278 research outputs found
Estimating long term behavior of flows without trajectory integration: the infinitesimal generator approach
The long-term distributions of trajectories of a flow are described by
invariant densities, i.e. fixed points of an associated transfer operator. In
addition, global slowly mixing structures, such as almost-invariant sets, which
partition phase space into regions that are almost dynamically disconnected,
can also be identified by certain eigenfunctions of this operator. Indeed,
these structures are often hard to obtain by brute-force trajectory-based
analyses. In a wide variety of applications, transfer operators have proven to
be very efficient tools for an analysis of the global behavior of a dynamical
system.
The computationally most expensive step in the construction of an approximate
transfer operator is the numerical integration of many short term trajectories.
In this paper, we propose to directly work with the infinitesimal generator
instead of the operator, completely avoiding trajectory integration. We propose
two different discretization schemes; a cell based discretization and a
spectral collocation approach. Convergence can be shown in certain
circumstances. We demonstrate numerically that our approach is much more
efficient than the operator approach, sometimes by several orders of magnitude
Excitation spectra for Andreev billiards of box and disk geometries
We study Andreev billiards of box and disk geometries by matching the wave functions at the interface of the normal and the superconducting region using the exact solutions of the Bogoliubov-de Gennes equation. The mismatch in the Fermi wave numbers and the effective masses of the normal system and the superconductor, as well as the tunnel barrier at the interface are taken into account. A Weyl formula (for the smooth part of the counting function of the energy levels) is derived. The exact quantum mechanical calculations show equally spaced singularities in the density of states. Based on the Bohr-Sommerfeld quantization rule a semiclassical theory is proposed to understand these singularities. For disk geometries two kinds of states can be distinguished: states either contribute through whispering gallery modes or are Andreev states strongly coupled to the superconductor. Controlled by two relevant material, parameters, three kinds of energy spectra exist in disk geometry. The first is dominated by Andreev reflections, the second, by normal reflections in an annular disk geometry. In the third case the coherence length is much larger than the radius of the superconducting region, and the spectrum is identical to that of a full disk geometry
Doped carbon nanotubes as a model system of biased graphene
Albeit difficult to access experimentally, the density of states (DOS) is a
key parameter in solid state systems which governs several important phenomena
including transport, magnetism, thermal, and thermoelectric properties. We
study DOS in an ensemble of potassium intercalated single-wall carbon nanotubes
(SWCNT) and show using electron spin resonance spectroscopy that a sizeable
number of electron states are present, which gives rise to a Fermi-liquid
behavior in this material. A comparison between theoretical and the
experimental DOS indicates that it does not display significant correlation
effects, even though the pristine nanotube material shows a Luttinger-liquid
behavior. We argue that the carbon nanotube ensemble essentially maps out the
whole Brillouin zone of graphene thus it acts as a model system of biased
graphene
Spectral degeneracy and escape dynamics for intermittent maps with a hole
We study intermittent maps from the point of view of metastability. Small
neighbourhoods of an intermittent fixed point and their complements form pairs
of almost-invariant sets. Treating the small neighbourhood as a hole, we first
show that the absolutely continuous conditional invariant measures (ACCIMs)
converge to the ACIM as the length of the small neighbourhood shrinks to zero.
We then quantify how the escape dynamics from these almost-invariant sets are
connected with the second eigenfunctions of Perron-Frobenius (transfer)
operators when a small perturbation is applied near the intermittent fixed
point. In particular, we describe precisely the scaling of the second
eigenvalue with the perturbation size, provide upper and lower bounds, and
demonstrate convergence of the positive part of the second eigenfunction
to the ACIM as the perturbation goes to zero. This perturbation and associated
eigenvalue scalings and convergence results are all compatible with Ulam's
method and provide a formal explanation for the numerical behaviour of Ulam's
method in this nonuniformly hyperbolic setting. The main results of the paper
are illustrated with numerical computations.Comment: 34 page
Exercise Increases Markers of Spermatogenesis in Rats Selectively Bred for Low Running Capacity
The oxidative stress effect of exercise training on testis function is under debate. In the present study we used a unique rat model system developed by artificial selection for low and high intrinsic running capacity (LCR and HCR, respectively) to evaluate the effects of exercise training on apoptosis and spermatogenesis in testis. Twenty-four 13-month-old male rats were assigned to four groups: control LCR (LCR-C), trained LCR (LCR-T), control HCR (HCR-C), and trained HCR (HCR-T). Ten key proteins connecting aerobic exercise capacity and general testes function were assessed, including those that are vital for mitochondrial biogenesis. The VO2 max of LCR-C group was about 30% lower than that of HCR-C rats, and the SIRT1 levels were also significantly lower than HCR-C. Twelve weeks of training significantly increased maximal oxygen consumption in LCR by nearly 40% whereas HCR remained unchanged. LCR-T had significantly higher levels of peroxisome proliferator-activated receptor-gamma coactivator-1 (PGC-1α), decreased levels of reactive oxygen species and increased acetylated p53 compared to LCR-C, while training produced no significant changes for these measures in HCR rats. BAX and Blc-2 were not different among all four groups. The levels of outer dense fibers -1 (Odf-1), a marker of spermatogenesis, increased in LCR-T rats, but decreased in HCR-TR rats. Moreover, exercise training increased the levels of lactate dehydrogenase C (LDHC) only in LCR rats. These data suggest that rats with low inborn exercise capacity can increase whole body oxygen consumption and running exercise capacity with endurance training and, in turn, increase spermatogenesis function via reduction in ROS and heightened activity of p53 in testes
Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces
Reproducing kernel Hilbert spaces (RKHSs) play an important role in many
statistics and machine learning applications ranging from support vector
machines to Gaussian processes and kernel embeddings of distributions.
Operators acting on such spaces are, for instance, required to embed
conditional probability distributions in order to implement the kernel Bayes
rule and build sequential data models. It was recently shown that transfer
operators such as the Perron-Frobenius or Koopman operator can also be
approximated in a similar fashion using covariance and cross-covariance
operators and that eigenfunctions of these operators can be obtained by solving
associated matrix eigenvalue problems. The goal of this paper is to provide a
solid functional analytic foundation for the eigenvalue decomposition of RKHS
operators and to extend the approach to the singular value decomposition. The
results are illustrated with simple guiding examples
Negative length orbits in normal-superconductor billiard systems
The Path-Length Spectra of mesoscopic systems including diffractive
scatterers and connected to superconductor is studied theoretically. We show
that the spectra differs fundamentally from that of normal systems due to the
presence of Andreev reflection. It is shown that negative path-lengths should
arise in the spectra as opposed to normal system. To highlight this effect we
carried out both quantum mechanical and semiclassical calculations for the
simplest possible diffractive scatterer. The most pronounced peaks in the
Path-Length Spectra of the reflection amplitude are identified by the routes
that the electron and/or hole travels.Comment: 4 pages, 4 figures include
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