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An analysis of computer diagnostic systems shows that most systems use answer data (product) for their analyses. This process of determining an error pattern, in addition, does little in the way of telling a teacher what should be done to help the child. This two-fold problem, extant in all computerized arithmetic diagnostic systems to date, prompted this study which sought other data sources in order to bring about more accurate computer analyses. A cognitive orientation suggested that the use of clinical diagnostic techniques should be explored as an alternative to error analysis. Essentially, these two approaches were compared. That is, to what extent does error pattern diagnosis (an essentially product oriented approach) and clinical mathematical diagnosis (a process oriented approach) interrelate? Participants for this study were five, eight year olds from southwest Virginia. These children completed a test that was developed by Van Lehn (1982). This test was analyzed for error patterns and the children were selected on the basis of their error patterns. These children were then tested in a clinical setting using a measure developed for this study in cooperation with a clinical mathematics diagnostician. The analysis was done on the results of these two measures and the protocols collected during the clinical interviews. The results indicated that there was no clear connection between the two types of diagnosis, but the analysis did yield a broader description of each individual participant. That is, error analysis or clinical mathematics alone does not completely describe an individual's knowledge of mathematics. / Master of Arts

Experimental modal analysis (EMA) techniques have become a popular method of studying the dynamic characteristics of structures. A survey of literature available reveals that experimental modal models resulting from EMA may suffer from inaccuracy due to a host of reasons. Every stage of EMA could be a potential source of errors - from suspension of the test structures, transduction to parameter estimation phase. Though time-domain methods are actively being investigated by many researchers and are in use, fast Fourier transform (FFT) methods, due to their speed and ease of implementation, are the most widely used in experimental modal analysis work. This work attempts to quantify errors that result from a typical modal test. Using a simple beam with free-free boundary conditions simulated, three different modal tests are performed. Each test differs from the other chiefly in the excitation method and FRF estimator used. Using finite element models as the reference, correlation between finite element and experimental models are performed. The ability of the EMA process to accurately estimate the modal parameters is established on the basis of level of correlation obtained for natural frequencies and mode shapes. Linear regression models are used to correlate test and analysis natural frequencies. The modal assurance criterion (MAC) is used to establish the accuracy of mode vectors from the modal tests. The errors are further quantified spatially (on a location-by-location basis) for natural frequencies and mode shapes resulting from the EMA process. Finally, conclusions are made regarding the accuracy of modal parameters obtained via FFT-based EMA techniques. / Master of Science

Estimators for the individual error variance were derived in a nonreplicated two-way classification by the use of the model x_ij = μ_i + β_ij + ε_ij, i=1,2,...n; j=1,2,...,r, where x_ij = observation on the i^th treatment of the j^th block, μ_i = true mean of the i^th treatment, β_j = bias of the j^th block, ε_ij = random error, distributed normally with means zero and variance σ²_j, and E(x_ij) = μ_i + β_j. The estimator σ̂²_t, for σ²_t, t=1,2,3,...,r, was derived for n ≥ 2 and r = 3, by applying the principle of maximum likelihood to a set of (n-1)(r-1) transformed variables usually ascribed to error. Equations were derived for the maximum likelihood estimators for n ≥ 2 and r ≥ 3. A general quadratic form was used and when four reasonable assumptions were applied, estimators of the variances were obtained in for form of Q_t = [r(r-1)∑_i(x_ij-x_i.-x_.t+x_..)²-∑_i∑_j(x_ij-x_i.-x_.j+x_..)²] ÷ [(n-1)(r-1)(r-2)] where x_i., x_.j and x_.. are the means of i^th treatment, j^th block and grand mean respectively. σ̂²_t and Q_t were shown to be identical when σ²_t was being estimated for the case n ≥ 2, r = 3. It was noted that the derived estimator Q_j is equal to the estimators proposed by Grubbs [J.A.S.A., Vol. 43 (1948)] and Ehrenberd [Biometrika, Vol 37. (1950).] It was shown that Q_t/σ² = [(r-1)²x_(n-1)²-x_(n-1)(r-2)²]/[(n-1)(r-1)(r-2)], a linear difference of two independent central chi-square variates. The statistic Q/E was derived such that Q_t/E = [(((r-1)²)/(1+(r-2)F))-1]/[(n-1)(r-1)(r-2)] with F, a central F-statistic with (n-1)(r-2) and (n-1) degrees of freedom in the numerator and denominator respectively and E =∑_i∑_j(x_ij-x_i.-x_.j+x_..)². It was noted that this statistic may be used to test H_o: σ²_t = σ²against one of H_a₁: σ²_t > σ²; H_a₂: σ²_t < σ² and H_a₃: σ²_t ≠ σ² assuming σ²_j = σ², j≠t, j=1,2,...,r. A final test was of homogeneity of variances when r = 3 and was based on - 2 ln λ = (n-1)[2 ln (n-1) + ln(Q₁Q₂+Q₁Q₃+Q₂Q₃) - 2 ln E + ln 4/3], where λ is a likelihood ratio and -2 ln λ is approximately distributed as x² with 2 degrees of freedom for large n. A more general statistic for testing homogeneity of variance for r ≥ 3 was proposed and its distribution discussed in a special case. / Ph. D.

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The research in this thesis was initially motivated by an outstanding problem posed by Argyros and Haydon. They used a generalised version of the Bourgain-Delbaen construction to construct a Banach space $XK$ for which the only bounded linear operators on $XK$ are compact perturbations of (scalar multiples of) the identity; we say that a space with this property has very few operators. The space $XK$ possesses a number of additional interesting properties, most notably, it has $ell_1$ dual. Since $ell_1$ possesses the Schur property, weakly compact and norm compact operators on $XK$ coincide. Combined with the other properties of the Argyros-Haydon space, it is tempting to conjecture that such a space must necessarily have very few operators. Curiously however, the proof that $XK$ has very few operators made no use of the Schur property of $ell_1$. We therefore arrive at the following question (originally posed in cite{AH}): must a HI, $mathcal{L}_{infty}$, $ell_1$ predual with few operators (every operator is a strictly singular perturbation of $lambda I$) necessarily have very few operators? We begin by giving a detailed exposition of the original Bourgain-Delbaen construction and the generalised construction due to Argyros and Haydon. We show how these two constructions are related, and as a corollary, are able to prove that there exists some $delta > 0$ and an uncountable set of isometries on the original Bourgain-Delbaen spaces which are pairwise distance $delta$ apart. We subsequently extend these ideas to obtain our main results. We construct new Banach spaces of Bourgain-Delbaen type, all of which have $ell_1$ dual. The first class of spaces are HI and possess few, but not very few operators. We thus have a negative solution to the Argyros-Haydon question. We remark that all these spaces have finite dimensional Calkin algebra, and we investigate the corollaries of this result. We also construct a space with $ell_1$ Calkin algebra and show that whilst this space is still of Bourgain-Delbaen type with $ell_1$ dual, it behaves somewhat differently to the first class of spaces. Finally, we briefly consider shift-invariant $ell_1$ preduals, and hint at how one might use the Bourgain-Delbaen construction to produce new, exotic examples.

published_or_final_version / abstract / Electrical and Electronic Engineering / Master / Master of Philosophy

This thesis studies abstract stochastic differential equations on Banach spaces. The well-posedness of abstract stochastic differential equations on such spaces is a recent result of van Neerven, Veraar and Weis, based on the theory of stochastic integration of Banach space valued processes constructed by the same authors. We study existence and uniqueness for solutions of stochastic differential equations with (possibly infinite) delay in their inputs on UMD Banach spaces. Such problems are also known as functional differential equations or delay differential equations. We show that the methods of van Neerven et al. extend to such problems if the initial history of the system lies in a space of a type introduced by Hale and Kato. The results are essentially of a fixed point type, both autonomous and non-autonomous cases are discussed and an example is given. We also study some long time properties of solutions to these stochastic differential equations on general Banach spaces. We show the existence of solutions to stochastic problems with almost periodicity in a weak or distributional sense. Results are again given for both autonomous and non-autonomous cases and depend heavily on estimates for R-bounds of operator families developed by Veraar. An example is given for a second order differential operator on a domain in ℝ^d. Finally we consider the existence of invariant measures for such problems. This extends recent work of van Gaans in Hilbert spaces to Banach spaces of type 2.

In this thesis, we investigate the pathwise regularity of partial sum process of general orthogonal series, and prove that the partial sum process is a geometric 2-rough process under the same condition as in Menshov-Rademacher Theorem. For Fourier series, the condition can be improved, and an equivalent condition on the limit function is identified.

Masters Report A report submitted in ful llment of the requirements for the degree of Master of Science (50/50) in the Centre for Telecommunication Access and Services (CeTAS) School of Electrical and Information Engineering Faculty of Engineering and the Built Environment February 2017 / Bounded distance decoding of Reed Solomon (RS) codes involves nding a unique codeword if there is at least one codeword within the given distance. A corrupted message having errors that is less than or equal to half the minimum distance cor- responds to a unique codeword, and therefore will decode errors correctly using the minimum distance decoder. However, increasing the decoding radius to be slightly higher than half of the minimum distance may result in multiple codewords within the Hamming sphere. The list decoding and syndrome extension methods provide a maximum error correcting capability whereby the radius of the Hamming ball can be extended for low rate RS codes. In this research, we study the probability of having unique codewords for (7; k) RS codes when the decoding radius is increased from the error correcting capability t to t + 1. Simulation results show a signi cant e ect of the code rates on the probability of having unique codewords. It also shows that the probability of having unique codeword for low rate codes is close to one. / MT2017