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.

Exponentially Convergent Multiscale Finite Element Method

We provide a concise review of the exponentially convergent multiscale finite element method(ExpMsFEM)for efficient model reduction of PDEs in heterogeneous media without scale separation and in high-frequency wave propagation.The ExpMsFEM is built on the non-overlapped domain decomposition in the classical MsFEM while enriching the approximation space systematically to achieve a nearly exponential convergence rate regarding the number of basis functions.Unlike most generalizations of the MsFEM in the literature,the ExpMsFEM does not rely on any partition of unity functions.In general,it is necessary to use function representations dependent on the right-hand side to break the algebraic Kolmogorov n-width barrier to achieve exponential convergence.Indeed,there are online and offline parts in the function representation provided by the ExpMsFEM.The online part depends on the right-hand side locally and can be computed in parallel efficiently.The offline part contains basis functions that are used in the Galerkin method to assemble the stiffness matrix;they are all independent of the right-hand side,so the stiffness matrix can be used repeatedly in multi-query scenarios.

The Maximum and Minimum Value of Exponential RandićIndices of Quasi-Tree Graph

The exponential Randić index has important applications in the fields of biology and chemistry. The exponential Randić index of a graph G is defined as the sum of the weights e 1 d( u )d( v ) of all edges uv of G, where d( u ) denotes the degree of a vertex u in G. The paper mainly provides the upper and lower bounds of the exponential Randić index in quasi-tree graphs, and characterizes the extremal graphs when the bounds are achieved.

EXPONENTIAL DICHOTOMIES,HOMOCLINIC ORBITS AND METHODS OF MELNIKOV

In this paper we construct, by using the theory of exponential dichotomies, a Melnikov-type function by which we can detect the existence of homoclinic orbits for the perturbed systems x = g(x) + epsilon h(t, x, epsilon). Our result of this paper may be complementary to that of K.J.Palmer([3]).

Exponential Dichotomies and Fredholm Operators of Dynamic Equations on Time Scales

For time-varying non-regressive linear dynamic equations on a time scale with bounded graininess, we introduce the concept of the associative operator with linear systems on time scales. The purpose of this research is the characterizations of the exponential dichotomy obtained in terms of Fredholm property of that associative operator. Particularly, we use Perron’s method, which was generalized on time scales by J. Zhang, M. Fan, H. Zhu in?[1], to show that if the associative operator is semi-Fredholm then the corresponding linear nonautonomous equation has an exponential dichotomy on both?T?+?and?T-.??Moreover, we also give the converse result that the linear systems have?an exponential dichotomy on both?T?+?andT-??then the associative operator is Fredholm on?T.

Exponential Dichotomies and Homoclinic Orbits from Heteroclinic Cycles

In this paper, we investigate the homoclinic bifurcations from a heteroclinic cycle by using exponential dichotomies. We give a Melnikov—type condition assuring the existence of homoclinic orbits form heteroclinic cycle. We improve some important results.

THE GROWTH OF SOLUTIONS TO HIGHER ORDER DIFFERENTIAL EQUATIONS WITH EXPONENTIAL POLYNOMIALS AS ITS COEFFICIENTS

By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.

A Numerical Investigation Based on Exponential Collocation Method for Nonlinear SITR Model of COVID-19

In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics,namely,susceptible(S),infected(I),treatment(T),and recovered(R).The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points.To indicate the usefulness of this method,we employ it in some cases.For error analysis of the method,the residual of the solutions is reviewed.The reported examples show that the method is reasonably efficient and accurate.

The Modi Exponentiated Exponential Distribution

In this study, a new four-parameter distribution called the Modi Exponentiated Exponential distribution was proposed and studied. The new distribution has three shape and one scale parameters. Its mathematical and statistical properties were investigated. The parameters of the new model were estimated using the method of Maximum Likelihood Estimation. Monte Carlo simulation was used to evaluate the performance of the MLEs through average bias and RMSE. The flexibility and goodness-of-fit of the proposed distribution were demonstrated by applying it to two real data sets and comparing it with some existing distributions.

Vertical deformation model on postseismic phase using exponential and logarithmic function based on InSAR

The Palu MW7.4 earthquake occurred on September 28, 2018, with the epicenter at 119.86°, 0.72°. The severe shaking caused severe damage in Central Sulawesi, especially in Palu. We conducted a postseismic deformation study to determine the deformation pattern and reduce future earthquakes’ impact.Interferometric Synthetic Aperture Radar(In SAR) data were processed using Li CSBAS to get the time series. The time series data were fitted to exponential and logarithmic functions to determine the mechanism of postseismic deformation. The exponential model identified the influence of the viscoelastic mechanism, and the logarithm identified the afterslip mechanism. The Palu earthquake was fitted to logarithmic and exponential, but the logarithmic was more significant than an exponential function.Afterslip mechanism predominates, and viscoelastic mechanisms play a minor role in this postseismic deformation.

Improved population mean estimator with exponential function under non-response

In this article,we consider a new family of exponential type estimators for estimating the unknown population mean of the study variable.We propose estimators taking advantage of the auxiliary variable information under the first and second non-response cases separately.The required theoretical comparisons are obtained and the numerical studies are conducted.In conclusion,the results show that the proposed family of estimators is the most efficient estimator with respect to the estimators in literature under the obtained conditions for both cases.

Trimmable bandgap reference circuit with exponential curvature compensation

This paper proposes an improved exponential curvature-compensated bandgap reference circuit to exploit the exponential relationship between the current gainβof the bipolar junction transistor(BJT)and the temperature as well as reduce the influence of resistance-temperature dependency.Considering the degraded circuit performance caused by the process deviation,the trimmable module of the temperature coefficient(TC)is introduced to improve the circuit stability.The circuit has the advantages of simple structure,high linear stability,high TC accuracy,and trimmable TC.It consumes an area of 0.09 mm^(2)when fabricated by using the 0.25-μm complementary metal-oxide-semiconductor(CMOS)process.The proposed circuit achieves the simulated power supply rejection(PSR)of about-78.7 dB@1 kHz,the measured TC of~4.7 ppm/℃over a wide temperature range from-55℃to 125℃with the 2.5-V single-supply voltage,and the tested line regulation of 0.10 mV/V.Such a high-performance bandgap reference circuit can be widely applied in high-precision and high-reliability electronic systems.

The Class of Atomic Exponential Basis Functions EFup_(n)(x,ω)-Development and Application

The purpose of this paper is to present the class of atomic basis functions(ABFs)which are of exponential type and are denoted by EFupn(x,ω).While ABFs of the algebraic type are already represented in the numerical modeling of various problems inmathematical physics and computationalmechanics,ABFs of the exponential type have not yet been sufficiently researched.These functions,unlike the ABFs of the algebraic type Fupn(x),contain the tension parameterω,which gives them additional approximation properties.Exponential monomials up to the nth degree can be described exactly by the linear combination of the functions EFupn(x,ω).The function EFupn for n=0 is called the“mother”ABF of the exponential type,i.e.,EFup0(x,ω)≡Eup(x,ω).In other words,the functions EFupn(x,ω)are elements of the linear vector space EUPn and retain all the properties of their“mother”function Eup(x,ω).Thus,this paper,in terms of its content and purpose,can be understood as a sequel of the article by Brajcic Kurbasa et al.,which shows the basic properties and application of the basis function Eup(x,ω).This paper presents,in an analogous way,the development and application of the exponential basis functions EFupn(x,ω).Here,for the first time,expressions for calculating the values of the functions EFupn(x,ω)and their derivatives are given in a form suitable for application in numerical analyses,which is shown in the verification examples of the approximations of known functions.

论人工智能生成内容的可版权性:以用户的独创性表达为视角

AIGC既包含来自机器的选择,也包含来自人类用户的选择。当后者满足独创性表达要求时,AIGC便足以被认定为作品。在判断用户的独创性表达时,裁判者应当关心人贡献了什么,而不是工具贡献了什么;应当关心人贡献了什么,而不是人没有贡献什么;应当关心人贡献的实质,而非纠结于其形式。在以上原则的指导下,既有的作品构成要件规则足以支持相当一部分AIGC获得作品资格。AIGC的所谓“随机性”,只是发生在用户指定范围内的“随机”,并不妨碍用户控制达到著作权法的要求。作为文本的提示词,具有转化为包括视觉表达在内的其他类型表达的可能性。若以摄影为参照系,AI与传统工具的比较不仅不能导致对作品资格的否定,反而能够凸显“工具贡献大并不意味着人的贡献不足”。著作权法的“宽进宽出”结构提示我们,将用户做出独创性表达的AIGC纳入著作权法图式是在认知层面最为经济的利益平衡分析框架。考虑到独立创作例外和版权救济手段的灵活性,承认AIGC获得作品资格的可能性并不会过度妨碍公众自由。

On the Fractional Derivatives with an Exponential Kernel

The present article mainly focuses on the fractional derivatives with an exponential kernel(“exponential fractional derivatives”for brevity).First,several extended integral transforms of the exponential fractional derivatives are proposed,including the Fourier transform and the Laplace transform.Then,the L2 discretisation for the exponential Caputo derivative with a∈(1,2)is established.The estimation of the truncation error and the properties of the coefficients are discussed.In addition,a numerical example is given to verify the correctness of the derived L2 discrete formula.

Finite Element Implementation of the Exponential Drucker-Prager Plasticity Model for Adhesive Joints

This paper deals with the numerical implementation of the exponential Drucker-Parger plasticitymodel in the commercial finite element software,ABAQUS,via user subroutine UMAT for adhesive joint simulations.The influence of hydrostatic pressure on adhesive strength was investigated by a modified Arcan fixture designed particularly to induce a different state of hydrostatic pressure within an adhesive layer.The developed user subroutine UMAT,which utilizes an associated plastic flow during a plastic deformation,can provide a good agreement between the simulations and the experimental data.Better numerical stability at highly positive hydrostatic pressure loads for a very high order of exponential function can also be achieved compared to when a non-associated flow is used.

Non-harmonic resonance of viscoelastic structures subjected to time-dependent exponentially decreasing transverse distributed loads

In this paper,the non-harmonic resonance of Bernoulli viscoelastic beams,Kirchhoff viscoelastic plates,Timoshenko viscoelastic beams,and Mindlin viscoelastic plates subjected to time-dependent exponentially decreasing transverse distributed load is investigated for the first time.The constitutive equations are expressed utilizing Boltzmann integral law with a constant bulk modulus.The displacement vector is approximated by employing the separation of variables method.The Laplace transformation is used to transfer equations from the time domain to the Laplace domain and vice versa.The novel point of the proposed method is to express,prove and calculate the critical time in which the displacement will be several times the displacement at time zero.In addition,this new method calculates the maximum deflection at the critical time,explicitly and exactly,without any need to follow the time-displacement curve with a low computational cost.Additionally,the proposed method introduces the critical range of time so that the responses are greater than the responses at time zero.

地理学的时代特征、二元论与可持续发展

地理学作为一门学科总体上与科学体系的发展亦步亦趋。20世纪自然地理学与人文地理学的分裂曾严重消解了地理学的整体性和统一性,同时专业分化与研究主题的纵深发展造成了学科的空心化。面对21世纪复杂性科学、地球系统科学和新的科学综合体的到来,在全球化和可持续发展的大背景下,我们应该在大学课堂上让自然与人文再度融合,通过学术语言的规范、学科理论的构建、研究范式的转换,将二元论的影响消融于无形,能以一种整体化、系统性的观念来应对人类的未来和地球的可持续性发展这一艰巨而复杂的任务。

精细积分方法的发展与扩展应用

钟万勰院士于1991年首先提出计算矩阵指数的精细积分方法,其要点是2N类算法和增量存储。精细积分方法可给出矩阵指数在计算机意义上的精确解,为常微分方程的数值计算提供了高精度、高稳定性的算法,现已成功应用于结构动力响应、随机振动、热传导以及最优控制等众多领域。本文首先介绍矩阵指数精细积分方法的提出、基本思想和发展;然后依次介绍在时不变/时变线性微分方程、非线性微分方程以及大规模问题求解中发展起来的各种精细积分方法,分析了其优缺点和适用范围;最后介绍了精细积分方法的基本思想在两点边值问题、椭圆函数和病态代数方程等问题的扩展应用,进一步展示了该思想的特色。

基于POE模型的工业机器人运动学参数二次辨识方法研究

针对工业机器人在高度制造领域精度不高的问题,本文提出了一种基于POE模型的工业机器人运动学参数二次辨识方法。阐述了基于指数积(Product of exponential, POE)模型的运动学误差模型构建方法,并建立基于POE误差模型的适应度函数;为实现高精度的参数辨识,提出了一种二次辨识方法,先利用改进灰狼优化算法(Improved grey wolf optimizer, IGWO)实现运动学参数误差的粗辨识,初步将Staubli TX60型机器人的平均位置误差和平均姿态误差分别从(0.648 mm, 0.212°)降低为(0.457 mm, 0.166°);为进一步提高机器人的精度性能,再通过LM(Levenberg-Marquard)算法进行参数误差的精辨识,最终将Staubli TX60型机器人平均位置误差和平均姿态误差进一步降低为(0.237 mm, 0.063°),机器人平均位置误差和平均姿态误差分别降低63.4%和70.2%。为了验证上述二次辨识方法的稳定性,随机选取5组辨识数据集和验证数据集进行POE误差模型的参数误差辨识,结果表明提出的二次辨识方法能够稳定、精确地辨识工业机器人运动学参数误差。