Approximate solution of Volterra-Fredholm integral equations using generalized barycentric rational interpolant

It is well-known that interpolation by rational functions results in a more accurate approximation than the polynomials interpolation.However,classical rational interpolation has some deficiencies such as uncontrollable poles and low convergence order.In contrast with the classical rational interpolants,the generalized barycentric rational interpolants which depend linearly on the interpolated values,yield infinite smooth approximation with no poles in real numbers.In this paper,a numerical collocation approach,based on the generalized barycentric rational interpolation and Gaussian quadrature formula,was introduced to approximate the solution of Volterra-Fredholm integral equations.Three types of points in the solution domain are used as interpolation nodes.The obtained numerical results confirm that the barycentric rational interpolants are efficient tools for solving Volterra-Fredholm integral equations.Moreover,integral equations with Runge’s function as an exact solution,no oscillation occurrs in the obtained approximate solutions so that the Runge’s phenomenon is avoided.

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.

Feynman Path Integral Using Operator Integration in Banach Space

Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.

CAUCHY TYPE INTEGRALS AND A BOUNDARY VALUE PROBLEM IN A COMPLEX CLIFFORD ANALYSIS

Clifford analysis is an important branch of modern analysis;it has a very important theoretical significance and application value,and its conclusions can be applied to the Maxwell equation,Yang-Mill field theory,quantum mechanics and value problems.In this paper,we first give the definition of a quasi-Cauchy type integral in complex Clifford analysis,and get the Plemelj formula for it.Second,we discuss the H?lder continuity for the Cauchy-type integral operators with values in a complex Clifford algebra.Finally,we prove the existence of solutions for a class of linear boundary value problems and give the integral representation for the solution.

Design of integral sliding mode guidance law based on disturbance observer

With the increasing precision of guidance,the impact of autopilot dynamic characteristics and target maneuvering abilities on precision guidance is becoming more and more significant.In order to reduce or even eliminate the autopilot dynamic operation and the target maneuvering influence,this paper suggests a guidance system model involving a novel integral sliding mode guidance law(ISMGL).The method utilizes the dynamic characteristics and the impact angle,combined with a sliding mode surface scheme that includes the desired line-ofsight angle,line-of-sight angular rate,and second-order differential of the angular line-of-sight.At the same time,the evaluation scenario considere the target maneuvering in the system as the external disturbance,and the non-homogeneous disturbance observer estimate the target maneuvering as a compensation of the guidance command.The proposed system’s stability is proven based on the Lyapunov stability criterion.The simulations reveale that ISMGL effectively intercepted large maneuvering targets and present a smaller miss-distance compared with traditional linear sliding mode guidance laws and trajectory shaping guidance laws.Furthermore,ISMGL has a more accurate impact angle and fast convergence speed.

Results Involving Partial Differential Equations and Their Solution by Certain Integral Transform

In this study,we aimto investigate certain triple integral transformand its application to a class of partial differentialequations.We discuss various properties of the new transformincluding inversion, linearity, existence, scaling andshifting, etc. Then,we derive several results enfolding partial derivatives and establish amulti-convolution theorem.Further, we apply the aforementioned transform to some classical functions and many types of partial differentialequations involving heat equations,wave equations, Laplace equations, and Poisson equations aswell.Moreover,wedraw some figures to illustrate 3-D contour plots for exact solutions of some selected examples involving differentvalues in their variables.

Wave propagation of a functionally graded plate via integral variables with a hyperbolic arcsine function

Several studies on functionally graded materials(FGMs)have been done by researchers,but few studies have dealt with the impact of the modification of the properties of materials with regard to the functional propagation of the waves in plates.This work aims to explore the effects of changing compositional characteristics and the volume fraction of the constituent of plate materials regarding the wave propagation response of thick plates of FGM.This model is based on a higher-order theory and a new displacement field with four unknowns that introduce indeterminate integral variables with a hyperbolic arcsine function.The FGM plate is assumed to consist of a mixture of metal and ceramic,and its properties change depending on the power functions of the thickness of the plate,such as linear,quadratic,cubic,and inverse quadratic.By utilizing Hamilton’s principle,general formulae of the wave propagation were obtained to establish wave modes and phase velocity curves of the wave propagation in a functionally graded plate,including the effects of changing compositional characteristics of materials.

The tangential k-Cauchy-Fueter type operator and Penrose type integral formula on the generalized complex Heisenberg group

The tangential k-Cauchy-Fueter operator and k-CF functions are counterparts of the tangential Cauchy–Riemann operator and CR functions on the Heisenberg group in the theory of several complex variables,respectively.In this paper,we introduce a Lie group that the Heisenberg group can be imbedded into and call it generalized complex Heisenberg.We investigate quaternionic analysis on the generalized complex Heisenberg.We also give the Penrose integral formula for k-CF functions and construct the tangential k-Cauchy-Fueter complex.

Existence of Solutions of a Convolution Integral Equation

In this study, we prove the of existence of solutions of a convolution Volterra integral equation in the space of the Lebesgue integrable function on the set of positive real numbers and with the standard norm defined on it. An operator P was assigned to the convolution integral operator which was later expressed in terms of the superposition operator and the nonlinear operator. Given a ball Br belonging to the space L it was established that the operator P maps the ball into itself. The Hausdorff measure of noncompactness was then applied by first proving that given a set M∈ B r the set is bounded, closed, convex and nondecreasing. Finally, the Darbo fixed point theorem was applied on the measure obtained from the set E belonging to M. From this application, it was observed that the conditions for the Darbo fixed point theorem was satisfied. This indicated the presence of at least a fixed point for the integral equation which thereby implying the existence of solutions for the integral equation.

Solvability and Construction of a Solution to the Fredholm Integral Equation of the First Kind

The issues of solvability and construction of a solution of the Fredholm integral equation of the first kind are considered. It is done by immersing the original problem into solving an extremal problem in Hilbert space. Necessary and sufficient conditions for the existence of a solution are obtained. A method of constructing a solution of the Fredholm integral equation of the first kind is developed. A constructive theory of solvability and construction of a solution to a boundary value problem of a linear integrodifferential equation with a distributed delay in control, generated by the Fredholm integral equation of the first kind, has been created.

Fuzzy Henstock-Kurzweil Triple Integral on a Type 1 Quasi-Fuzzy Parallelepipedal Domain

In this article, we propose by using the Hausdorff distance Simpson’s rule for the triple integral of a fuzzy-valued function and the error bound of this method, one of the variables of which is fuzzy. In addition, thin δ-fine partitions are introduced. The integration domain is a quasi-fuzzy parallelipiped. A numerical example is presented in order to show the application and the significance of the method.

Trajectory tracking guidance of interceptor via prescribed performance integral sliding mode with neural network disturbance observer

This paper investigates interception missiles’trajectory tracking guidance problem under wind field and external disturbances in the boost phase.Indeed,the velocity control in such trajectory tracking guidance systems of missiles is challenging.As our contribution,the velocity control channel is designed to deal with the intractable velocity problem and improve tracking accuracy.The global prescribed performance function,which guarantees the tracking error within the set range and the global convergence of the tracking guidance system,is first proposed based on the traditional PPF.Then,a tracking guidance strategy is derived using the integral sliding mode control techniques to make the sliding manifold and tracking errors converge to zero and avoid singularities.Meanwhile,an improved switching control law is introduced into the designed tracking guidance algorithm to deal with the chattering problem.A back propagation neural network(BPNN)extended state observer(BPNNESO)is employed in the inner loop to identify disturbances.The obtained results indicate that the proposed tracking guidance approach achieves the trajectory tracking guidance objective without and with disturbances and outperforms the existing tracking guidance schemes with the lowest tracking errors,convergence times,and overshoots.

Application of Choquet Integral-Importance-Performance Analysis and TOPSIS Methods in Approaching the Preference Factors of Calligraphy and Seal Engraving Imagery

Classical Chinese characters,presented through calligraphy,seal engraving,or painting,can exhibit different aesthetics and essences of Chinese characters,making them the most important asset of the Chinese people.Calligraphy and seal engraving,as two closely related systems in traditional Chinese art,have developed through the ages.Due to changes in lifestyle and advancements in modern technology,their original functions of daily writing and verification have gradually diminished.Instead,they have increasingly played a significant role in commercial art.This study utilizes the Evaluation Grid Method(EGM)and the Analytic Hierarchy Process(AHP)to research the key preference factors in the application of calligraphy and seal engraving imagery.Different from the traditional 5-point equal interval semantic questionnaire,this study employs a non-equal interval semantic questionnaire with a golden ratio scale,distinguishing the importance ratio of adjacent semantic meanings and highlighting the weighted emphasis on visual aesthetics.Additionally,the study uses Importance-Performance Analysis(IPA)and Technique for Order Preference by Similarity to Ideal Solution(TOPSIS)to obtain the key preference sequence of calligraphy and seal engraving culture.Plus,the Choquet integral comprehensive evaluation is used as a reference for IPA comparison.It is hoped that this study can provide cultural imagery references and research methods,injecting further creativity into industrial design.

经颅磁声电刺激对健康与帕金森病大鼠神经回路中beta振荡的影响

背景:经颅磁声电刺激是一种基于磁声耦合电效应的无创、高精度脑神经聚焦刺激方法,可调节神经节律振荡活动,从而影响大脑的运动、认知等功能。目的:探究经颅磁声电刺激对健康与帕金森病大鼠神经回路中beta振荡的影响。方法:①动物实验:采用随机数字表法将24只Wistar大鼠随机分为4组,每组6只:正常对照组不进行任何干预;正常刺激组大脑前额叶皮质接受经颅磁声电刺激(空间峰值脉冲平均强度为13.33 W/cm^(2),基波频率为0.4 MHz,基波周期数为1000,脉冲重复频率为200 Hz);模型对照组、模型刺激组通过腹腔注射1-甲基-4-苯基-1,2,3,6-四氢吡啶建立帕金森病模型,造模成功后,模型对照组大脑前额叶皮质接受经颅磁声电假刺激,模型刺激组大脑前额叶皮质接受经颅磁声电刺激,每天刺激时长2.0 min,刺激结束停留8-10 min后,采集大鼠执行T迷宫过程中局部场电位信号并同时记录行为学正确率,对比分析各组局部场电位信号能量的时频分布情况和行为差异,当大鼠正确率连续3 d高于80%后停止刺激实验和T迷宫实验。②建模仿真实验:构建经颅磁声电刺激下皮质-基底神经节回路模型,分别改变超声发射周期(5,10,20 ms)、超声发射占空比(30%,50%,90%)和感应电流密度(20,50,100μA/cm^(2)),比较不同刺激参数下健康与帕金森病大鼠beta振荡的功率谱密度值。结果与结论:①动物实验:正常对照组大鼠的空间学习能力强于模型对照组(P<0.001),正常刺激组大鼠的空间学习能力强于正常对照组(P<0.05),模型刺激组大鼠的空间学习能力强于模型对照组(P<0.01);正常对照组beta振荡能量分布较为集中,正常刺激组较正常对照组beta振荡信号能量有所降低,模型对照组与模型刺激组beta振荡能量广泛分布且能量值显著高于正常对照组、正常刺激组,并且模型刺激组beta振荡信号能量明显低于模型对照组;②建模仿真实验:不加刺激时,健康大鼠beta频段功率谱密度峰值(30 dB)显著低于帕金森病大鼠(55 dB);施加经颅磁声电刺激后,两组大鼠beta频段功率谱密度值普遍降低;beta频段功率谱密度峰值与超声发射周期呈正相关、与感应电流密度呈负相关,当超声发射占空比为50%时功率谱密度峰值最低;③结果表明:经颅磁声电刺激可抑制健康与帕金森病大鼠的beta振荡,进而改善大鼠的运动功能与决策认知功能。

草甸草原植物beta多样性高光谱遥感估算方法

目前,全球生物多样性的状态非常堪忧,因此,利用光谱技术估算生物多样性成为了生态学家和遥感学家共同关注的热点。目前关于植物alpha多样性的研究很多,而关于beta多样性的研究较少,仍存在一些问题值得去探索。为了探究利用遥感技术估算植物beta多样性的最佳光谱指数以及最佳影像空间分辨率,以草甸草原为研究区,基于无人机高光谱遥感影像,从光谱距离、光谱角度、生物多样性概念三个方面计算了6种beta多样性估算指数(4种为我们构建的新指数,2种为已有的指数),并采用Mantel tests和相关系数筛选最佳的光谱指数。然后将筛选出来的指数应用于不同空间分辨率的影像,以期得到最佳观测尺度。另外,为了提高指数的估算能力,对比了一阶导数变换和Savitzky-Golay滤波两种光谱变换方法,以及相关系数法、连续投影法、竞争性自适应重加权法三种特征波段选择方法。结果表明,不论是采用亚尺度观测(像元大小<样方大小)还是等尺度观测(像元大小=样方大小),最佳的光谱指数均为光谱距离指数,且光谱距离指数在不同影像空间分辨率下均表现良好。在草原地区,当影像空间分辨率约为0.25 m时,该指数可以取得最佳的估算结果。经一阶导数变换并用相关系数法提取特征波段后构建的光谱距离指数与beta多样性拥有最强的相关性,今后可利用该指数构建估算模型或者直接表征beta多样性。该研究对于科学的选取光谱指数和影像空间分辨率去估算植物beta多样性具有一定的指导意义。

晶种辅助蒸气转化Al-SBA-15合成Beta纳米晶团聚体

采用晶种辅助蒸气转化(SASC)策略,将有序介孔分子筛Al-SBA-15颗粒直接转化为Beta纳米晶团聚体(BNA).除规则微孔外,BNA含有2种多级介孔结构.研究了骨架组装与多级孔结构的关系.通过调节热分解条件,将P123胶束选择性分解成碳颗粒并分散于Al-SBA-15介孔中,在颗粒内部构建了均匀的三维纳米空间网格.通过浸渍Beta晶种和四乙基铵阳离子(TEA+)强化分子筛成核,采用蒸气处理方法,将Al-SBA-15无定形孔壁在纳米空间中缓慢转化为Beta纳米晶体.通过调节合成参数,使纳米晶体沿Al-SBA-15孔壁方向规则排列,形成有序介观结构.此外,经高温焙烧去除碳颗粒后,BNA内部生成了穴状介孔.实验结果表明,低的溶剂用量可显著降低传质速度,提高骨架组装质量,从而使BNA酸性增强.优异的多级孔结构和较强的酸性使BNA在大分子反应中具有应用潜力.

高原鼠兔干扰对高寒草甸植物物种和功能性状beta多样性的影响

采用野外空间多点同步取样,分析了高原鼠兔干扰对高寒草甸植物物种beta多样性和植物功能性状beta多样性的影响,确定了高原鼠兔干扰下高寒草甸植物物种和功能性状beta多样性的变化途径,分别提出了高原鼠兔干扰区域内,基于植物物种多样性和功能性状多样性的高寒草甸植物多样性维持策略。结果表明,高原鼠兔干扰使高寒草甸植物物种相似性显著降低了28.1%,植物功能相似性降低了28.7%。尽管高原鼠兔干扰没有改变高寒草甸植物物种和功能性状beta多样性的变化途径,且对植物物种和功能性状的嵌套组分不存在显著影响,但高原鼠兔干扰显著降低了植物物种和功能性状周转组分所占的比例,降幅分别为36.6%和34.3%。高原鼠兔干扰区域内,高寒草甸植物物种beta多样性的变化以周转为主导(周转占比81.4%;嵌套占比:18.6%),植物功能性状beta多样性的变化以嵌套为主导(嵌套占比64.9%;周转占比35.1%)。因此,针对划定的高原鼠兔干扰区,需要同时保护区域内所有高原鼠兔栖息地(多位点保护),以达到维持植物物种多样性的目的,而可以仅通过保护该区域内植物功能性状丰富的位点,即可维持较高的植物功能多样性。

Integral imaging-based tabletop light field 3D display with large viewing angle

Light field 3D display technology is considered a revolutionary technology to address the critical visual fatigue issues in the existing 3D displays.Tabletop light field 3D display provides a brand-new display form that satisfies multi-user shared viewing and collaborative works,and it is poised to become a potential alternative to the traditional wall and portable display forms.However,a large radial viewing angle and correct radial perspective and parallax are still out of reach for most current tabletop light field 3D displays due to the limited amount of spatial information.To address the viewing angle and perspective issues,a novel integral imaging-based tabletop light field 3D display with a simple flat-panel structure is proposed and developed by applying a compound lens array,two spliced 8K liquid crystal display panels,and a light shaping diffuser screen.The compound lens array is designed to be composed of multiple three-piece compound lens units by employing a reverse design scheme,which greatly extends the radial viewing angle in the case of a limited amount of spatial information and balances other important 3D display parameters.The proposed display has a radial viewing angle of 68.7°in a large display size of 43.5 inches,which is larger than the conventional tabletop light field 3D displays.The radial perspective and parallax are correct,and high-resolution 3D images can be reproduced in large radial viewing positions.We envision that this proposed display opens up possibility for redefining the display forms of consumer electronics.

响应面模型用于Beta分子筛催化剂蜂窝状整体式成型的研究

Beta分子筛由于具有独特的三维十二元环孔道结构,可提供更多活性组分附着位点,是在实际应用中具有优异性能的工业催化剂。在工业使用中,蜂窝整体式催化剂具有通量高、流场均匀、压降小、不易堵塞的优点,是气固或者液固催化反应的最优解决方案。但由于Beta分子筛吸水率高,收缩率大,导致蜂窝成型困难,严重限制了Beta分子筛的工业应用。基于此,选择拟薄水铝石(SB粉)为黏结剂、硝酸为胶溶剂、田菁粉为助挤剂,在水粉比为0.6的条件下制备一系列蜂窝整体式成型Beta分子筛催化剂,基于响应面法,评价了各原料的配比对蜂窝整体式催化剂机械强度的影响,得到了催化剂的机械强度与3个因素(黏结剂用量、胶溶剂用量、助挤剂用量)之间关系的二阶多项式近似模型函数。将试验结果与预测值进行对比,证明了预测模型的可靠性,并利用该模型找到最优的Beta分子筛蜂窝成型条件为SB粉质量分数36.1%、硝酸质量分数8.4%、田菁粉质量分数4.7%。

Intelligent Small Sample Defect Detection of Concrete Surface Using Novel Deep Learning Integrating Improved YOLOv5

Dear Editor,This letter presents an intelligent small sample defect detection of concrete surface using novel deep learning integrating the improved YOLOv5 based on the Wasserstein GAN(WGAN)enhancement algorithm.The proposed method is capable of producing top-notch data sets to address the issues of insufficient samples and substandard quality.