ELLIPTIC EQUATIONS IN DIVERGENCE FORM WITH DISCONTINUOUS COEFFICIENTS IN DOMAINS WITH CORNERS

We study equations in divergence form with piecewise Cαcoefficients.The domains contain corners and the discontinuity surfaces are attached to the edges of the corners.We obtain piecewise C^(1,α) estimates across the discontinuity surfaces and provide an example to illustrate the issue regarding the regularity at the corners.

Elliptical encirclement control capable of reinforcing performances for UAVs around a dynamic target

Most researches associated with target encircling control are focused on moving along a circular orbit under an ideal environment free from external disturbances.However,elliptical encirclement with a time-varying observation radius,may permit a more flexible and high-efficacy enclosing solution,whilst the non-orthogonal property between axial and tangential speed components,non-ignorable environmental perturbations,and strict assignment requirements empower elliptical encircling control to be more challenging,and the relevant investigations are still open.Following this line,an appointed-time elliptical encircling control rule capable of reinforcing circumnavigation performances is developed to enable Unmanned Aerial Vehicles(UAVs)to move along a specified elliptical path within a predetermined reaching time.The remarkable merits of the designed strategy are that the relative distance controlling error can be guaranteed to evolve within specified regions with a designer-specified convergence behavior.Meanwhile,wind perturbations can be online counteracted based on an unknown system dynamics estimator(USDE)with only one regulating parameter and high computational efficiency.Lyapunov tool demonstrates that all involved error variables are ultimately limited,and simulations are implemented to confirm the usability of the suggested control algorithm.

Legendre-Jacobi’s Elliptic Integrals Shed Light on the Luminosity Distance in Cosmology

This article concerns the integral related to the transverse comoving distance and, in turn, to the luminosity distance both in the standard non-flat and flat cosmology. The purpose is to determine a straightforward mathematical formulation for the luminosity distance as function of the transverse comoving distance for all cosmology cases with a non-zero cosmological constant by adopting a different mindset. The applied method deals with incomplete elliptical integrals of the first kind associated with the polynomial roots admitted in the comoving distance integral according to the scientific literature. The outcome shows that the luminosity distance can be obtained by the combination of an analytical solution followed by a numerical integration in order to account for the redshift. This solution is solely compared to the current Gaussian quadrature method used as basic recognized algorithm in standard cosmology.

Elliptically polarized high-order harmonic generation of Ar atom in an intense laser field

High-order harmonic generation(HHG) of Ar atom in an elliptically polarized intense laser field is experimentally investigated in this work.Interestingly,the anomalous ellipticity dependence on the laser ellipticity(ε) in the lower-order harmonics is observed,specifically in the 13rd-order,which displays a maximal harmonic intensity at ε ≈ 0.1,rather than at ε = 0 as expected.This contradicts the general trend of harmonic yield,which typically decreases with the increase of laser ellipticity.In this study,we attribute this phenomenon to the disruption of the symmetry of the wave function by the Coulomb effect,leading to the generation of a harmonic with high ellipticity.This finding provides valuable insights into the behavior of elliptically polarized harmonics and opens up a potential way for exploring new applications in ultrafast spectroscopy and light–matter interactions.

THE WEIGHTED KATO SQUARE ROOT PROBLEMOF ELLIPTIC OPERATORS HAVING A BMOANTI-SYMMETRICPART

Let n≥2 and let L be a second-order elliptic operator of divergence form with coefficients consisting of both an elliptic symmetric part and a BMO anti-symmetric part in ℝ^(n).In this article,we consider the weighted Kato square root problem for L.More precisely,we prove that the square root L^(1/2)satisfies the weighted L^(p)estimates||L^(1/2)(f)||L_(ω)^p(R^(n))≤C||■f||L_(ω)^p(R^(n);R^(n))for any p∈(1,∞)andω∈Ap(ℝ^(n))(the class of Muckenhoupt weights),and that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,2+ε)andω∈Ap(ℝ^(n))∩RH_(2+ε/p),(R^(n))(the class of reverse Hölder weights),whereε∈(0,∞)is a constant depending only on n and the operator L,and where(2+ε/p)'denotes the Hölder conjugate exponent of 2+ε/p.Moreover,for any given q∈(2,∞),we give a sufficient condition to obtain that||■f||L_(ω)^p(R^(n);R^(n))≤C||L^(1/2)(f)||L_(ω)^p(R^(n))for any p∈(1,q)andω∈A_(p)(R^(n))∩pRH_(q/p),(R^(n)).As an application,we prove that when the coefficient matrix A that appears in L satisfies the small BMO condition,the Riesz transform∇L^(−1/2)is bounded on L_(ω)^(p)(ℝ^(n))for any given p∈(1,∞)andω∈Ap(ℝ^(n)).Furthermore,applications to the weighted L^(2)-regularity problem with the Dirichlet or the Neumann boundary condition are also given.

Sensitive Information Security Based on Elliptic Curves

The elliptic curve cryptography algorithm represents a major advancement in the field of computer security. This innovative algorithm uses elliptic curves to encrypt and secure data, providing an exceptional level of security while optimizing the efficiency of computer resources. This study focuses on how elliptic curves cryptography helps to protect sensitive data. Text is encrypted using the elliptic curve technique because it provides great security with a smaller key on devices with limited resources, such as mobile phones. The elliptic curves cryptography of this study is better than using a 256-bit RSA key. To achieve equivalent protection by using the elliptic curves cryptography, several Python libraries such as cryptography, pycryptodome, pyQt5, secp256k1, etc. were used. These technologies are used to develop a software based on elliptic curves. If built, the software helps to encrypt and decrypt data such as a text messages and it offers the authentication for the communication.

Remote sensing image encryption algorithm based on novel hyperchaos and an elliptic curve cryptosystem

Remote sensing images carry crucial ground information,often involving the spatial distribution and spatiotemporal changes of surface elements.To safeguard this sensitive data,image encryption technology is essential.In this paper,a novel Fibonacci sine exponential map is designed,the hyperchaotic performance of which is particularly suitable for image encryption algorithms.An encryption algorithm tailored for handling the multi-band attributes of remote sensing images is proposed.The algorithm combines a three-dimensional synchronized scrambled diffusion operation with chaos to efficiently encrypt multiple images.Moreover,the keys are processed using an elliptic curve cryptosystem,eliminating the need for an additional channel to transmit the keys,thus enhancing security.Experimental results and algorithm analysis demonstrate that the algorithm offers strong security and high efficiency,making it suitable for remote sensing image encryption tasks.

The Regularity of Solutions to Mixed Boundary Value Problems of Second-Order Elliptic Equations with Small Angles

This paper considers the regularity of solutions to mixed boundary value problems in small-angle regions for elliptic equations. By constructing a specific barrier function, we proved that under the assumption of sufficient regularity of boundary conditions and coefficients, as long as the angle is sufficiently small, the regularity of the solution to the mixed boundary value problem of the second-order elliptic equation can reach any order.

Besov Estimates for Sub-Elliptic Equations in the Heisenberg Group

In this article, we deal with weak solutions to non-degenerate sub-elliptic equations in the Heisenberg group, and study the regularities of solutions. We establish horizontal Calderón-Zygmund type estimate in Besov spaces with more general assumptions on coefficients for both homogeneous equations and non-homogeneous equations. This study of regularity estimates expands the Calderón-Zygmund theory in the Heisenberg group.

ELLIPTIC MOTIONS ON MODAL CONVERSION ULTRASONIC MOTORS

The forming of elliptic motions on the modal conversion ultrasonic motors （MCUMs） is discussed. The principles of the modal conversion are investigated based on the coupling with the stator and the rotor, and using an independent coupler. The elliptical locus observed on the longitudinal-torslonal vibration converter with oblique slits is analyzed by using vibration theory. A method for the modal conversion is proposed by using the local mode of a substructure On a main structure. The method can be used to design the modal conversion type ultrasonic motors.

Viscous warm pressure bulging process of AZ31B magnesium alloy with different ellipticity dies

Based on the bulging principle of different ellipticity dies, the methyl vinyl silicone rubber with excellent thermal stability and heat transfer performance was chosen as the viscous medium. The finite element analysis and experiments of viscous warm pressure bulging （VWPB） of AZ31B magnesium alloy were conducted to analyze the influence of different ellipticity dies on the formability of AZ31B magnesium alloy. At the same time, based on the grid strain rule, the forming limit diagram （FLD） of VWPB of AZ31B magnesium alloy was obtained through measuring the strain of bulging specimens. The results showed that at the temperature range of viscous medium thermal stability, the viscous medium can fit the geometry variation of sheet and generate non-uniform pressure field, and as the die ellipticity increases, the difference value of non-uniform pressure reduces. Meanwhile, according to the FLD, the relationship between part complexity and ultimate deformation was investigated.

A Class of the Singularly Perturbed Boundary Value Problems for Elliptic Equation with a Super Surface of Turning Point in n-dimensional Space

In this paper, we study a class singular perturbed elliptic equation boundary value problem with a super surface of turning point in n-dimensional space by using the method of multiple scales and the comparison theorem. The uniformly valid asymptotic approxmations of solutions for the boundary value problem is constructed.

The Singularly Perturbed Boundary Value Problems for Elliptic Equation with Turning Point

The singularly perturbed elliptic equation boundary value problem with turning point is considered. Using the method of multiple scales and the comparison theorem, the asymptotic behavior of solution for the boundary value problem is studied.

An Elliptical-Beam Reflector Antenna forSatellite Communication System

An offset elliptical reflector antenna suitable for satellite application was designed and investigated when it was fed by a rectangular horn partially filled.with a dielectric..The.reflector antenna exhibits high gain, low cross polarization. low sidelines and an elliptical beam. Al- though this study has been carried out in view of possible satellite applications, it is clear that this. antenna. is also suitable for use in radar antennas.

Positive Solutions of a Semilinear Elliptic Equation with Coefficient That Changes Sign

We prove the existence of a positive solution to the problem-Δu=a(x)f(u), x∈Ω, u(x)=0,x∈Ω,where Ω is a bounded domain in R n with smooth boundary, a(x) is allowed to change sign.

A Class of Nonlocal Boundary Value Problems for Elliptic Systems in Unbounded Domains

A class of nonlocal boundary value probl em s for elliptic systems in the unbounded domains are considered. Under suitable c onditions, the existence of solution and the comparison theorem for the boundary value problems are studied.

基于Elliptical Copula函数的相关性模型研究

针对沪深股指构建了两种基于Elliptical Copula函数的相关性模型,并利用参数估计的结果计算其相关性指标.结果表明,Elliptical Copula函数在金融相关性分析中比传统方法合理有效,其中学生氏t-copula函数在服从厚尾分布的相关性模型中比高斯Copula更具实际意义.

Representation Theory of Three-Dimensional Elliptical Crack Under Dynamic Loading

The relation between the normal displacement on the surface of a dynamical elliptical crack and the normal stress over the crack surface was studied. The three dimensional elastodynamic equations and Fourier Laplace transforms are used. Based on the influence function and the inversion of integral transforms, one can find that if the distribution of normal displacement on the surface of a dynamic elliptical crack is a polynomial of degree n in x 1 and x 2 , then the normal pressure acting over the ellipse is also a polynomial P n(x 1,x 2) of the same degree in x 1 and x 2 .

Understanding the effect of elliptical polarization in surface nuclear magnetic resonance method

Currently, surface nuclear magnetic resonance （SNMR） method is the only geophysical method that detects groundwater directly. In this paper, we investigate the effect of elliptical polarization in the perpendicular excitation magnetic field. The effect of elliptical polarization is clearly visible in our ellipticity calculation and it can cause strong distortion to the excitation field in the presence of high subsurface conductivities. By examining the co-rotating and counter-rotating components of the field, we show that elliptical polarization affects transmitting and receiving processes differently and that a clear phase lag exists between transmitter loop and receiver loop. Finally, we derive the response function of coincident loops and calculate proton tip angles, the kernel function and SNMR response curves of a 1D aquifer model. Based on the simulations, we conclude that the elliptical polarization and phase lag can significantly affect SNMR response and it is essential to include elliptical polarization in SNMR modeling and data interpretation.

A CLASS OF NONLOCAL BOUNDARY VALUE PROBLEMS OF NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS

The nonlocal boundary value problems for nonlinear elliptic systems in the unbounded domain are considered. Under suitable conditions the existence of solution and comparison theorem for the boundary value problems are studied.