Local quasi-interpolation by cubic C1 splines on type-6 tetrahedral partitions

^** Email: sorokina{at}math.uga.edu^*** Corresponding author. Email: zeilfeld{at}euklid.math.uni-mannheim.de We describe an approximating scheme based on cubic C¹ splineson type-6 tetrahedral partitions using data on volumetric grids.The quasi-interpolating piecewise polynomials are directly determinedby setting their Bernstein–Bézier coefficientsto appropriate combinations of the data values. Hence, eachpolynomial piece of the approximating spline is immediatelyavailable from local portions of the data, without using prescribedderivatives at any point of the domain. The locality of themethod and the uniform boundedness of the associated operatorprovide an error bound, which shows that the approach can beused to approximate and reconstruct trivariate functions. Simultaneously,we show that the derivatives of the quasi-interpolating splinesyield nearly optimal approximation order. Numerical tests withup to 17 x 10⁶ data sites show that the method can be used forefficient approximation.     

Optimal constrained polynomials approximation of hyperbolas based on Lupaş q‐Bézier curves

This paper presents an explicit optimal polynomial for approximating the quadratic Lupaş q‐Bézier curve. We first prove that the quadratic Lupaş q‐Bézier curve represents a hyperbola or a parabola. Then we research the approximation of quadratic Lupaş q‐Bézier curves by polynomials. Since the denominator of quadratic Lupaş q‐Bézier curves is a linear function, the explicit optimal constrained approximation is obtained. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed method.     

On a problem of Duke–Erdős–Rödl on cycle-connected subgraphs

In this short note, we prove that for β<1/5 every graph G with n vertices and n^2−β edges contains a subgraph G^′ with at least cn^2−2β edges such that every pair of edges in G^′ lie together on a cycle of length at most 8. Moreover edges in G^′ which share a vertex lie together on a cycle of length at most 6. This result is best possible up to the constant factor and settles a conjecture of Duke, Erdős, and Rödl.     

Necessary conditions in Phragmén–Lindelöf type estimates and decomposition of holomorphic functions

We study Phragmén–Lindelöf type estimates for subharmonic and holomorphic functions in one variable and discuss the relation between the two cases. Our main result gives for the type of results which we consider necessary and almost sufficient conditions under which these principles hold.     

Existence of positive solutions for integral boundary value problems of fractional differential equations on infinite interval

In this paper, we consider a class of nonlinear fractional differential equations on the infinite interval with the integral boundary conditions By using Krasnoselskii fixed point theorem, the existence results of positive solutions for the boundary value problem in three cases and , are obtained, respectively. We also give out two corollaries as applications of the existence theorems and some examples to illustrate our results. Copyright © 2016 John Wiley & Sons, Ltd.