Hausdorff dimension of repellors in low sensitive systems

Methods to estimate the Hausdorff dimension of invariant sets of scattering systems are presented. Based on the levels' hierarchical structure of the time delay function, these techniques can be used in systems whose future-invariant-set codimensions are approximately equal to or greater than one. The discussion is illustrated by a numerical example of a scatterer built with four hard spheres located at the vertices of a regular tetrahedron.Comment: 9 pages, 5 figures, accepted in Physics Letters

Mixmaster chaos

The significant discussion about the possible chaotic behavior of the mixmaster cosmological model due to Cornish and Levin [J.N. Cornish and J.J. Levin, Phys. Rev. Lett. 78 (1997) 998; Phys. Rev. D 55 (1997) 7489] is revisited. We improve their method by correcting nontrivial oversights that make their work inconclusive to precisely confirm their result: ``The mixmaster universe is indeed chaotic''.Comment: 9 pages, 4 figure

Surface Effects in Magnetic Microtraps

We have investigated Bose-Einstein condensates and ultra cold atoms in the vicinity of a surface of a magnetic microtrap. The atoms are prepared along copper conductors at distances to the surface between 300 um and 20 um. In this range, the lifetime decreases from 20 s to 0.7 s showing a linear dependence on the distance to the surface. The atoms manifest a weak thermal coupling to the surface, with measured heating rates remaining below 500 nK/s. In addition, we observe a periodic fragmentation of the condensate and thermal clouds when the surface is approached.Comment: 4 pages, 4 figures; v2: corrected references; v3: final versio

Fractional Langevin equation

We investigate fractional Brownian motion with a microscopic random-matrix model and introduce a fractional Langevin equation. We use the latter to study both sub- and superdiffusion of a free particle coupled to a fractal heat bath. We further compare fractional Brownian motion with the fractal time process. The respective mean-square displacements of these two forms of anomalous diffusion exhibit the same power-law behavior. Here we show that their lowest moments are actually all identical, except the second moment of the velocity. This provides a simple criterion which enables to distinguish these two non-Markovian processes.Comment: 4 page

Treatment of backscattering in a gas of interacting fermions confined to a one-dimensional harmonic atom trap

An asymptotically exact many body theory for spin polarized interacting fermions in a one-dimensional harmonic atom trap is developed using the bosonization method and including backward scattering. In contrast to the Luttinger model, backscattering in the trap generates one-particle potentials which must be diagonalized simultaneously with the two-body interactions. Inclusion of backscattering becomes necessary because backscattering is the dominant interaction process between confined identical one-dimensional fermions. The bosonization method is applied to the calculation of one-particle matrix elements at zero temperature. A detailed discussion of the validity of the results from bosonization is given, including a comparison with direct numerical diagonalization in fermionic Hilbert space. A model for the interaction coefficients is developed along the lines of the Luttinger model with only one coupling constant $K$. With these results, particle densities, the Wigner function, and the central pair correlation function are calculated and displayed for large fermion numbers. It is shown how interactions modify these quantities. The anomalous dimension of the pair correlation function in the center of the trap is also discussed and found to be in accord with the Luttinger model.Comment: 19 pages, 5 figures, journal-ref adde

Highly anisotropic Bose-Einstein condensates: crossover to lower dimensionality

We develop a simple analytical model based on a variational method to explain the properties of trapped cylindrically symmetric Bose-Einstein condensates (BEC) of varying degrees of anisotropy well into regimes of effective one dimension (1D) and effective two dimension (2D). Our results are accurate in regimes where the Thomas-Fermi approximation breaks down and they are shown to be in agreement with recent experimental data.Comment: 4 pages, 2 figures; significantly more new material added; title and author-list changed due to changes in conten

Spin-triplet superconducting pairing due to local (Hund's rule, Dirac) exchange

We discuss general implications of the local spin-triplet pairing among fermions induced by local ferromagnetic exchange, example of which is the Hund's rule coupling. The quasiparticle energy and their wave function are determined for the three principal phases with the gap, which is momentum independent. We utilize the Bogolyubov-Nambu-De Gennes approach, which in the case of triplet pairing in the two-band case leads to the four-components wave function. Both gapless modes and those with an isotropic gap appear in the quasiparticle spectrum. A striking analogy with the Dirac equation is briefly explored. This type of pairing is relevant to relativistic fermions as well, since it reflects the fundamental discrete symmetry-particle interchange. A comparison with the local interband spin-singlet pairing is also made.Comment: 16 pages, LaTex, submitted to Phys. Rev.

Chaos in a double driven dissipative nonlinear oscillator

We propose an anharmonic oscillator driven by two periodic forces of different frequencies as a new time-dependent model for investigating quantum dissipative chaos. Our analysis is done in the frame of statistical ensemble of quantum trajectories in quantum state diffusion approach. Quantum dynamical manifestation of chaotic behavior, including the emergence of chaos, properties of strange attractors, and quantum entanglement are studied by numerical simulation of ensemble averaged Wigner function and von Neumann entropy.Comment: 9 pages, 18 figure

EndoTime: non-categorical timing estimates for luteal endometrium

STUDY QUESTION Can the accuracy of timing of luteal phase endometrial biopsies based on urinary ovulation testing be improved by measuring the expression of a small number of genes and a continuous, non-categorical modelling approach? SUMMARY ANSWER Measuring the expression levels of six genes (IL2RB, IGFBP1, CXCL14, DPP4, GPX3 and SLC15A2) is sufficient to obtain substantially more accurate timing estimates and to assess the reliability of timing estimates for each sample. WHAT IS KNOWN ALREADY Commercially available endometrial timing approaches based on gene expression require large gene sets and use a categorical approach that classifies samples as pre-receptive, receptive or post-receptive. STUDY DESIGN, SIZE, DURATION Gene expression was measured by RTq-PCR in different sample sets, comprising a total of 664 endometrial biopsies obtained 4–12 days after a self-reported positive home ovulation test. A further 36 endometrial samples were profiled by RTq-PCR as well as RNA-sequencing. PARTICIPANTS/MATERIALS, SETTING, METHODS A computational procedure, named ‘EndoTime’, was established that models the temporal profile of each gene and estimates the timing of each sample. Iterating these steps, temporal profiles are gradually refined as sample timings are being updated, and confidence in timing estimates is increased. After convergence, the method reports updated timing estimates for each sample while preserving the overall distribution of time points. MAIN RESULTS AND THE ROLE OF CHANCE The Wilcoxon rank-sum test was used to confirm that ordering samples by EndoTime estimates yields sharper temporal expression profiles for held-out genes (not used when determining sample timings) than ordering the same expression values by patient-reported times (GPX3: P  0.05). LARGE SCALE DATA The RTq-PCR data files are available via the GitHub repository for the EndoTime software at https://github.com/AE-Mitchell/EndoTime, as is the code used for pre-processing of RTq-PCR data. The RNA-sequencing data are available on GEO (accession GSE180485). LIMITATIONS, REASONS FOR CAUTION Timing estimates are informed by glandular gene expression and will only represent the temporal state of other endometrial cell types if in synchrony with the epithelium. Methods that estimate the day of ovulation are still required as these data are essential inputs in our method. Our approach, in its current iteration, performs batch correction such that larger sample batches impart greater accuracy to timing estimations. In theory, our method requires endometrial samples obtained at different days in the luteal phase. In practice, however, this is not a concern as timings based on urinary ovulation testing are associated with a sufficient level of noise to ensure that a variety of time points will be sampled. WIDER IMPLICATIONS OF THE FINDINGS Our method is the first to assay the temporal state of luteal-phase endometrial samples on a continuous domain. It is freely available with fully shared data and open-source software. EndoTime enables accurate temporal profiling of any gene in luteal endometrial samples for a wide range of research applications and, potentially, clinical use

Stability and collapse of localized solutions of the controlled three-dimensional Gross-Pitaevskii equation

On the basis of recent investigations, a newly developed analytical procedure is used for constructing a wide class of localized solutions of the controlled three-dimensional (3D) Gross-Pitaevskii equation (GPE) that governs the dynamics of Bose-Einstein condensates (BECs). The controlled 3D GPE is decomposed into a two-dimensional (2D) linear Schr\"{o}dinger equation and a one-dimensional (1D) nonlinear Schr\"{o}dinger equation, constrained by a variational condition for the controlling potential. Then, the above class of localized solutions are constructed as the product of the solutions of the transverse and longitudinal equations. On the basis of these exact 3D analytical solutions, a stability analysis is carried out, focusing our attention on the physical conditions for having collapsing or non-collapsing solutions.Comment: 21 pages, 14 figure