SOME INEQUALITIES OF HERMITE-HADAMARD TYPE FOR s-CONVEX FUNCTIONS

In this paper several inequalities of the left-hand side of Hermite-Hadamard’s inequality are obtained for s-convex functions.

A comparison of piecewise cubic Hermite interpolating polynomials,cubic splines and piecewise linear functions for the approximation of projectile aerodynamics

Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and nu- merical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set, and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the short- coming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a gl-cubic Hermite in- terpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global G2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an al- ternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15（1） （2007）, pp. 41-53].

Comprehensive Reliability Allocation Method for CNC Lathes Based on Cubic Transformed Functions of Failure Mode and Effects Analysis

Reliability allocation of computerized numerical controlled（CNC）lathes is very important in industry.Traditional allocation methods only focus on high-failure rate components rather than moderate failure rate components,which is not applicable in some conditions.Aiming at solving the problem of CNC lathes reliability allocating,a comprehensive reliability allocation method based on cubic transformed functions of failure modes and effects analysis（FMEA）is presented.Firstly,conventional reliability allocation methods are introduced.Then the limitations of direct combination of comprehensive allocation method with the exponential transformed FMEA method are investigated.Subsequently,a cubic transformed function is established in order to overcome these limitations.Properties of the new transformed functions are discussed by considering the failure severity and the failure occurrence.Designers can choose appropriate transform amplitudes according to their requirements.Finally,a CNC lathe and a spindle system are used as an example to verify the new allocation method.Seven criteria are considered to compare the results of the new method with traditional methods.The allocation results indicate that the new method is more flexible than traditional methods.By employing the new cubic transformed function,the method covers a wider range of problems in CNC reliability allocation without losing the advantages of traditional methods.

Generalizations of Hermite-Hadamard Type Inequalities Involving S-convex Functions

Some new Hermite-Hadamard type's integral equations and inequalities are established. The results in [3] and [6] which refined the upper bound of distance between the middle and left of the typical Hermite-Hadamard's integral inequality are generalized.

Stability of a cubic functional equation in intuitionistic random normed spaces

In this paper, the stability of a cubic functional equation in the setting of intuitionistic random normed spaces is proved. We first introduce the notation of intuitionistic random normed spaces. Then, by virtue of this notation, we study the stability of a cubic functional equation in the setting of these spaces under arbitrary triangle norms. Furthermore, we present the interdisciplinary relation among the theory of random spaces, the theory of intuitionistic spaces, and the theory of functional equations.

Approximation of the Cubic Functional Equations in Lipschitz Spaces

Let G be an Abelian group and letρ：G×G→[0,∞） be a metric on G. Let E be a normed space. We prove that under some conditions if f：G→E is an odd function and Cx：G→E defined by Cx（y）：=2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y） is a cubic function for all x∈G, then there exists a cubic function C：G→E such that f？C is Lipschitz. Moreover, we investigate the stability of cubic functional equation 2 f （x＋y）＋2 f （x-y）＋12 f （x）-f （2x＋y）-f （2x-y）=0 on Lipschitz spaces.

AN INTEGRATION METHOD WITH FITTING CUBIC SPLINE FUNCTIONS TO A NUMERICAL MODEL OF 2ND-ORDER SPACE-TIME DIFFERENTIAL REMAINDER——FOR AN IDEAL GLOBAL SIMULATION CASE WITH PRIMITIVE ATMOSPHERIC EQUATIONS

In this paper,the forecasting equations of a 2nd-order space-time differential remainder are deduced from the Navier-Stokes primitive equations and Eulerian operator by Taylor-series expansion.Here we introduce a cubic spline numerical model(Spline Model for short),which is with a quasi-Lagrangian time-split integration scheme of fitting cubic spline/bicubic surface to all physical variable fields in the atmospheric equations on spherical discrete latitude-longitude mesh.A new algorithm of"fitting cubic spline—time step integration—fitting cubic spline—……"is developed to determine their first-and2nd-order derivatives and their upstream points for time discrete integral to the governing equations in Spline Model.And the cubic spline function and its mathematical polarities are also discussed to understand the Spline Model’s mathematical foundation of numerical analysis.It is pointed out that the Spline Model has mathematical laws of"convergence"of the cubic spline functions contracting to the original functions as well as its 1st-order and 2nd-order derivatives.The"optimality"of the 2nd-order derivative of the cubic spline functions is optimal approximation to that of the original functions.In addition,a Hermite bicubic patch is equivalent to operate on a grid for a 2nd-order derivative variable field.Besides,the slopes and curvatures of a central difference are identified respectively,with a smoothing coefficient of 1/3,three-point smoothing of that of a cubic spline.Then the slopes and curvatures of a central difference are calculated from the smoothing coefficient 1/3 and three-point smoothing of that of a cubic spline,respectively.Furthermore,a global simulation case of adiabatic,non-frictional and"incompressible"model atmosphere is shown with the quasi-Lagrangian time integration by using a global Spline Model,whose initial condition comes from the NCEP reanalysis data,along with quasi-uniform latitude-longitude grids and the so-called"shallow atmosphere"Navier-Stokes primitive equations in the spherical coordinates.The Spline Model,which adopted the Navier-Stokes primitive equations and quasi-Lagrangian time-split integration scheme,provides an initial ideal case of global atmospheric circulation.In addition,considering the essentially non-linear atmospheric motions,the Spline Model could judge reasonably well simple points of any smoothed variable field according to its fitting spline curvatures that must conform to its physical interpretation.

Cost Effective Operating Strategy for Unit Commitment and Economic Dispatch of Thermal Power Plants with Cubic Cost Functions Using TLBO Algorithm

This paper deals with a Unit Commitment (UC) problem of a power plant aimed to find the optimal scheduling of the generating units involving cubic cost functions. The problem has non convex generator characteristics, which makes it very hard to handle the corresponding mathematical models. However, Teaching Learning Based Optimization (TLBO) has reached a high efficiency, in terms of solution accuracy and computing time for such non convex problems. Hence, TLBO is applied for scheduling of generators with higher order cost characteristics, and turns out to be computationally solvable. In particular, we represent a model that takes into account the accurate higher order generator cost functions along with ramp limits, and turns to be more general and efficient than those available in the literature. The behavior of the model is analyzed through proposed technique on modified IEEE-24 bus system.

GREEN’S FUNCTION AND EFFECTIVE ELASTIC STIFFNESS TENSOR FOR ARBITRARY AGGREGATES OF CUBIC CRYSTALS

A closed but approximate formula of Green’s function for an arbitrary aggregate of cubic crystallites is given to derive the e?ective elastic sti?ness tensor of the polycrystal. This formula, which includes three elastic constants of single cubic crystal and ?ve texture coe?cients, accounts for the e?ects of the orientation distribution function (ODF) up to terms linear in the tex- ture coe?cients. Thus it is expected that our formula would be applicable to arbitrary aggregates with weak texture or to materials such as aluminum whose single crystal has weak anisotropy. Three examples are presented to compare predictions from our formula with those from Nishioka and Lothe’s formula and Synge’s contour integral through numerical integration. As an applica- tion of Green’s function, we brie?y describe the procedure of deriving the e?ective elastic sti?ness tensor for an orthorhombic aggregate of cubic crystallites. The comparison of the computational results given by the ?nite element method and our e?ective elastic sti?ness tensor is made by an example.

OPTIMAL ERROR BOUNDS FOR THE CUBIC SPLINE INTERPOLATION OF LOWER SMOOTH FUNCTIONS(1)

In this paper,the kernel of the cubic spline interpolation is given.An optimal error bound for the cu- bic spline interpolation of lower smooth functions is obtained.

Some Integral Inequalities of Simpson Type for Strongly Extended s-Convex Functions

The main purpose of this survey paper is to point out some very recent developments on Simpson’s inequality for strongly extended s-convex function. Firstly, the concept of strongly extended s-convex function is introduced. Next a new identity is also established. Finally, by this identity and H?lder’s inequality, some new Simpson type for the product of strongly extended s-convex function are obtained.

Source Quantitative Identification by Reference-Based Cubic Blind Deconvolution Algorithm

The semi-blind deconvolution algorithm improves the separation accuracy by introducing reference information.However,the separation performance depends largely on the construction of reference signals.To improve the robustness of the semi-blind deconvolution algorithm to the reference signals and the convergence speed,the reference-based cubic blind deconvolution algorithm is proposed in this paper.The proposed algorithm can be combined with the contribution evaluation to provide trustworthy guidance for suppressing satellite micro-vibration.The normalized reference-based cubic contrast function is proposed and the validity of the new contrast function is theoretically proved.By deriving the optimal step size of gradient iteration under the new contrast function,we propose an efficient adaptive step optimization method.Furthermore,the contribution evaluation method based on vector projection is presented to implement the source contribution evaluation.Numerical simulation analysis is carried out to validate the availability and superiority of this method.Further tests given by the simulated satellite experiment and satellite ground experiment also confirm the effectiveness.The signals of control moment gyroscope and flywheel were extracted,respectively,and the contribution evaluation of vibration sources to the sensitive load area was realized.This research proposes a more accurate and robust algorithm for the source separation and provides an effective tool for the quantitative identification of the mechanical vibration sources.

OPTIMAL ERROR BOUNDS FOR THE CUBIC SPLINE INTERPOLATION OF LOWER SMOOTH FUNCTION(Ⅱ)

Abstract In this paper, by using the explicit expression of the kernel of the cubic spline interpolation, the optimal error bounds for the cubic spline interpolation of lower soomth functions are obtained.

采用三次样条函数拟合蚊媒密度概率分布及其风险评估

目的研究登革热蚊媒MOI风险的概率分布,为更加科学精准开展MOI风险评估提供新方法。方法采用三次样条函数、累积概率分布和Python语言编程对广州市2016—2019年各月MOI监测数据进行拟合分析。结果本研究绘制了广州市1-12月的MOI风险评估概率分布表和分布图。1-12月各月累积概率小于0.01和0.05的MOI临界值分别在7.54~60.12、5.02~34.10之间。该临界值均在1月最低,其次是12月和2月;均在6月份最高,其次是5月和7月。1-12月各月MOI风险等级为传播、暴发和流行的概率分别在5.07%~66.60%、0.25%~36.19%、0.00%~16.80%之间。传播、暴发和流行的风险概率最低值均出现在1月,其次均在2月;最高值分别出现在7月、7月和6月,其次是6月、6月和5月。结论广州市各月的MOI风险概率分布具有显著的季节性消长规律和特征,在开展登革热蚊媒监测评估与控制工作中值得重视。三次样条函数和累积概率分布拟合法为开展MOI风险评估提供了新的分析方法。

基于改进RRT算法的移动机器人路径规划研究

快速扩展随机树算法(rapidly-exploring random trees,RRT)规划移动机器人路径时,存在搜索盲目性强、搜索时间长、收敛速度慢、路径冗余点多且不平滑等问题。鉴于此,提出一种改进的RRT路径规划算法。首先,针对传统RRT算法盲目搜索以及局部极值的问题,提出概率目标偏置与人工势场结合的采样策略,引导随机树的扩展;其次,针对随机树扩展的避障能力差的问题,提出基于安全距离的碰撞检测以及动态变步长扩展策略;最后,针对路径上冗余点多以及曲率不连续的问题,提出考虑安全距离的剪枝优化和三次B样条曲线对初始路径进行拟合优化。仿真结果表明,在不同地图的路径规划中,相比于传统RRT算法,增强了通过狭窄通道能力,优化了路径的平滑性,搜索时间、迭代次数、路径长度分别减少约70%、40%、15%;相比于RRT衍生算法RRT-Connect,搜索时间、路径长度分别减少约25%、10%。

膳食叶黄素和玉米黄质与老年人低认知功能的剂量-反应关系

目的 使用限制性立方样条回归模型探讨膳食叶黄素和玉米黄质摄入量与老年人低认知功能的剂量-反应关系。方法 基于美国国家健康与营养调查(National Health and Nutrition Examination Survey, NHANES)数据库,选取年龄≥60岁且具有完整认知功能测试的老年人群。应用Logistic回归模型和限制性立方样条回归模型评估膳食叶黄素和玉米黄质摄入量与老年人低认知功能的关联和剂量-反应关系。结果 共纳入2 695名研究对象,其中认知功能正常组1 629例,低认知功能组1 066例。与认知功能正常组比较,低认知功能组年龄在70~<80岁和≥80岁的人群所占比例明显较高,糖尿病及中风发生比例也明显增高,此外,膳食叶黄素和玉米黄质摄入量以及叶黄素和玉米黄质总摄入量明显降低,差异均有统计学意义(P<0.05)。限制性立方样条回归分析显示,以性别分层,随着叶黄素和玉米黄质总摄入量的持续增加,男性和女性认知功能下降风险均呈近似“L”形的非线性变化(P非线性<0.01),但是在同样“L”形趋势的情况下,叶黄素和玉米黄质总摄入量是男性认知功能下降的易感因素。结论 叶黄素和玉米黄质总摄入量与老年人认知功能下降风险呈近似“L”形的非线性剂量-反应关系。

基于Roberts算子的激光夜视图像自动分割方法

激光夜视图像分辨率较低,边缘信息难以检测,导致图像分割过程存在效率低、精度较差等问题,为此设计基于Roberts算子的激光夜视图像自动分割方法。使用相机采集夜间场景色彩信息,采用加权平均法重构图像数据,输出完整的激光夜视图像。拟合图像周边相邻像素,利用三次样条插值函数平滑处理图像,运用Roberts算子计算图像梯度,实现激光夜视图像边缘检测,得到图像插值后的平滑信息,提升夜视图像分辨率。通过模糊C均值聚类明确聚类范围,引入混沌粒子群算法实现图像分割。实验结果表明,所提方法分割差异率在3%以下,图像分割Dice相似性系数在0.97以上,图像分割平均耗时为9 s。

基于样条权函数神经网络的人脸识别研究

文章对人脸识别技术进行了调查,对比分析了传统识别技术以及基于深度学习识别技术的不同,指出由于深度学习的网络结构较为复杂,在人脸识别过程中容易出现算法过拟合的现象。针对上述过拟合问题,文章对基于权函数神经网络的人脸识别算法进行了研究,此类神经网络结构简单,仅有输入、输出两层结构,将人脸图片作为输入信息,网络训练目标为计算三次样条权函数,仿真实验证明该类网络具有较好的识别准确度和识别效率。

REFINEMENTS OF TRANSFORMATION INEQUALITIES FOR ZERO-BALANCED HYPERGEOMETRIC FUNCTIONS

In the article, we present some refinements of three classes of transformation inequalities for zero-balanced hypergeometric functions by use of the updated monotonicity criterion for the quotient of power series.