The contraction number of a multigrid method for solving the Poisson equation

Summary The treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence. For the linear equations which arise from the discretization of the Poisson equation, a convergence factor 0,5 is established independent of the shape of the domain and of the regularity of the solution.Dedicated to Professor Dr.Dr.h.c. Lothar Collatz on the occasion of his 70 th birthday     

A generalization of the Stein-Rosenberg theorem to Banach spaces

Summary In the first part of this note we prove a generalization of the Stein-Rosenberg theorem; the context is that of real Banach spaces with a normal reproducing cone and the operators involved are positive and completely continuous. Our generalization of the Stein-Rosenberg theorem improves the modern version of it as stated by F. Robert in [5, §2]. In the second part, we discuss briefly how our results are related to other versions of the Stein-Rosenberg theorem. In the last section we describe a situation to which the results in the first part can be applied.     

Effective computation of periodic orbits and bifurcation diagrams in delay equations

Summary By employing a numerical method which uses only rather classical tools of Numerical Analysis such as Newton's method and routines for ordinary differential equations, unstable periodic solutions of differential-difference equations can be computed. The method is applied to determine bifurcation diagrams with backward bifurcation.Dedicated to Professor Lothar Collatz on the occasion of his 70th birthdayThis paper has been read at the Conference of Numerical Mathematicians, Zeist, The Netherlands, October 12–15, 1979     

On the order of composite multistep methods for ordinary differential equations

Summary In this paper, a general class ofk-step methods for the numerical solution of ordinary differential equations is discussed. It is shown that methods with order of consistencyq have order of convergence (q+1) if a very simple condition is satisfied. This result gives a new aspect to previous results of Spijker; it also serves as a starting point for a new theory of cyclick-step methods, completing an approach of Donelson and Hansen. It facilitates the practical determination of high-order cyclick-step methods, especially of stiffly stable,k-step methods.     

Variational principles and convergence of finite element approximations of a holonomic elastic-plastic problem

Summary It is shown that a boundary-value problem based on a holonomic elastic-plastic constitutive law may be formulated equivalently as a variational inequality of the second kind. A regularised form of the problem is analysed, and finite element approximations are considered. It is shown that solutions based on finite element approximation of the regularised problem converge.     

Thirteen ways to estimate global error

Summary Various techniques that have been proposed for estimating the accumulated discretization error in the numerical solution of differential equations, particularly ordinary differential equations, are classified, described, and compared. For most of the schemes either an outline of an error analysis is given which explains why the scheme works or a weakness of the scheme is illustrated.This research is partially supported by NSF Grant No. MCS-8107046     

An iterative substructuring algorithm for equilibrium equations

Summary The topic of iterative substructuring methods, and more generally domain decomposition methods, has been extensively studied over the past few years, and the topic is well advanced with respect to first and second order elliptic problems. However, relatively little work has been done on more general constrained least squares problems (or equivalent formulations) involving equilibrium equations such as those arising, for example, in realistic structural analysis applications. The potential is good for effective use of iterative algorithms on these problems, but such methods are still far from being competitive with direct methods in industrial codes. The purpose of this paper is to investigate an order reducing, preconditioned conjugate gradient method proposed by Barlow, Nichols and Plemmons for solving problems of this type. The relationships between this method and nullspace methods, such as the force method for structures and the dual variable method for fluids, are examined. Convergence properties are discussed in relation to recent optimality results for Varga's theory ofp-cyclic SOR. We suggest a mixed approach for solving equilibrium equations, consisting of both direct reduction in the substructures and the conjugate gradient iterative algorithm to complete the computations.Dedicated to R. S. Varga on the occasion of his 60th birthdayResearch completed while pursuing graduate studies sponsored by the Department of Mathematical Sciences, US Air Force Academy, CO, and funded by the Air Force Institute of Technology, WPAFB, OHResearch supported by the Air Force under grant no. AFOSR-88-0285 and by the National Science Foundation under grant no. DMS-89-02121     

Application of Different Zeolites for Improvement of the Characteristics of a pH‐FET Biosensor Based on Immobilized Urease

Different modifications of the zeolites Na⁺‐Beta and LTA were applied for improving the working characteristics of a urea biosensor. The bioselective membrane of the biosensor was based on urease and different zeolites co‐immobilized with bovine serum albumin on the surface of a pH‐FET. It was shown that the biosensors modified with the zeolites H⁺‐Beta30 and H⁺‐Beta50 are characterized by increased sensitivity to urea. The influence of the zeolite concentration on the sensitivity of the biosensors was studied. The optimal concentration of the zeolites H⁺‐Beta30 and H⁺‐Beta50 in the bioselective membrane was 15 %. Different variants of co‐immobilization of urease and zeolite H⁺‐Beta30 were studied and the optimal method was selected. Thus, a general conclusion is that the urea biosensor sensitivity can be improved using zeolite H⁺‐Beta30 for urease immobilization in the bioselective membrane.     

Microwave-assisted sol-gel synthesis of CeO2–NiO nanocomposite based NO2 gas sensor for selective detection at lower operating temperature

The progress in the development of gas sensors has considerably grown using some novel nanomaterials of metal, metal oxide and composite. In the current study, we intended and evaluated the properties of nanomaterials like CeO₂, NiO, and CeO₂–NiO composite and its application as NO₂ gas sensor. Sensing of low concentration of NO₂ gas at optimum functional temperature was succeeded using CeO₂–NiO nanocomposites (NCs) film. The working temperature ranges in between 100 and 225 ?°C. Highly crystalline nanomaterials (CeO₂, NiO and CeO₂–NiO) have been prepared by applying microwave-assisted sol-gel route. The as-prepared nanomaterials are characterized for their structure, size, morphology and constitution by X-ray diffraction (XRD), scanning electron microscopy (SEM) and energy dispersive X-ray analysis. XRD studies of nanoparticles reveal the formation of nanoscale CeO₂ and NiO with crystallite size 26, 23 ?nm, respectively. Both are having a face centered cubic structure. The nanocomposite (NC) Ce:Ni ?= ?60:40 has crystallite size of 13 ?nm. XRD study of NCs shows assimilation of Ni metal into the ceria and proves physical similarities of two phases. It can be observed from SEM that prepared NC has a porous surface which enables more surface active sites for adsorbing oxygen. The optical properties are measured with the help of UV–Vis. Spectroscopy. Optical band gaps of 3.19, 3.41 and 2.9 ?eV were observed for CeO_2, NiO nanoparticles (NPs) and CeO₂–NiO NC, respectively. Gas sensing properties state that the NC material shows a higher gas response % of 67.34% for NO₂ gas (25 ?ppm) at comparatively low operating temperature (125 ?°C). It gives response time as (～28 ?s) and the recovery (～54 ?s). NiO incorporation in CeO₂ results in a decline of operating temperature of NC and improves the sensing features.     

Representation of Bounded Convex Sets By Rational Convex Hull Of Its Gamma-Extreme Points

H. Minkowski proved C= convext C whenever C is a compact convex subset of a finite-dimensional linear space. If C is bounded but not closed, this representation does not hold anymore. In this case, we introduce the set of so-called γ-extreme points extγC of C and show C = convext γ C = raco extγ C, where raco M denotes the rational convex hull of M.