On the Leibniz rule and Laplace transform for fractional derivatives

Taylor series is a useful mathematical tool when describing and constructing a function. With the series representation, some properties of fractional calculus can be revealed clearly. This paper investigates two typical applications: Lebiniz rule and Laplace transform. It is analytically shown that the commonly used Leibniz rule cannot be applied for Caputo derivative. Similarly, the well-known Laplace transform of Riemann-Liouville derivative is doubtful for n-th continuously differentiable function. By the aid of this series representation, the exact formula of Caputo Leibniz rule and the explanation of Riemann-Liouville Laplace transform are presented. Finally, three illustrative examples are revisited to confirm the obtained results

Spin-Dependent Hubbard Model and a Quantum Phase Transition in Cold Atoms

We describe an experimental protocol for introducing spin-dependent lattice structure in a cold atomic fermi gas using lasers. It can be used to realize Hubbard models whose hopping parameters depend on spin and whose interaction strength can be controlled with an external magnetic field. We suggest that exotic superfluidities will arise in this framework. An especially interesting possibility is a class of states that support coexisting superfluid and normal components, even at zero temperature. The quantity of normal component varies with external parameters. We discuss some aspects of the quantum phase transition that arises at the point where it vanishes.Comment: 9 pages, 7 figures; added/corrected references in [11] and [44

GeneFishing to reconstruct context specific portraits of biological processes.

Rapid advances in genomic technologies have led to a wealth of diverse data, from which novel discoveries can be gleaned through the application of robust statistical and computational methods. Here, we describe GeneFishing, a semisupervised computational approach to reconstruct context-specific portraits of biological processes by leveraging gene-gene coexpression information. GeneFishing incorporates multiple high-dimensional statistical ideas, including dimensionality reduction, clustering, subsampling, and results aggregation, to produce robust results. To illustrate the power of our method, we applied it using 21 genes involved in cholesterol metabolism as "bait" to "fish out" (or identify) genes not previously identified as being connected to cholesterol metabolism. Using simulation and real datasets, we found that the results obtained through GeneFishing were more interesting for our study than those provided by related gene prioritization methods. In particular, application of GeneFishing to the GTEx liver RNA sequencing (RNAseq) data not only reidentified many known cholesterol-related genes, but also pointed to glyoxalase I (GLO1) as a gene implicated in cholesterol metabolism. In a follow-up experiment, we found that GLO1 knockdown in human hepatoma cell lines increased levels of cellular cholesterol ester, validating a role for GLO1 in cholesterol metabolism. In addition, we performed pantissue analysis by applying GeneFishing on various tissues and identified many potential tissue-specific cholesterol metabolism-related genes. GeneFishing appears to be a powerful tool for identifying related components of complex biological systems and may be used across a wide range of applications

On the Causality and Stability of the Relativistic Diffusion Equation

This paper examines the mathematical properties of the relativistic diffusion equation. The peculiar solution which Hiscock and Lindblom identified as an instability is shown to emerge from an ill-posed initial value problem. These do not meet the mathematical conditions required for realistic physical problems and can not serve as an argument against the relativistic hydrodynamics of Landau and Lifshitz.Comment: 6 page

In-depth investigation into the transient humidity response at the body-seat interface on initial contact using a dual temperature and humidity sensor

Relative humidity (RH) at the body-seat interface is considered an important factor in both sitting comfort and generation of health concerns such as skin lesions. Technical difficulties appear to have limited research aimed at the detailed and simultaneous exploration of RH and temperature changes at the body-seat interface; using RH sensors without the capability to record temperature where RH is recorded. To explore the causes of a spike in RH consistently produced on first contact between body and seat surface, we report data from the first use of dual temperature and RH (HTU21D) sensors in this interface. Following evaluation of sensor performance, the effect of local thermal changes on RH was investigated. The expected strong negative correlation between temperature and RH (R 2 = =0.94) supported the importance of considering both parameters when studying impact of sitting on skin health. The influence of sensor movement speed (higher velocity approach: 0.32 cm/s ± 0.01 cm/s; lower velocity approach: 0.17 cm/s ± 0.01 cm/s) into a static RH region associated with a higher local temperature were compared with data gathered by altering the rate of a person sitting. In all cases, the faster sitting down (or equivalent) generated larger RH outcomes: e.g., in human sitting 53.7% ± 3.3% RH (left mid-thigh), 56.4% ± 5.1% RH (right mid-thigh) and 53.2% ± 2.7% RH (Coccyx). Differences in size of RH change were seen across the measurement locations used to study the body-seat interface. The initial sitting contact induces a transient RH response (duration ≤ 40 s) that does not accurately reflect the microenvironment at the body-seat interface. It is likely that any movement during sitting would result in similar artefact formation. As a result, caution should be taken when investigating RH performance at any enclosed interface when the surfaces may have different temperatures and movement may occur

Uncertainty of the South China Sea prediction using NSCAT and NCEP winds during tropical storm Ernie 1996

Journal of Geophysical Research, American Geophysical Unio