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

Simultaneous Measurement for Strain and Temperature Using Fiber Bragg Gratings and Multimode Fibers

An all-fiber sensor capable of simultaneous measurement of temperature and strain is newly presented. The sensing head is formed by a fiber Bragg grating combined with a section of multimode fiber that acts as a Mach-Zehnder interferometer for temperature and strain discrimination. The strain and temperature coefficients of multimode fibers vary with the core sizes and materials. This feature can be used to improve the strain and temperature resolution by suitably choosing the multimode fiber. For a 10 pm wavelength resolution, a resolution of 9.21 μ∈ in strain and 0.26°C in temperature can be achieved

Convergence Theory of Learning Over-parameterized ResNet: A Full Characterization

ResNet structure has achieved great empirical success since its debut. Recent work established the convergence of learning over-parameterized ResNet with a scaling factor $\tau=1/L$ on the residual branch where $L$ is the network depth. However, it is not clear how learning ResNet behaves for other values of $\tau$. In this paper, we fully characterize the convergence theory of gradient descent for learning over-parameterized ResNet with different values of $\tau$. Specifically, with hiding logarithmic factor and constant coefficients, we show that for $\tau\le 1/\sqrt{L}$ gradient descent is guaranteed to converge to the global minma, and especially when $\tau\le 1/L$ the convergence is irrelevant of the network depth. Conversely, we show that for $\tau>L^{-\frac{1}{2}+c}$, the forward output grows at least with rate $L^c$ in expectation and then the learning fails because of gradient explosion for large $L$. This means the bound $\tau\le 1/\sqrt{L}$ is sharp for learning ResNet with arbitrary depth. To the best of our knowledge, this is the first work that studies learning ResNet with full range of $\tau$.Comment: 31 page

Superradiance Lattice

We show that the timed Dicke states of a collection of three-level atoms can form a tight-binding lattice in momentum space. This lattice, coined the superradiance lattice (SL), can be constructed based on electromagnetically induced transparency (EIT). For a one-dimensional SL, we need the coupling field of the EIT system to be a standing wave. The detuning between the two components of the standing wave introduces an effective uniform force in momentum space. The quantum lattice dynamics, such as Bloch oscillations, Wannier-Stark ladders, Bloch band collapsing and dynamic localization can be observed in the SL. The two-dimensional SL provides a flexible platform for Dirac physics in graphene. The SL can be extended to three and higher dimensions where no analogous real space lattices exist with new physics waiting to be explored.Comment: 6pages, 4 figure

Data-driven Radiative Hydrodynamic Modeling of the 2014 March 29 X1.0 Solar Flare

Spectroscopic observations of solar flares provide critical diagnostics of the physical conditions in the flaring atmosphere. Some key features in observed spectra have not yet been accounted for in existing flare models. Here we report a data-driven simulation of the well-observed X1.0 flare on 2014 March 29 that can reconcile some well-known spectral discrepancies. We analyzed spectra of the flaring region from the Interface Region Imaging Spectrograph (IRIS) in MgII h&k, the Interferometric BIdimensional Spectropolarimeter at the Dunn Solar Telescope (DST/IBIS) in H$\alpha$ 6563 \AA\ and CaII 8542 \AA, and the Reuven Ramaty High Energy Solar Spectroscope Imager (RHESSI) in hard X-rays. We constructed a multi-threaded flare loop model and used the electron flux inferred from RHESSI data as the input to the radiative hydrodynamic code RADYN to simulate the atmospheric response. We then synthesized various chromospheric emission lines and compared them with the IRIS and IBIS observations. In general, the synthetic intensities agree with the observed ones, especially near the northern footpoint of the flare. The simulated MgII line profile has narrower wings than the observed one. This discrepancy can be reduced by using a higher microturbulent velocity (27 km/s) in a narrow chromospheric layer. In addition, we found that an increase of electron density in the upper chromosphere within a narrow height range of $\approx$800 km below the transition region can turn the simulated MgII line core into emission and thus reproduce the single peaked profile, which is a common feature in all IRIS flares.Comment: 14 pages, 18 figures, accepted in Ap

Core-Offset Small-Core-Diameter Dispersion Compensation Fiber Interferometer and its Applications in Fiber Sensors

We propose a core-offset small core diameter dispersion compensation fiber (DCF) interferometer and investigate its applications in fiber sensors. If the transverse force is applied to a short section of the DCF, there is almost no crosstalk on the transmission spectrum between the extinction ratio variation induced by the transverse force and the wavelength shift caused by the longitudinal strain or ambient temperature, which can be applied to measure both transverse and longitudinal strain, or both transverse strain and temperature, simultaneously. The proposed sensors have the advantages of low cost, simple and compact structure, and good reproducibility

Temperature- and Phase-Independent Lateral Force Sensor based on a Core-Offset Multi-Mode Fiber Interferometer

A novel lateral force sensor based on a core-offset multi-mode fiber (MMF) interferometer is reported. High extinction ratio can be obtained by misaligning a fused cross section between the single-mode fiber (SMF) and MMF. With the variation of the lateral force applied to a short section of the MMF, the extinction ratio changes while the interference phase remains almost constant. The change of the extinction ratio is independent of temperature variations. The proposed force sensor has the advantages of temperature- and phase-independency, high extinction ratio sensitivity, good repeatability, low cost, and simple structure. Moreover, the core-offset MMF interferometer is expected to have applications in fiber filters and tunable phase-independent attenuators