Two hyperfinite constructions of the brownian bridge

Infinitesimal Analysis is used to give two constructions of the brownian bridge process. In the first construction a hyperfinite tied down random walk is used and a brownian bridge is obtained via the standard part map. As a consequence it is shown that the brownian bridge is the weak limit of a sequence of normalized tied down random walks. The second construction is based on a hyperfinite uniform empirical process. This construction gives an almost trivial proof of Donsker's Invariance Principle for the uniform empirical process     

Assessing the heterogeneity of a disease spread through a community

A multivariate non-parametric test and a semi-parametric regression model via counting process are proposed for detecting the heterogeneity of a disease spread through a community. The infection rates are allowed to depend on time in an arbitrary manner. Infectious data usually are not completely observed, nevertheless only partial information of the epidemic is needed for the suggested methods. The testing procedures and the associated methods of analysis are illustrated with reference to epidemics of respiratory disease on the Island of Tristan da Cunha in the South Atlantic     

Solvability of the clamped Euler–Bernoulli beam equation

In this study, solvability of the initial boundary value problem for general form Euler–Bernoulli beam equation which includes also moving point-loads is investigated. The complete proof of an existence and uniqueness properties of the weak solution of the considered equation with Dirichlet type boundary conditions is derived. The method used here is based on Galerkin approximation which is the main tool for the weak solution theory of linear evolution equations as well as in derivation of a priori estimate for the approximate solutions. All steps of the proposed technique are explained in detail.     

A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation

This paper presents a direct solution technique for solving the generalized pantograph equation with variable coefficients subject to initial conditions, using a collocation method based on Bernoulli operational matrix of derivatives. Only small dimension of Bernoulli operational matrix is needed to obtain a satisfactory result. Numerical results with comparisons are given to confirm the reliability of the proposed method for generalized pantograph equations.     

A heuristic approach for deriving the priority vector in AHP

This paper proposes an approach for deriving the priority vector from an inconsistent pair-wise comparison matrix through the nearest consistent matrix and experts judgments, which enables balancing the consistency and experts judgments. The developed algorithm for achieving a nearest consistent matrix is based on a logarithmic transformation of the pair-wise comparison matrix, and follows an iterative feedback process that identifies an acceptable level of consistency while complying with experts preferences. Three numerical examples are examined to illustrate applications and advantages of the developed approach.     

An exploration of students’ errors in derivatives in a university of technology

This paper reports on an exploration of errors that were displayed by students who studied mathematics in chemical engineering in derivatives of various functions such as algebraic, exponential, logarithmic and trigonometric functions. The participants of this study were a group of twenty students who were at risk in an extended curriculum programme in a university of technology in Western Cape, South Africa. The researcher used a qualitative case study approach and collected data from students’ written work. This research uses action, process, object, and schema (APOS) theory to classify errors into categories and to analyse and interpret the data collected. The students displayed five different kinds of errors, namely, conceptual, interpretation, linear extrapolation, procedural and arbitrary. The use of APOS theory as a framework revealed that several students’ errors might be caused by over-generalisation of mathematical rules and properties such as the power rule of differentiation and distributive property in manipulation of algebraic expressions. This study suggests that teaching of the standard rules of differentiation should put emphasis on its restrictions to eliminate common errors that normally crop up due to over-generalisation of certain differentiation rules.     

Comparing confidence limits for short-run process incapability index Cpp

Process incapability index C_pp has been proposed in the manufacturing industry to assess process incapability. In industries it is sometimes unable to get large samples, and, hence, the CAN (consistent and asymptotically normal) property of the unbiased estimator for C_pp is missing. In this paper, six bootstrap methods are applied to construct upper confidence bounds (UCBs) of C_pp for short-urn production processes where sample size is small; standard bootstrap (SB), Bayesian bootstrap (BB), bootstrap pivotal (BP), percentile bootstrap (PB), bias-corrected percentile bootstrap (BCPB), and bias-corrected and accelerated bootstrap (BCa). A numerical simulation study is conducted in order to demonstrate the performance of the six various estimation methods. We further investigate the accuracy of the six methods by calculating the relative coverage (defined as the ratio of coverage percentage to average length of UCB). Detailed discussions of simulation results for seven short-run processes are presented. Finally, one real example from Ford Company’s Windsor Casting Plant is used to illustrate the six interval estimation methods.     

The optimization of hydrothermal process of MoS_2 nanosheets and their good microwave absorption performances

In this study,flower-like MoS_2 constructed by nanosheets was synthesized by a simple hydrothermal method.The hydrothermal process was optimized and the effects of hydrothermal condition,including reaction temperature,reaction time and the ratio of Mo source to S source(Mo:S) in precursor,on microwave absorption performances and dielectric properties were investigated.Our results showed that when the reaction temperature was 180℃,the reaction time was 18 h,and the Mo:S was 1:3.5,the synthesized MoS_2 had the best performance:Its minimum reflection loss could reach-55.78 dB,and the corresponding matching thickness was 2.30 mm with a wide effective bandwidth of 5.17 GHz.Further researches on the microwave absorption mechanism revealed that in addition to the destructive interference of electromagnetic waves,various polarization phenomena such as defect dipole polarization were the main reasons for microwave loss.We believe that MoS_2 is a candidate for a practical microwave absorbent.