13,118 research outputs found
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 on the residual branch where is the network
depth. However, it is not clear how learning ResNet behaves for other values of
. In this paper, we fully characterize the convergence theory of gradient
descent for learning over-parameterized ResNet with different values of .
Specifically, with hiding logarithmic factor and constant coefficients, we show
that for gradient descent is guaranteed to converge to the
global minma, and especially when the convergence is irrelevant
of the network depth. Conversely, we show that for ,
the forward output grows at least with rate in expectation and then the
learning fails because of gradient explosion for large . This means the
bound 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 .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 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 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
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