Luego de establecer el que las matrices innegativas e indescomponibles nxn forman un subconjunto abierto y denso en el espacio de las innegativas nxn se demuestra la diferenciablidad de la función que a cada matriz tal le asigna su raíz de Perron - Frobenius.

Liquid Chromatography/ Mass Spectrometry (LC/MS) is used to interface the separating power of LC with the sensitivity and specificity of MS for the determination of trace levels of organic compounds in a variety of matrices. The technique is finding increasing application in the field of environmental and pharmaceutical analysis. Particle Beam LC/MS (PB/LC/MS) uses a particle beam interface to connect the LC to the MS. This interface design has the advantage of being able to produce "classical" electron impact (El) spectra which can then be searched against commercial MS libraries. The aim of this work was to apply PB/LC/MS to a range of new problems in environmental analysis and evaluate the usefulness of this technique. PB/LC/MS was used to determine compounds that cannot easily be analysed by more conventional techniques such as gas chromatography with mass spectrometry (GC/MS) or liquid chromatography with UV/vis detection (LC/UV). For example, some polycyclic aromatic hydrocarbons (PAH) are too involatile to analyse by GC/MS, some commonly prepared isocyanate derivatives cannot be accurately identified by LC/UV and some classes of pesticides are thermally labile and so cannot be determined by GC/MS.The work presented in this thesis examines the factors affecting the sensitivity and performance of PB/LC/MS and comparisons are made with other analytical methods. Compound classes examined are polycyclic aromatic hydrocarbons (PAH), pesticides and isocyanate derivatives in a variety of environmental matrices. Methods for improving the sensitivity of PB/LC/MS are investigated and the results of these experiments used to compare the different models are used to explain PB/MS behaviour. Conclusions regarding the accuracy of these models are then made. The ability of PB/MS to provide useful El MS for identification purposes in complex environmental matrices is also investigated.

Multi-dimensional (n-D) systems can be described by matrices whose elements are polynomial in more than one indeterminate. These systems arise in the study of partial differential equations and delay differential equations for example, and have attracted great interest over recent years. Many of the available results have been developed by generalising the corresponding results from the well known 1-D theory. However, this is not always the best approach since there are many differences between 1-D, 2-D and n-D (n > 2) polynomial matrices. This is due mainly to the underlying polynomial ring structure.

We give overcrowding estimates for the Sine(beta) process, the bulk point process limit of the Gaussian beta-ensemble. We show that the probability of having exactly n points in a fixed interval is given by e(-beta/2n2) log(n)+O(n(2)) as n -> infinity. We also identify the next order term in the exponent if the size of the interval goes to zero.

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The problem of determining the structure of linear transformations on the algebra of n-square matrices over the complex field is discussed by M. Marcus and B. N. Moyls in the paper ''Linear Transformations on Algebras of Matrices". The authors were able to characterize linear transformations which preserve one or more of the following properties of n-square matrices; rank, determinant and eigenvalues. The problem of obtaining a similar characterization of transformations as given by M. Marcus and B. N. Moyls but for a wider class of fields is considered in this thesis. In particular, their characterization of rank preserving transformations holds for an arbitrary field. One of the results on determinant preserving transformations obtained by M. Marcus and B. N. Moyls states that if a linear transformation T maps unimodular matrices into unimodular matrices, then T preserves determinants. Since this result does not necessarily hold for algebras of matrices over finite fields, the discussion on the characterization of determinant preserving transformations is limited to algebras of matrices over infinite fields. / Science, Faculty of / Mathematics, Department of / Graduate

The analysis of structures which contain catenary cables is made difficult by the non-linear force-deformation relationships of the cables. For all but the smallest deflections it is not possible to linearize these relationships without causing significant inaccuracies. Newton's Method solves non-linear equations by solving a succession of linearized problems, the answer converging to the solution of the non-linear problem. Newton's Method so used to analyze cable-containing structures results in a succession of linear stiffness analysis problems. As a result, conventional stiffness analysis computer programs may be modified without great difficulty to solve cable structures by Newton's Method. The use of Newton's Method to solve cable structures forms the body of this thesis. The two basic innovations necessary, which are the provision of methods for calculating the end-forces of a cable in an arbitrary position, and for evaluating the stiffness matrix of a cable, are presented. Also discussed are the co-ordinate transformations necessary to describe the cable stiffness matrix and cable end forces in a Global Co-ordinate System. The virtues of the method are demonstrated in two example problems, and the theoretical basis for Newton's Method is examined. Finally, the value of the method presented is briefly discussed. / Applied Science, Faculty of / Civil Engineering, Department of / Graduate

An acyclic orientation of a graph is an assignment of a direction to each edge in a way that does not form any directed cycles. Acyclic orientations of a complete bipartite graph are in bijection with a class of matrices called lonesum matrices, which can be uniquely reconstructed from their row and column sums. We utilize this connection and other properties of lonesum matrices to determine an analytic form of the generating function for the length of the longest path in an acyclic orientation on a complete bipartite graph, and then study the distribution of the length of the longest path when the acyclic orientation is random. We use methods of analytic combinatorics, including analytic combinatorics in several variables (ACSV), to determine asymptotics for lonesum matrices and other related classes. / Includes bibliography. / Dissertation (PhD)--Florida Atlantic University, 2021. / FAU Electronic Theses and Dissertations Collection

Motivated by dimension reduction in regression analysis and signal detection, we investigate order determination for large dimensional matrices with spiked structures in which the dimensions of the matrices are proportional to the sample sizes. Because the asymptotic behaviors of the estimated eigenvalues differ completely from those in fixed dimension scenarios, we then discuss the largest possible order, say q, we can identify and introduce criteria for different settings of q. When q is assumed to be fixed, we propose a "valley-cliff" criterion with two versions - one based on the original differences of eigenvalues and the other based on the transformed differences - to reduce the effect of ridge selection in the criterion. This generic method is very easy to implement and computationally inexpensive, and it can be applied to various matrices. As examples, we focus on spiked population models, spiked Fisher matrices and factor models with auto-covariance matrices. For the case of divergent q, we propose a scale-adjusted truncated double ridge ratio (STDRR) criterion, where a scale adjustment is implemented to deal with the bias in scale parameter for large q. Again, examples include spiked population models, spiked Fisher matrices. Numerical studies are conducted to examine the finite sample performances of the method and to compare it with existing methods. As for theoretical contributions, we investigate the limiting properties, including convergence in probability and central limit theorems, for spiked eigenvalues of spiked Fisher matrices with divergent q. Keywords: Auto-covariance matrix, factor model, finite-rank perturbation, Fisher matrix, principal component analysis (PCA), phase transition, random matrix theory (RMT), ridge ratio, spiked population model.

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