertainty, then the value of the Lyapunov time seems relatively smooth across some regions of initial condition space, while in other regions it fluctuates wildly on scales as small as we can reliably measure using numerical methods. This probably indicates a fractal structure of Lyapunov exponents measured across initial condition space. Then, we add the four inner terrestrial planets and several post-Newtonian corrections such as general relativity into the model. In this more realistic Sun + eight-planet system, we find that the above structure of chaos for the outer planets becomes uniformly chaotic for almost all planets and almost all initial conditions, with a Lyapunov time-scale of about 5-20 Myr. This seems to indicate that the addition of the inner planets adds more chaos to the system. Finally, we show that if we instead remove the outer planets and look at the isolated five-body system of the Sun + four terrestrial planets, then the terrestrial planets alone show no evidence of chaos at all, over a large range of initial conditions inside the observational error volume. We thus conclude that the uniformity of chaos in the outer planets comes not from the inner planets themselves, but from the interplay between the outer and inner ones. Interestingly, however, there exist rare and isolated initial conditions for which one individual outer planetary orbit may appear integrable over a 200-Myr time-scale, while all the other planets simultaneously appear chaotic. © 2010 The Authors. Journal compilation © 2010 RAS

Partner symmetries of the complex Monge-Ampere equation yield hyper-Kahler metrics without continuous symmetries

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Amp\`ere equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. We identify the real and imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the K\"ahler potential which directly leads to a Legendre transformation and to a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge-Amp\`ere equation and obtain hyper-K\"ahler metrics with anti-self-dual Riemann curvature 2-form that admit no Killing vectors.Comment: submitted to J. Phys.