Many experimental problems in the natural sciences result in data which can best be represented by linear combinations of exponentials of the form f(t) = ∑[with p above and k=1 below] α_k e^-λ_kt. Among such problems are those dealing with growth, decay, ion concentration, and survival and mortality. Also, in general, the solution to any problem which may be represented by linear differential equations with constant coefficients is a linear combination of exponentials. In most problems like those which have been mentioned, the parameters α_k and λ_k have biological or physical significance. Therefore, in fitting the. function f(t) to the data it is not only necessary that the function approximate the data closely, but it is also necessary that the parameters α_k and λ_k be accurately estimated. Furthermore, a measure of the accuracy of the estimation of the parameters is required. A new estimation procedure for linear combinations of exponentials is developed in this paper. Unlike the iterative maximum likelihood and least-squares methods for estimating the parameters for such a model, the new procedure is noniterative and can be easily applied. Also, in contrast to other non-iterative methods, error estimates are available for the parameter estimates yielded by the new procedure. In the model for the new procedure the points t_i at which observations are taken are assumed to be equally spaced and the number of such points is specified to be an integral multiple of the number of parameters to be estimated. Moreover, each observation is specified to have expectation f (t_i), where f is the function mentioned earlier. The coefficients α_k are assumed to be non-zero and the exponents λ_k are assumed to be distinct and positive. Then in the derivation of new procedure, the observations are reduced to as many sums as there are parameters to be estimated. Each of these sums is equated to its expected value and the resultant equations are solved for estimators of the parameters. The estimators from the new procedure are shown to be asymptotically normally distributed as either the number of points at which observations are taken or the number of observations made at each such point approaches infinity. The asymptotic variances obtained are used to form approximate confidence limits for the α_k and λ_k. The statistical properties of the estimators are also studied. It is found that they are consistent, but not in general unbiased or efficient. Asymptotic efficiencies are calculated tor a few sets of parameter values and a bias approximation is obtained for two special cases. The new method is also shown to be optimum relative to certain similar methods and necessary conditions for the new procedure to lead to admissible estimates are studied. In the last portion of the thesis a sampling study is reported for observations generated with a model containing only one exponential term and with errors which are normally distributed. The small sample biases and variances for the estimates computed from these observations are given and the effects of changes in the parameters in the model are investigated. Then some actual experimental data are fitted using both the new procedure and some alternative methods. The final chapter in the body of the thesis contains a critical evaluation of the new procedure relative to other estimation methods. / Ph. D.

A research project submitted to the Faculty of Science, School of Education, University of the Witwatersrand, in partial fulfilment of the requirements for the degree of Master of Science by combination of coursework and research, 2017 / This study investigates learners’ performance in solving arithmetic equivalences and arithmetic and algebraic equations and was influenced by the notion of the didactic cut (Filloy & Rojano, 1989). Data was collected from two township schools in Johannesburg using a written test. With a Vygotskian perspective on learning, learners’ performance was investigated in two ways: through a response pattern analysis of 106 test scripts as well as through an error analysis on 46 scripts. The response pattern analysis identified seven clusters of responses, each of which suggested a different performance pattern. Two clusters of responses suggest evidence of the didactic cut and that learners struggled with the concept of negativity. A purposive sample of 46 test scripts was analysed further to investigate the actual errors that learners made. Common errors within the two most relevant response pattern analyses were also investigated. Using a combination of typological and inductive methods to categorise learners’ errors, equality and negativity errors were most prominent. Findings revealed that there were very few learners who used arithmetic strategies to solve arithmetic equations and that instead, they used algebraic procedures. The most unexpected finding was that learners appear to memorise the structure of solutions and hence manipulate their procedures in order to obtain familiar structured solutions. Key words: Equality, equal sign, solving linear equations, negativity, learner error, response patterns / XL2018

by Tang Chi Nang. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1995. / Includes bibliographical references (leaves 104-[109]). / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Gaussian elimination --- p.2 / Chapter 1.1.1 --- Numerical stability --- p.2 / Chapter 1.2 --- Block Gaussian elimination --- p.3 / Chapter 1.2.1 --- Numerical stability --- p.4 / Chapter 1.3 --- Elimination graph --- p.4 / Chapter 1.4 --- Elimination ordering --- p.5 / Chapter 1.5 --- Computation and storage requirement --- p.6 / Chapter 1.6 --- Outline of the thesis --- p.7 / Chapter 2 --- Weighted graph elimination --- p.8 / Chapter 2.1 --- Weighted elimination graph --- p.8 / Chapter 2.2 --- Sparse Gaussian elimination --- p.9 / Chapter 2.3 --- Computation and storage requirement --- p.12 / Chapter 2.3.1 --- Computation requirement --- p.12 / Chapter 2.3.2 --- Storage requirement --- p.14 / Chapter 2.4 --- Elimination ordering --- p.15 / Chapter 2.5 --- Repeated structure --- p.18 / Chapter 3 --- Main theory --- p.21 / Chapter 3.1 --- Motivation --- p.21 / Chapter 3.2 --- Notations --- p.22 / Chapter 3.2.1 --- Connectivity --- p.23 / Chapter 3.2.2 --- Separator --- p.23 / Chapter 3.2.3 --- Equivalence --- p.24 / Chapter 3.3 --- Repetition separator --- p.25 / Chapter 3.4 --- Repetition elimination process --- p.30 / Chapter 3.5 --- Multiple Separators --- p.32 / Chapter 3.6 --- Feasibility --- p.33 / Chapter 3.6.1 --- Two-separator case --- p.34 / Chapter 3.6.2 --- General case --- p.39 / Chapter 3.6.3 --- Successive repetition elimination process (SREP) --- p.41 / Chapter 3.7 --- Generalized repetition elimination process --- p.42 / Chapter 3.7.1 --- Extra edges --- p.42 / Chapter 3.7.2 --- Acyclic edges --- p.43 / Chapter 3.7.3 --- Generalized repetition separator --- p.45 / Chapter 4 --- Application in queueing analysis --- p.52 / Chapter 4.1 --- Markov Chain Reduction Principle --- p.54 / Chapter 4.1.1 --- Numerical stability --- p.57 / Chapter 4.2 --- Multi-class MMPP/M/1/L queue --- p.57 / Chapter 4.2.1 --- Single-class case (QBD case) --- p.58 / Chapter 4.2.2 --- Preemptive LCFS case --- p.63 / Chapter 4.2.3 --- Non-preemptive LCFS case --- p.70 / Chapter 4.2.4 --- FCFS case --- p.72 / Chapter 4.2.5 --- Extension to phase type service time --- p.77 / Chapter 4.3 --- 2-class priority system --- p.77 / Chapter 5 --- Choosing the right algorithm --- p.85 / Chapter 5.1 --- MMPP/M/1/L system with bursty arrival --- p.86 / Chapter 5.1.1 --- Algorithm Comparison --- p.89 / Chapter 5.1.2 --- Numerical Examples --- p.90 / Chapter 5.2 2 --- -class priority system --- p.90 / Chapter 5.2.1 --- Algorithm Comparison --- p.95 / Chapter 5.2.2 --- Numerical Examples --- p.95 / Chapter 5.3 --- Conclusion --- p.95 / Chapter 6 --- Conclusion --- p.98 / Chapter 6.1 --- Further research --- p.99 / Chapter A --- List of frequently-used notations --- p.101 / Chapter A.l --- System of equations and Digraph --- p.101 / Chapter A.2 --- General-purpose functions --- p.102 / Chapter A.3 --- Single repetition separator --- p.102 / Chapter A.4 --- Sequence of repetition separators --- p.103 / Chapter A.5 --- Compatibility --- p.103 / Bibliography --- p.104

Bibliography: p. 36.

Thesis (Ph. D.)--University of Joensuu, 2006. / Includes bibliographical references (p. 25-27).

In order to introduce the investigation contemplated in this thesis, let us consider the differential equation d4y+ 3Σ zj (aj + bjzα) djy = 0(1)z4 dz4 j = 0 dzjHere, the variable z is regarded as complex, as are the coefficients aj, bj, (j=0,1,..., n-1) and a is an arbitrary positive integer. It is also assumed that the difference of no two roots of the indicial equation about z = 0 is congruent to zero modulo α .This problem arises in some current Bio-medical problems.Ball State UniversityMuncie, IN 47306

In order to introduce the investigation contemplated in this thesis, let us consider the differential equation d3y d2y dyz3 ____+ z2___(b0 + blzm) + z - (c0 + clzm) dz3 dz2 dz+ (d0 + dlzm + d2z2m) y = 0Here, m is an arbitrary positive integer and the variable z is complex as are the constantsbi,ci (i=0,1) and di (i=0,1,2) with d2≠0. It is also assumed that the difference of no two roots of the indicial equation about z = 0 is congruent to zero modulo m.Ball State UniversityMuncie, IN 47306

Ball State University LibrariesLibrary services and resources for knowledge buildingMasters ThesesThere is no abstract available for this thesis.

Univ., Diss.--Joensuu, 2006.

Thesis (Ph. D. ) -- McMaster University, 1998. / Includes bibliographical references. Also available via World Wide Web.