Stable manifolds and homoclinic points near resonances in the restricted three-body problem

The restricted three-body problem describes the motion of a massless particle under the influence of two primaries of masses $1-\mu$ and $\mu$ that circle each other with period equal to $2\pi$. For small $\mu$, a resonant periodic motion of the massless particle in the rotating frame can be described by relatively prime integers $p$ and $q$, if its period around the heavier primary is approximately $2\pi p/q$, and by its approximate eccentricity $e$. We give a method for the formal development of the stable and unstable manifolds associated with these resonant motions. We prove the validity of this formal development and the existence of homoclinic points in the resonant region. In the study of the Kirkwood gaps in the asteroid belt, the separatrices of the averaged equations of the restricted three-body problem are commonly used to derive analytical approximations to the boundaries of the resonances. We use the unaveraged equations to find values of asteroid eccentricity below which these approximations will not hold for the Kirkwood gaps with $q/p$ equal to 2/1, 7/3, 5/2, 3/1, and 4/1. Another application is to the existence of asymmetric librations in the exterior resonances. We give values of asteroid eccentricity below which asymmetric librations will not exist for the 1/7, 1/6, 1/5, 1/4, 1/3, and 1/2 resonances for any $\mu$ however small. But if the eccentricity exceeds these thresholds, asymmetric librations will exist for $\mu$ small enough in the unaveraged restricted three-body problem

Dimer and N\'eel order-parameter fluctuations in the spin-fluid phase of the s=1/2 spin chain with first and second neighbor couplings

The dynamical properties at T=0 of the one-dimensional (1D) s=1/2 nearest-neighbor (nn) XXZ model with an additional isotropic next-nearest-neighbor (nnn) coupling are investigated by means of the recursion method in combination with techniques of continued-fraction analysis. The focus is on the dynamic structure factors S_{zz}(q,\omega) and S_{DD}(q,\omega), which describe (for q=\pi) the fluctuations of the N\'eel and dimer order parameters, respectively. We calculate (via weak-coupling continued-fraction analysis) the dependence on the exchange constants of the infrared exponent, the renormalized bandwidth of spinon excitations, and the spectral-weight distribution in S_{zz}(\pi,\omega) and S_{DD}(\pi,\omega), all in the spin-fluid phase, which is realized for planar $nn$ anisotropy and sufficiently weak nnn coupling. For some parameter values we find a discrete branch of excitations above the spinon continuum. They contribute to S_{zz}(q,\omega) but not to S_{DD}(q,\omega).Comment: RevTex file (7 pages), 8 figures (uuencoded ps file) available from author

Systematic Mapping of the Hubbard Model to the Generalized t-J Model

The generalized t-J model conserving the number of double occupancies is constructed from the Hubbard model at and in the vicinity of half-filling at strong coupling. The construction is realized by a self-similar continuous unitary transformation. The flow equation is closed by a truncation scheme based on the spatial range of processes. We analyze the conditions under which the t-J model can be set up and we find that it can only be defined for sufficiently large interaction. There, the parameters of the effective model are determined.Comment: 16 pages, 13 figures included. v2: Order of sections changed. Calculation and discussion of apparent gap in Section IV.A correcte

Constructive control of quantum systems using factorization of unitary operators

We demonstrate how structured decompositions of unitary operators can be employed to derive control schemes for finite-level quantum systems that require only sequences of simple control pulses such as square wave pulses with finite rise and decay times or Gaussian wavepackets. To illustrate the technique it is applied to find control schemes to achieve population transfers for pure-state systems, complete inversions of the ensemble populations for mixed-state systems, create arbitrary superposition states and optimize the ensemble average of dynamic observables.Comment: 28 pages, IoP LaTeX, principal author has moved to Cambridge University ([email protected]

Electrowinning of Nickel from ammonical sulphate bath and effect of acetone on morphology of nickel deposit and its correlation with kinetic parameters

The electrodeposition of nickel from nickel sulphate bath was studied in ammonia medium. The electrolytic conditions for nickel deposition was optimized at room temperature. The effect of acetone on current efficiency, morphology, stability and particle size of deposited nickel powder was studied. The effect of organic additive Tribenzyl ammonium chloride (TBAC) on the morphology of nickel powder was also studied. The kinetics of electrodeposition was studied and the results were utilized in developing mathematical model. During electrodeposition the current efficiency was found to increase with increase in acetone concentration up to 15% V/V in bath solution. On further increase of acetone concentration in bath solution current efficiency decreases. The stability of the electrowon deposited nickel powder was found to be in the range of 85 to 89 %. Powder morphology was found to be dentritic, porous and irregular. The morphology was also found to be underdeveloped dentritic to rounded aggregate as the concentration of organic additive TBAC increases. The average particle size of the deposited powder was found to be decreasing as the concentration of the acetone increases. The average size of the particle is in the range of 13-16 m

On the Exponentials of Some Structured Matrices

In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra isomorphism and basic Lie theory. When used in conjunction with structure preserving similarities, such as Givens rotations, these techniques extend to dimensions bigger than four.Comment: 19 page

Adapted continuous unitary transformation to treat systems with quasiparticles of finite lifetime

An improved generator for continuous unitary transformations is introduced to describe systems with unstable quasiparticles. Its general properties are derived and discussed. To illustrate this approach we investigate the asymmetric antiferromagnetic spin-1/2 Heisenberg ladder which allows for spontaneous triplon decay. We present results for the low energy spectrum and the momentum resolved spectral density of this system. In particular, we show the resonance behavior of the decaying triplon explicitly.Comment: 40 pages, 12 figure

The extended Malkus-Robbins dynamo as a perturbed Lorenz system

Recent investigations of some self-exciting Faraday-disk homopolar dynamo ([1-4]) have yielded the classic Lorenz equations as a special limit when one of the principal bifurcation parameters is zero. In this paper we focus upon one of those models [3] and illustrate what happens to some of the lowest order unstable periodic orbits as this parameter is increased from zero