Approximation Properties of Vectorial Exponential Functions

This contribution is dedicated to the celebration of Rémi Abgrall’s accomplishments in Applied Mathematics and Scientific Computing during the conference“Essentially Hyperbolic Problems:Unconventional Numerics,and Applications”.With respect to classical Finite Elements Methods,Trefftz methods are unconventional methods because of the way the basis functions are generated.Trefftz discontinuous Galerkin(TDG)methods have recently shown potential for numerical approximation of transport equations[6,26]with vectorial exponential modes.This paper focuses on a proof of the approximation properties of these exponential solutions.We show that vectorial exponential functions can achieve high order convergence.The fundamental part of the proof consists in proving that a certain rectangular matrix has maximal rank.

Elastic Kirchhoff migration of vectorial wave-fields

Based on Kuo and Dai＇s vectorial wave-field extrapolation equations, we derive new Kirchhoff migration equations by introducing unit vectors which represent the ray directions at the imaging points of the reflected P- and PS converted-waves. Furthermore, using the slope of the events on shot records and a ray racing procedure, mirror-image reflection points are found and the reflection data are smeared along the Fresnel zone. The migration method proposed in this paper solves two troublesome imaging problems caused by limited receiving aperture and migration artifacts resulting from wave propagation at the velocities of non original wave type. The migration method is applied successfully with model data, demonstrating that the new method is effective and correct.

Vectorial Hermite-Laguerre-Gaussian beams beyond the paraxial approximation

Starting from the vectorial Rayleigh-Sommerfeld integrals, the free-space propagation expressions for vectorial Hermit-Lagucrre-Gaussian （HLG） beams beyond the paraxial approximation are derived. The far-field expressions and the scalar paraxial results are given as special cases of our general expressions. The intensity distributions of vectorial nonparaxial HLG beams are studied and illustrated with numerical examples.

A Semi-Vectorial Hybrid Morphological Segmentation of Multicomponent Images Based on Multithreshold Analysis of Multidimensional Compact Histogram

In this work, we propose an original approach of semi-vectorial hybrid morphological segmentation for multicomponent images or multidimensional data by analyzing compact multidimensional histograms based on different orders. Its principle consists first of segment marginally each component of the multicomponent image into different numbers of classes fixed at K. The segmentation of each component of the image uses a scalar segmentation strategy by histogram analysis;we mainly count the methods by searching for peaks or modes of the histogram and those based on a multi-thresholding of the histogram. It is the latter that we have used in this paper, it relies particularly on the multi-thresholding method of OTSU. Then, in the case where i) each component of the image admits exactly K classes, K vector thresholds are constructed by an optimal pairing of which each component of the vector thresholds are those resulting from the marginal segmentations. In addition, the multidimensional compact histogram of the multicomponent image is computed and the attribute tuples or ‘colors’ of the histogram are ordered relative to the threshold vectors to produce (K + 1) intervals in the partial order giving rise to a segmentation of the multidimensional histogram into K classes. The remaining colors of the histogram are assigned to the closest class relative to their center of gravity. ii) In the contrary case, a vectorial spatial matching between the classes of the scalar components of the image is produced to obtain an over-segmentation, then an interclass fusion is performed to obtain a maximum of K classes. Indeed, the relevance of our segmentation method has been highlighted in relation to other methods, such as K-means, using unsupervised and supervised quantitative segmentation evaluation criteria. So the robustness of our method relatively to noise has been tested.

Nonparaxial propagation of a vectorial apertured off-axis Lorentz beam

Based on the vectorial Ftayleigh-Sommerfeld integral formula and the complex Gaussian expansion of the hardedge aperture function, an analytical propagation expression for a nonparaxial vectorial off-axis Lorentz beam passing through a rectangular aperture is derived. The unapertured case, the far field expression and the scalar paraxial result are also presented as special cases of the general formulae, respectively. Some numerical examples are also given to show the propagation characteristics of a nonparaxial vectorial off-axis Lorentz beam through a rectangular aperture. It is indicated that the f parameter, the off-axis displacement and the truncation parameter all play an important role in determining nonparaxial propagation behaviour.

Spatial pattern analysis of forest trees based on the vectorial mark

Analysis of spatial patterns to describe the spatial correlation between a tree location and marks(i.e.,structural variables),can reveal stand history,population dynamics,competition and symbiosis.However,most studies of spatial patterns have concentrated on tree location and tree sizes rather than on crown asymmetry especially with direct analysis among marks characterizing facilitation and competition among of trees,and thus cannot reveal the cause of the distributions of tree locations and quantitative marks.To explore the spatial correlation among quantitative and vectorial marks and their implication on population dynamics,we extracted vertical and horizontal marks(tree height and crown projection area)characterizing tree size,and a vectorial mark(crown displacement vector characterizing the crown asymmetry)using an airborne laser scanning point cloud obtained from two forest stands in Oxfordshire,UK.Quantitatively and vectorially marked spatial patterns were developed,with corresponding null models established for a significance test.We analyzed eight types of univariate and bivariate spatial patterns,after first proposing four types.The accuracy of the pattern analysis based on an algorithm-segmented point cloud was compared with that of a truly segmented point cloud.The algorithm-segmented point cloud managed to detect 70–86%of patterns correctly.The eight types of spatial patterns analyzed the spatial distribution of trees,the spatial correlation between tree size and facilitated or competitive interactions of sycamore and other species.These four types of univariate patterns jointly showed that,at smaller scales,the trees tend to be clustered,and taller,with larger crowns due to the detected facilitations among trees in the study area.The four types of bivariate patterns found that at smaller scales there are taller trees and more facilitation among sycamore and other species,while crown size is mostly homogeneous across scales.These results indicate that interspecific facilitation and competition mainly affect tree height in the study area.This work further confirms the connection of tree size with individual facilitation and competition,revealing the potential spatial structure that previously was hard to detect.

VECTORIAL EKELAND VARIATIONAL PRINCIPLE AND CYCLICALLY ANTIMONOTONE EQUILIBRIUM PROBLEMS

In this article, we extend the cyclic antimonotonicity from scalar bifunctions to vector bifunctions. We find out a cyclically antimonotone vector bifunction can be regarded as a family of cyclically antimonotone scalar bifunctions. Using a pre-order principle(see Qiu, 2014), we prove a new version of Ekeland variational principle(briefly, denoted by EVP), which is quite different from the previous ones, for the objective function consists of a family of scalar functions. From the new version, we deduce several vectorial EVPs for cyclically antimonotone equilibrium problems, which extend and improve the previous results. By developing the original method proposed by Castellani and Giuli, we deduce a number of existence results(no matter scalar-valued case,or vector-valued case), when the feasible set is a sequentially compact topological space or a countably compact topological space. Finally, we propose a general coercivity condition. Combining the general coercivity condition and the obtained existence results with compactness conditions, we obtain several existence results for equilibrium problems in noncompact settings.

Generation of diffraction-free vectorial elliptic hollow beams with space-varying inhomogeneous polarizations

Diffraction-free vectorial elliptic hollow beams(vEHBs)are generated by an optical system composed of a short elliptic hollow fiber(EHF)and an axicon.Each beam has a closed elliptic annular intensity profile and space-varying polarization states in its diffraction-free distance of more than 1 m.The generated beams have a counter-clockwise or clockwise periodically-rotated inhomogeneous polarization.And the spin angular momentum(SAM)of the vEHBs is 1ħor-1ħwhich is consistent with the type of dual-mode in the EHF and the periodic polarization rotations of the vEHBs.The vEHBs have potential applications in optically trapping and micromanipulating the micro-or nano-particles,quantum information transmission,and Bose-Einstein condensates,etc.

Far-field divergence of a vectorial plane wave diffracted by a circular aperture from the vectorial structure

Based on the vectorial structure of an electromagnetic wave, the analytical and concise expressions for the TE and TM terms of a veetorial plane wave diffracted by a circular aperture are derived in the far-field. The expressions of the energy flux distributions of the TE term, the TM term and the diffracted plane wave are also presented. The ratios of the power of the TE and TM terms to that of the diffracted plane wave are examined in the far-field. In addition, the far-field divergence angles of the TE term, the TM term and the diffracted plane wave, which are related to the energy flux distribution, are investigated. The different energy flux distributions of the TE and TM terms result in the discrepancy of their divergence angles. The influences of the linearly polarized angle and the radius of the circular aperture on the far-field divergence angles of the TE term, the TM term and the diffracted plane wave are discussed in detail. This research may promote the recognition of the optical propagation through a circular aperture.

Full-vectorial analysis of optical waveguides by the finite difference method based on polynomial interpolation

Based on the polynomial interpolation, a new finite difference （FD） method in solving the full-vectorial guidedmodes for step-index optical waveguides is proposed. The discontinuities of the normal components of the electric field across abrupt dielectric interfaces are considered in the absence of the limitations of scalar and semivectorial approximation, and the present PD scheme can be applied to both uniform and non-uniform mesh grids. The modal propagation constants and field distributions for buried rectangular waveguides and optical rib waveguides are presented. The hybrid nature of the vectorial modes is demonstrated and the singular behaviours of the minor field components in the corners are observed. Moreover, solutions are in good agreement with those published early, which tests the validity of the present approach.

Vectorial Bed load Equations and River Width Dynamic Adjustment Modeling

Based on the analysis of forces acting on non-cohesive sediment particle under non-equilibrium con- dition,a vectorial equation of coarse particle movement on arbitrarily riverbed and the mathematical model used to simulate river width adjust are developed,some flume experiments for the time evolution of straight river channel are done.

A NEW DERIVATION OF THE VECTORIAL EQUATION OF MOTION FOR A TEST PARTICLE IN THE SCHWARZCHILD FIELD

A new derivation of the vectorial equation of motion for a test particle in the Schwarzchild field is given which greatly simplifies the procedure given by C. A. Murray[1]

A Simulation Model for Sexual and Vectorial Transmission of Zika Virus(ZIKV)

Nowadays the Zika virus (ZIKV) has been one of the most studied vector-borne diseases due to the considerable outbreaks that have generated around the world as well as due to the new transmission mechanisms and health complications originated. According to statistics of the INS-Colombia for July 2016, 68% of the population infected by ZIKV (confirmed cases) are pregnant women. Furthermore, the Quindío department belongs to the states with more than 50% of the total infected persons being pregnant women. Taking into account those characteristics, a theoretical model is proposed and analyzed to describe the population dynamics considering the sexual and vectorial transmission of ZIKV, with special emphasis in the consequences of the non-vectorial transmission in the population. The obtained results with simulations through the beta parameter indicate that the probability of sexual transmission between susceptible women and infected men points out the importance of campaigns to inculcate prevention measures for the safe sexual relationships between ZIKV infected population.

Intelligent vectorial surrogate modeling framework for multi-objective reliability estimation of aerospace engineering structural systems

To improve the computational efficiency and accuracy of multi-objective reliability estimation for aerospace engineering structural systems,the Intelligent Vectorial Surrogate Modeling(IVSM)concept is presented by fusing the compact support region,surrogate modeling methods,matrix theory,and Bayesian optimization strategy.In this concept,the compact support region is employed to select effective modeling samples;the surrogate modeling methods are employed to establish a functional relationship between input variables and output responses;the matrix theory is adopted to establish the vector and cell arrays of modeling parameters and synchronously determine multi-objective limit state functions;the Bayesian optimization strategy is utilized to search for the optimal hyperparameters for modeling.Under this concept,the Intelligent Vectorial Neural Network(IVNN)method is proposed based on deep neural network to realize the reliability analysis of multi-objective aerospace engineering structural systems synchronously.The multioutput response function approximation problem and two engineering application cases(i.e.,landing gear brake system temperature and aeroengine turbine blisk multi-failures)are used to verify the applicability of IVNN method.The results indicate that the proposed approach holds advantages in modeling properties and simulation performances.The efforts of this paper can offer a valuable reference for the improvement of multi-objective reliability assessment theory.

A General Vectorial Ekeland's Variational Principle with a P-distance

In this paper, by using p-distances on uniform spaces, we establish a general vectorial Ekeland variational principle （in short EVP）, where the objective function is defined on a uniform space and taking values in a pre-ordered real linear space and the perturbation involves a p-distance and a monotone function of the objective function. Since p-distances are very extensive, such a form of the perturbation in deed contains many different forms of perturbations appeared in the previous versions of EVP. Besides, we only require the objective function has a very weak property, as a substitute for lower semi-continuity, and only require the domain space （which is a uniform space） has a very weak type of completeness, i.e., completeness with respect to a certain p-distance. Such very weak type of completeness even includes local completeness when the uniform space is a locally convex topological vector space. From the general vectorial EVP, we deduce a general vectorial Caristi＇s fixed point theorem and a general vectorial Takahashi＇s nonconvex minimization theorem. Moreover, we show that the above three theorems are equivalent to each other. We see that the above general vectorial EVP includes many particular versions of EVP, which extend and complement the related known results.

Generalized Vectorial Quasi-Equilibrium Problem onW-Space

Generalized vectorial quasi-equilibrium problem and generalized set-valued quasi-equilibrium problem are introduced and the existence theorems of generalized vectorial quasi-equilihrium problem on W-space are obtained. As corollaries, existence theorems of four kinds of quasi-equilibrium problem are obtained, and these results generalize and improve correspond results in [1-7] .

Vectorial optical fields： recent advances and future prospects

Driven by their potential applications, vectorial optical fields with spatially inhomogeneous states of polarization within the cross section have drawn significant attention recently. This work intends to review some of the latest development of this rapidly growing field of optics and offer a general overview of the current status of this field in a few areas. Mathematical descriptions of generalized vectorial optical fields are provided along with several special examples. A time-reversal methodology for the creation of a wide variety of exotic optical focal fields with prescribed characteristics within the focal volume is presented. Recently developed methods for the generation of vectorial optical fields that utilize fiber lasers,digital lasers, vectorial optical field generator, metasurfaces or photoalignment liquid crystals are summarized. The interactions of these vectorial optical fields with various micro-and nano-structures are presented and the prospects of their potential applications are discussed. The connection of vectorial optical fields with higher dimensionality in quantum information is summarized.

Full vectorial feature of second-harmonic generation with full Poincaré beams

We demonstrate the full vectorial feature of second-harmonic generation(SHG), i.e., from infrared full Poincaré beams to visible full Poincaré beams, based on two cascading type I phase-matching beta barium borate crystals of orthogonal optical axes. We visualize the structured features of the vectorial SHG wave by using Stokes polarimetry and show the interesting doubling effect of the polarization topological index, i.e., a low-order full Poincaré beam is converted to a high-order one. However, the polarization singularities of both C points and L lines are found to keep invariant during the SHG process. Our scheme could offer a deeper understanding on the interaction of vectorial light fields with media and can be generalized to other nonlinear optical effects.

Revealing complex optical phenomena through vectorial metrics

Advances in vectorial polarization-resolved imaging are bringing new capabilities to applications ranging from fundamental physics through to clinical diagnosis.Imaging polarimetry requires determination of the Mueller matrix(MM)at every point,providing a complete description of an object’s vectorial properties.Despite forming a comprehensive representation,the MM does not usually provide easily interpretable information about the object’s internal structure.Certain simpler vectorial metrics are derived from subsets of the MM elements.These metrics permit extraction of signatures that provide direct indicators of hidden optical properties of complex systems,while featuring an intriguing asymmetry about what information can or cannot be inferred via these metrics.We harness such characteristics to reveal the spin Hall effect of light,infer microscopic structure within laser-written photonic waveguides,and conduct rapid pathological diagnosis through analysis of healthy and cancerous tissue.This provides new insight for the broader usage of such asymmetric inferred vectorial information.

Vectorial holography-mediated growth of plasmonic metasurfaces

Nowadays,the electromagnetic properties of artificial photonic materials can be well-tuned via designs over their composition and geometries.However,engineering the properties of artificial materials at the nanoscale is challenging and costly.Here we demonstrate a facile and low-cost method for fabricating large-area silver nanoparticle metasurfaces(AgNPMSs)by using the vectorial holography-mediated growth technique.The AgNPMS,which can be regarded as a hologram device,possesses excellent chiroptical properties.The vectorial holographic technique may open avenues for fabricating novel chiroptical metamaterials with large degrees of freedom,which can be further used for beam steering,photocatalysis,biosensing,etc.