Theoretical Measuring for Negative Chromatic Dispersion Curves of Photonic Crystal Fiber by Gaussian Function

Negative dispersion curves in a typical type of high negative chromatic dispersion photonic crystal fiber(PCF) have been investigated in this paper. The depended class of (PCF) has double-core structure (core- region: which has inner core and outer core) with a honeycomb photonic lattice in the cladding region. Negative dispersion curves deviated from core-region of this type of fibers will be investigated. The investigation has depended an estimation process using an approximation function to create a mathematical model that enables us to measure negative dispersion curves. The influence of inner-core parameters (dcore d1 and d2) on dispersion curves has been investigated by varying the values of these parameters.  Negative dispersion curves that were introduced by a previous study using finite-difference frequency-domain (FDFD)method for this class of(PCFs) are directly included in this work in order  to measure matching ratio with our results.   Gaussian approximation function has been considered to estimate our mathematical model. Keywords: Photonic crystal fiber, Theoretical model, Negative chromatic dispersion, Gaussian function

Reconstruction of piecewise-smooth multivariate functions from Fourier data

In some applications, one is interested in reconstructing a function $f$ from its Fourier series coefficients. The problem is that the Fourier series is slowly convergent if the function is non-periodic, or is non-smooth. In this paper, we suggest a method for deriving high order approximation to $f$ using a Pad\'e-like method. Namely, by fitting some Fourier coefficients of the approximant to the given Fourier coefficients of $f$. Given the Fourier series coefficients of a function on a rectangular domain in $\mathbb{R}^d$, assuming the function is piecewise smooth, we approximate the function by piecewise high order spline functions. First, the singularity structure of the function is identified. For example in the 2-D case, we find high accuracy approximation to the curves separating between smooth segments of $f$. Secondly, simultaneously we find the approximations of all the different segments of $f$. We start by developing and demonstrating a high accuracy algorithm for the 1-D case, and we use this algorithm to step up to the multidimensional case.Comment: 22 pages, 21 figure

Origin of the Verwey transition in magnetite: Group theory, electronic structure, and lattice dynamics study

The Verwey phase transition in magnetite has been analyzed using the group theory methods. It is found that two order parameters with the symmetries $X_3$ and $\Delta_5$ induce the structural transformation from the high-temperature cubic to the low-temperature monoclinic phase. The coupling between the order parameters is described by the Landau free energy functional. The electronic and crystal structure for the cubic and monoclinic phases were optimized using the {\it ab initio} density functional method. The electronic structure calculations were performed within the generalized gradient approximation including the on-site interactions between 3d electrons at iron ions -- the Coulomb element $U$ and Hund's exchange $J$. Only when these local interactions are taken into account, the phonon dispersion curves, obtained by the direct method for the cubic phase, reproduce the experimental data. It is shown that the interplay of local electron interations and the coupling to the lattice drives the phonon order parameters and is responsible for the opening of the gap at the Fermi energy. Thus, it is found that the metal-insulator transition in magnetite is promoted by local electron interactions, which significantly amplify the electron-phonon interaction and stabilize weak charge order coexisting with orbital order of the occupied $t_{2g}$ states at Fe ions. This provides a scenario to understand the fundamental problem of the origin of the Verwey transition in magnetite.Comment: 17 pages, 5 figures, 8 tables. Accepted version to be published in Phys. Rev.

Improvements to the TCVD method to segment hand-drawn sketches

Tangent and Corner Vertices Detection (TCVD) is a method to detect corner vertices and tangent points in sketches using parametric cubic curves approximation, which is capable to detect corners with a high accuracy and a very low false positive rate, and also to detect tangent points far above other methods in literature. In this article, we present several improvements to TCVD method in order to establish mathematical conditions to detect corners and make the obtaining of curves independent from the scale, what increases the success ratio in transitions between lines and curves. The new conditions for obtaining corners use the radius as the inverse of the curvature, and the second derivative of the curvature. For the detection of curves, a new descriptor is presented, avoiding the parameters dependent of scale used in TCVD method. In order to obtain the performance of the implemented improvements, several tests have been carried out using a dataset which contains sketches more complex than those used for validation of TCVD algorithm (sketches with more curves and tangent points and sketches of different sizes). For corners detection, the accuracy obtained was pretty similar to that obtained with the previous TCVD, however, for curves and tangent points detection the accuracy increases significantly.Spanish Ministry of Science and Education and the FEDER Funds, through HYMAS project (Ref. DPI2010-19457) and INIA project VIS-DACSA (Ref. RTA2012-00062-C04-03) partially supported this work.Albert Gil, FE.; Aleixos Borrás, MN. (2017). Improvements to the TCVD method to segment hand-drawn sketches. Pattern Recognition. 63:416-426. https://doi.org/10.1016/j.patcog.2016.10.024S4164266

Trajectory definition with high relative accuracy (HRA) by parametric representation of curves in nano-positioning systems

Nanotechnology applications demand high accuracy positioning systems. Therefore, in order to achieve sub-micrometer accuracy, positioning uncertainty contributions must be minimized by implementing precision positioning control strategies. The positioning control system accuracy must be analyzed and optimized, especially when the system is required to follow a predefined trajectory. In this line of research, this work studies the contribution of the trajectory definition errors to the final positioning uncertainty of a large-range 2D nanopositioning stage. The curve trajectory is defined by curve fitting using two methods: traditional CAD/CAM systems and novel algorithms for accurate curve fitting. This novel method has an interest in computer-aided geometric design and approximation theory, and allows high relative accuracy (HRA) in the computation of the representations of parametric curves while minimizing the numerical errors. It is verified that the HRA method offers better positioning accuracy than commonly used CAD/CAM methods when defining a trajectory by curve fitting: When fitting a curve by interpolation with the HRA method, fewer data points are required to achieve the precision requirements. Similarly, when fitting a curve by a least-squares approximation, for the same set of given data points, the HRA method is capable of obtaining an accurate approximation curve with fewer control points

A high-order study of the quantum critical behavior of a frustrated spin-12\frac{1}{2} antiferromagnet on a stacked honeycomb bilayer

We study a frustrated spin-$\frac{1}{2}$ $J_{1}$--$J_{2}$--$J_{3}$--$J_{1}^{\perp}$ Heisenberg antiferromagnet on an $AA$-stacked bilayer honeycomb lattice. In each layer we consider nearest-neighbor (NN), next-nearest-neighbor, and next-next-nearest-neighbor antiferromagnetic (AFM) exchange couplings $J_{1}$, $J_{2}$, and $J_{3}$, respectively. The two layers are coupled with an AFM NN exchange coupling $J_{1}^{\perp}\equiv\delta J_{1}$. The model is studied for arbitrary values of $\delta$ along the line $J_{3}=J_{2}\equiv\alpha J_{1}$ that includes the most highly frustrated point at $\alpha=\frac{1}{2}$, where the classical ground state is macroscopically degenerate. The coupled cluster method is used at high orders of approximation to calculate the magnetic order parameter and the triplet spin gap. We are thereby able to give an accurate description of the quantum phase diagram of the model in the $\alpha\delta$ plane in the window $0 \leq \alpha \leq 1$, $0 \leq \delta \leq 1$. This includes two AFM phases with N\'eel and striped order, and an intermediate gapped paramagnetic phase that exhibits various forms of valence-bond crystalline order. We obtain accurate estimations of the two phase boundaries, $\delta = \delta_{c_{i}}(\alpha)$, or equivalently, $\alpha = \alpha_{c_{i}}(\delta)$, with $i=1$ (N\'eel) and 2 (striped). The two boundaries exhibit an "avoided crossing" behavior with both curves being reentrant

Heterogeneity effects on flow and transport within a shallow fluvial aquifer

The effects of aquifer heterogeneity on flow and transport are considered numerically at two scales using high resolution groundwater models. Heterogeneity effects on river loss were evaluated at the kilometer scale using stochastic, geostatistical models with grid cells on the order of several meters. It was found that river loss decreased directly with an increase in the extent of heterogeneity and that homogeneous approximations resulted in increased loss estimates. Heterogeneity effects on transport were simulated at the scale of several meters using a homogeneous approximation, traditional geostatistical models and a new integrated method of aquifer characterization. The integrated method combines geophysics and geostatistics to create a more realistic approximation of subsurface features. Using grid cells of several centimeters, transport was simulated for multiple heterogeneity realizations in three directions through the models to evaluate potential anisotropy of the transport rates. The resulting breakthrough curves for the homogeneous and traditional geostatistical models showed no directional anisotropy but the integrated models showed anisotropic behavior consistent with the bedding direction as well as non-Fickian transport rates