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

Test of quantum nonlocality via vector meson decays to KSKSK_SK_S

In the system of a pair of quantum-entangled neutral kaons from meson decays, when one kaon collapses into the $K_S$ state, the other will collapse instantaneously into the $K_L$ state due to entanglement and nonlocality. However, if the alternative hypothesis is correct and there's a time window during which one kaon is unaware that the other has decayed, some quantum mechanically prohibited $K_SK_S$ decays may occur. We calculate the branching ratios of $K_SK_S$ in vector meson decays under the locality hypothesis and compare them with experimental results. We show that the branching ratio of $J/\psi\rightarrow K_SK_S$ under locality assumption is already excluded by the BESIII experimental upper limit. Additional experimental results are proposed to perform this test in the future.Comment: 7 pages, 3 figures, 1 tabl

Application of stratified emulsion splitting technology in highly hard nuclear cataract surgery

AIM:To explore safety and effectiveness of stratified emulsion splitting technology in highly hard nuclear cataract ultrasonic emulsification. METHODS: Totally 43 cases(47 eyes)of highly hard nuclear cataract was performed cataract extraction combined IOL implantation with stratified emulsion splitting cataract ultrasonic emulsification technology. RESULTS:Postoperative visual acuity 1 day, 1 week and 1 month best-corrected visual acuity ≥0.5 was accounted for 70.21%, 87.23% and 89.36%. Intraoperative capsule rupture was found in 2 eyes. Corneal mild edema was found in 7 eyes(14.89%)after the first l day post-operation. CONCLUSION:Stratified emulsion splitting technique has a practical application with little tissue damage, low capsular rupture rate and few complications for highly hard nuclear cataract patients and can bring a desirable outcome

Null and time-like geodesics in Kerr-Newman black hole exterior

We study the null and time-like geodesics of the light and the neutral particles respectively in the exterior of Kerr-Newman black holes. The geodesic equations are known to be written as a set of first-order differential equations in Mino time from which the angular and radial potentials can be defined. We classify the roots for both potentials, and mainly focus on those of the radial potential with an emphasis on the effect from the charge of the black holes. We then obtain the solutions of the trajectories in terms of the elliptical integrals and the Jacobian elliptic functions for both null and time-like geodesics, which are manifestly real functions of the Mino time that the initial conditions can be explicitly specified. We also describe the details of how to reduce those solutions into the cases of the spherical orbits. The effect of the black hole's charge decreases the radii of the spherical motion of the light and the particle for both direct and retrograde motions. In particular, we focus on the light/particle boomerang of the spherical orbits due to the frame dragging from the back hole's spin with the effect from the charge of the black hole. To sustain the change of the azimuthal angle of the light rays, say for example $\Delta \phi=\pi$ during the whole trip, the presence of the black hole's charge decreases the radius of the orbit and consequently reduces the needed values of the black hole's spin. As for the particle boomerang, the particle's inertia renders smaller change of the angle $\Delta \phi$ as compared with the light boomerang. Moreover, the black hole's charge also results in the smaller angle change $\Delta \phi$ of the particle than that in the Kerr case. The implications of the obtained results to observations are discussed.Comment: 50 pages, 18 figure

Metagenomic analysis of microbial consortia enriched from compost: new insights into the role of Actinobacteria in lignocellulose decomposition

Additional file 11: Table S7. Summary of de novo assembly results (37 k)

Quantitative imaging of bilirubin by photoacoustic microscopy

Noninvasive detection of both bilirubin concentration and its distribution is important for disease diagnosis. Here we implemented photoacoustic microscopy (PAM) to detect bilirubin distribution. We first demonstrate that our PAM system can measure the absorption spectra of bilirubin and blood. We also image bilirubin distributions in tissue-mimicking samples, both without and with blood mixed. Our results show that PAM has the potential to quantitatively image bilirubin in vivo for clinical applications

Poly[[[silver(I)-μ-1,4-bis­[(imidazol-1-yl)meth­yl]benzene-κ2 N 3:N 3′-silver(I)-μ-1,4-bis­[(imidazol-1-yl)meth­yl]benzene-κ2 N 3:N 3′] 4,4′-diazenediyldibenzoate] dihydrate]

In the title compound, [Ag2(C14H14N4)2](C14H8N2O4)·2H2O, each of the two unique Ag+ ions is two-coordinated by two N atoms from two different 1,4-bis­[(imidazol-1-yl)meth­yl]benzene ligands in an almost linear fashion [N—Ag—N = 170.34 (10) and 160.25 (10)°]. The 4,4′-diazenediyldibenzoate anions do not coordinate to Ag. O—H⋯O hydrogen bonds stabilize the crystal structure

Role of cholinergic anti-inflammatory pathway in regulating host response and its interventional strategy for inflammatory diseases

The cholinergic anti-inflammatory pathway (CAP) is a neurophysiological mechanism that regulates the immune system. The CAP inhibits inflammation by suppressing cytokine synthesis via release of acetylcholine in organs of the reticuloendothelial system, including the lungs, spleen, liver, kidneys and gastrointestinal tract. Acetylcholine can interact with α 7 nicotinic acetylcholine receptors (α 7 nAchR) expressed by macrophages and other cytokine producing cells, down-regulate pro-inflammatory cytokine synthesis and prevent tissue damage. Herein is a review of the neurophysiological mechanism in which the CAP regulates inflammatory response, as well as its potential interventional strategy for inflammatory diseases