Geodesic continued fractions and LLL

We discuss a proposal for a continued fraction-like algorithm to determine simultaneous rational approximations to $d$ real numbers $\alpha_1,\ldots,\alpha_d$. It combines an algorithm of Hermite and Lagarias with ideas from LLL-reduction. We dynamically LLL-reduce a quadratic form with parameter $t$ as $t\downarrow0$. The new idea in this paper is that checking the LLL-conditions consists of solving linear equations in $t$

Landau Level Mixing and Solenoidal Terms in Lowest Landau Level Currents

We calculate the lowest Landau level (LLL) current by working in the full Hilbert space of a two dimensional electron system in a magnetic field and keeping all the non-vanishing terms in the high field limit. The answer a) is not represented by a simple LLL operator and b) differs from the current operator, recently derived by Martinez and Stone in a field theoretic LLL formalism, by solenoidal terms. Though that is consistent with the inevitable ambiguities of their Noether construction, we argue that the correct answer cannot arise naturally in the LLL formalism.Comment: 12 pages + 2 figures, Revtex 3.0, UIUC preprint P-94-04-029, (to appear in Mod. Phys. Lett. B

Why the lowest Landau level approximation works in strongly type II superconductors

Higher than the lowest Landau level contributions to magnetization and specific heat of superconductors are calculated using Ginzburg - Landau equations approach. Corrections to the excitation spectrum around solution of these equations (treated perturbatively) are found. Due to symmetries of the problem leading to numerous cancellations the range of validity of the LLL approximation in mean field is much wider then a naive range and extends all the way down to $H = {H_{c2}(T)}/13$. Moreover the contribution of higher Landau levels is significantly smaller compared to LLL than expected naively. We show that like the LLL part the lattice excitation spectrum at small quasimomenta is softer than that of usual acoustic phonons. This enhanses the effect of fluctuations. The mean field calculation extends to third order, while the fluctuation contribution due to HLL is to one loop. This complements the earlier calculation of the LLL part to two loop order.Comment: 20 pages, Latex file, three figure

Localized matter-waves patterns with attractive interaction in rotating potentials

We consider a two-dimensional (2D) model of a rotating attractive Bose-Einstein condensate (BEC), trapped in an external potential. First, an harmonic potential with the critical strength is considered, which generates quasi-solitons at the lowest Landau level (LLL). We describe a family of the LLL quasi-solitons using both numerical method and a variational approximation (VA), which are in good agreement with each other. We demonstrate that kicking the LLL mode or applying a ramp potential sets it in the Larmor (cyclotron) motion, that can also be accurately modeled by the VA.Comment: 13 pages, 11 figure

Infinite computable version of Lovasz Local Lemma

Lov\'asz Local Lemma (LLL) is a probabilistic tool that allows us to prove the existence of combinatorial objects in the cases when standard probabilistic argument does not work (there are many partly independent conditions). LLL can be also used to prove the consistency of an infinite set of conditions, using standard compactness argument (if an infinite set of conditions is inconsistent, then some finite part of it is inconsistent, too, which contradicts LLL). In this way we show that objects satisfying all the conditions do exist (though the probability of this event equals~$0$). However, if we are interested in finding a computable solution that satisfies all the constraints, compactness arguments do not work anymore. Moser and Tardos recently gave a nice constructive proof of LLL. Lance Fortnow asked whether one can apply Moser--Tardos technique to prove the existence of a computable solution. We show that this is indeed possible (under almost the same conditions as used in the non-constructive version)

Vortex distribution in the Lowest Landau Level

We study the vortex distribution of the wave functions minimizing the Gross Pitaevskii energy for a fast rotating condensate in the Lowest Landau Level (LLL): we prove that the minimizer cannot have a finite number of zeroes thus the lattice is infinite, but not uniform. This uses the explicit expression of the projector onto the LLL. We also show that any slow varying envelope function can be approximated in the LLL by distorting the lattice. This is used in particular to approximate the inverted parabola and understand the role of ``invisible'' vortices: the distortion of the lattice is very small in the Thomas Fermi region but quite large outside, where the "invisible" vortices lie.Comment: 4 pages, 1 figur

The Lefthanded Local Lemma characterizes chordal dependency graphs

Shearer gave a general theorem characterizing the family \LLL of dependency graphs labeled with probabilities $p_v$ which have the property that for any family of events with a dependency graph from \LLL (whose vertex-labels are upper bounds on the probabilities of the events), there is a positive probability that none of the events from the family occur. We show that, unlike the standard Lov\'asz Local Lemma---which is less powerful than Shearer's condition on every nonempty graph---a recently proved `Lefthanded' version of the Local Lemma is equivalent to Shearer's condition for all chordal graphs. This also leads to a simple and efficient algorithm to check whether a given labeled chordal graph is in \LLL.Comment: 12 pages, 1 figur