Two inequalities for differentiable mappings and applications to weighted trapezoidal formula,weighted midpoint formula and random variable

In this paper, we establish two inequalities for differentiable convex mappings and differentiable concave mappings which are connected with Fejér’s inequality holding for convex mappings and concave mappings. Some error estimates for the weighted trapezoidal formula and the weighted midpoint formula are given.     

Computational aspects of Boolean cubature

Boolean methods of interpolation [1,4] have been applied to construct multivariate quadrature rules for periodic functions of Korobov classes which are comparable with lattice rules of numerical integration [6,7]. In particular, we introducedd-variate Boolean trapezoidal rules [3,4] andd-variate Boolean midpoint rules [2,4]. The basic tools for constructing Boolean midpoint rules are Boolean midpoint sums. It is the purpose of this paper to use a modification of these Boolean midpoint sums to compute Boolean trapezoidal rules in an efficient way.     

Fejer and Hermite-Hadamard type inequalities for superquadratic functions

In this paper we use basic properties of superquadratic functions to obtain new inequalities including Fejer's type and Hermite-Hadamard type inequalities. For superquadratic functions which are also convex, we get refinements of known results.     

Sharp Integral Inequalities of the Hermite–Hadamard Type

We consider a family of two-point quadrature formulae and establish sharp estimates for the remainders under various regularity conditions. Improved forms of certain integral inequalities due to Hermite and Hadamard, Iyengar, Milovanovi and Pecari , and others are obtained as special cases. Our results can also be interpreted as analogues to a theorem of Ostrowski on the deviation of a function from its averages. Furthermore, we establish a generalization of a result of Fink concerning L^p estimates for the remainder of the trapezoidal rule and present the best constants in the error bounds.     

Inequalities on the Triangle

The article deals with generalizations of the inequalities for convex functions on the triangle. The Jensen and the Hermite-Hadamard inequality are included in the study. Considering a convex function on the triangle, we obtain a generalization of the Jensen-Mercer inequality, and a refinement of the Hermite-Hadamard inequality.     

Higher-order Newton–Cotes rules with end corrections

We present higher-order quadrature rules with end corrections for general Newton–Cotes quadrature rules. The construction is based on the Euler–Maclaurin formula for the trapezoidal rule. We present examples with 6 well-known Newton–Cotes quadrature rules. We analyze modified end corrected quadrature rules, which consist on a simple modification of the Newton–Cotes quadratures with end corrections. Numerical tests and stability estimates show the superiority of the corrected rules based on the trapezoidal and the midpoint rules.     

Some generalizations of Hadamard’s-type inequalities through differentiability for s-convex functions and their applications

In this paper, a general form of integral inequalities of Hermite-Hadamard’s type through differentiability for s-convex function in second sense and whose all derivatives are absolutely continuous are established. The generalized integral inequalities contributes some better estimates than some already presented. The inequalities are then applied to numerical integration and some special means.     

Efficient transient noise analysis in circuit simulation

Stochastic differential algebraic equations (SDAEs) arise as a mathematical model for electrical network equations that are influenced by additional sources of Gaussian white noise. We discuss adaptive linear multi-step methods for their numerical integration, in particular stochastic analogues of the trapezoidal rule and the two-step backward differentiation formula, and we obtain conditions that ensure mean-square convergence of this methods. For the case of small noise we present a strategy for controlling the step-size in the numerical integration. It is based on estimating the mean-square local errors and leads to step-size sequences that are identical for all computed paths. Test results illustrate the performance of the presented methods. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Smoothing effects on the IMR and ITR

This paper is a study of the effects of smoothing on the implicit midpoint rule (IMR) and the implicit trapezoidal rule (ITR) with implications for extrapolation of the numerical solution of ordinary differential equations. We extend the study of the well-known smoothing formula of Gragg to a two-step smoothing formula and compare the effectiveness of their use with the IMR and ITR for nonstiff and strongly stiff cases. We present an analysis of the Prothero-Robinson problem and as well as experimental results on linear and nonlinear problems.     

逻辑系统'Luk中命题积分真度的若干等式与不等式

对' Lukasiewicz逻辑系统,利用序结构知识和赋值函数保并、交、补、蕴涵运算的性质研究了命题的积分真度,推出了若干关于积分真度的等式与不等式,修正完善了积分真度的交推理规则,给出了积分真度的等式与不等式的一些应用,使较复杂的积分真度计算得以简化,或进行较合理的估值.     

高精度数值积分公式的构造及其应用

通过对一个给定的数值积分公式进行加速、改进,得到了两类新的精度更高的数值积分公式.然后将其进行复合,得到复合公式,并将复合公式推广到计算二重积分.最后进行了数值实验,数值计算结果表明:两类新的数值积分公式都具有比给定的公式更高阶的精度和更快的收敛速度.     

Refining the Hermite-Hadamard inequalities for operator convex maps in multiple operator variables

Intensive studies aiming to extend the Hermite-Hadamard inequalities and to explore some properties and applications of these inequalities have been carried out recently. The contribution of this paper falls within this framework. We investigate here some refinements of the Hermite-Hadamard inequalities for operator convex maps involving multiple operator arguments.

     

BCn-symmetric polynomials

We consider two important families of BC_n-symmetric polynomials, namely Okounkov's interpolation polynomials and Koornwinder's orthogonal polynomials. We give a family of difference equations satisfied by the former as well as generalizations of the branching rule and Pieri identity, leading to a number of multivariate q-analogues of classical hypergeometric transformations. For the latter, we give new proofs of Macdonald's conjectures, as well as new identities, including an inverse binomial formula and several branching rule and connection coefficient identities. We also derive families of ordinary symmetric functions that reduce to the interpolation and Koornwinder polynomials upon appropriate specialization. As an application, we consider a number of new integral conjectures associated to classical symmetric spaces.     

Trapezoidal and midpoint splittings for initial-boundary value problems

In this paper we consider various multi-component splittings based on the trapezoidal rule and the implicit midpoint rule. It will be shown that an important requirement on such methods is internal stability. The methods will be applied to initial-boundary value problems. Along with a theoretical analysis, some numerical test results will be presented.

     

Some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables

Several inequalities for differentiable convex, wright-convex and quasi-convex mapping are obtained respectively that are connected with the celebrated Hermite-Hadamard integral inequality. Also, some error estimates for weighted Trapezoid formula and higher moments of random variables are given.     

The Hermite-Hadamard inequality in Beckenbach's setting

The classical Hermite-Hadamard inequality gives a lower and an upper estimations for the integral average of convex functions defined on compact intervals, involving the midpoint and the endpoints of the domain. The aim of the present paper is to extend this inequality and to give analogous results when the convexity notion is induced by Beckenbach families. The key tool of the investigations is based on some general support theorems that are obtained via the pure geometric properties of Beckenbach families and can be considered as generalizations of classical support and chord properties of ordinary convex functions. The Markov-Krein-type representation of Beckenbach families is also investigated.     

A general multidimensional Hermite-Hadamard type inequality

Using a stochastic approach, we establish a multidimensional version of the classical Hermite-Hadamard inequalities which holds for convex functions on general convex bodies. The result is closely related to the Dirichlet problem.     

Bilabelled increasing trees and hook-length formulae

We introduce two different kinds of increasing bilabellings of trees, for which we provide enumeration formulae. One of the bilabelled tree families considered is enumerated by the reduced tangent numbers and is in bijection with a tree family introduced by Poupard [11]. Both increasing bilabellings naturally lead to hook-length formulae for trees and forests; in particular, one construction gives a combinatorial interpretation of a formula for labelled unordered forests obtained recently by Chen et al. [1].     

On Trigonometric Blending Interpolation and Cubature Formulae

We discuss error representations for Hermite-Lagrange trigonometric interpolation introduced in Dryanov and Petrov (Interpolation and L ₁-approximation by trigonometric polynomials and blending functions, J. Approx. Theory 164, 1049–1064 (2012)) and obtain one-sided trigonometric quadratures for approximate integration of one-dimensional integrals. Next, we study error representations of multivariate Hermite-Lagrange transfinite trigonometric interpolation and derive one-sided trigonometric blending interpolants to multivariate functions, under some restrictions. Then, we construct one-sided transfinite cubature formulae for approximate integration of multivariate integrals. We construct also cubature formulae with positive coefficients, based on line integrals and exact in a vector space of trigonometric blending functions with prescribed order.     

Harmonic analysis for star graphs and the spherical coordinate trapezoidal rule

Novel ideas in harmonic analysis are used to analyze the trapezoidal rule integration for two spheres. Sampling in spherical coordinates links three levels of harmonic analysis. Eigenfunctions of a nonstandard manifold Laplacian descend by restriction, first to a differential graph Laplacian, and then to difference operators. Trapezoidal rule integration with appropriate sampling is exact on eigenspaces of the manifold Laplacian, a fact which leads to trapezoidal rule error estimates on Sobolev-style spaces of functions. Singular functions with accurate trapezoidal rule integrals are identified, and a simplified analysis of smooth function numerical integration is provided.