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 purpose of this thesis is to integrate the most up-to-date information on the subject of linear programming into a comprehensive and understandable treatise for the consideration of management. The value of this study, then, is determined by the effectiveness of its presentation so that management may grasp an ample understanding of the subject.

Thesis (M.A.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The problem discussed in this paper is that of finding a best approximation to a given real-valued function f(x) over a continuum by means of finding a best approximation over a discrete set of points. We are also seeking to find a numerical method of finding a best approximation over our discrete set of points. A best approximation is one which minimizes the maximum deviation of our approximation from our given function f(x). We first discuss the concept of linear programming. In this paper we are not so much concerned with the theory behind linear programming as we are with the method to solve a linear programming problem, namely the simplex method. We discuss the simplex method from the point of view of a programmer, noting how results are continually updated until an optimal solution to t he problem is found. The only theoretical aspect of linear programming which we discuss is the notion of duals and the relationship between the solution of a primal and a dual problem. This becomes very important later in the paper when we try to formulate a best a proximat ion problem as a linear programming problem. Next we discuss the theoretical aspects of best approximation over a continuum. We prove existence, uniqueness, and, most important for our purposes, characterization. Our approximating functions are assumed to form a Chebyschev set throughout this paper. Finally we discuss best approximation over a discrete set of points. We first prove that the characterization theorem holds for problems of this type. Now that we have a way to tell whether our approximation is the best that can be obtained, we turn our attention to the relationship between the best approximation problem over a continuum and a discrete set of points. We prove in a quite general context that the best approximation over a discrete set of points converges uniformly to the solution to the problem over the continuum. We then retrace our steps and establish similar results for the particular case of polynomial approximation. After this we try to find out about the rate at which this convergence takes place. In general this question has no answer for its depends on the smoothness of the functions involved; if, however, we assume the fun ctions satisfy a Holder condition we may obtain some bounds on the rate of convergence. Finally, we reformulate the best approximation problem, showing how it can be considered as a linear programming problem which we already have a means of solving. / 2031-01-01

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A software package for linear programming has been developed using the revised simplex and dual simplex algorithms. The design of the program incorporates an experimental change in the dual simplex algorithm. If the entered problem is not primal feasible, a modified dual simplex algorithm is used. The traditional dual simplex method requires an initial dual feasible basis and maintains feasibility throughout its application. The experimental change is to ignore this criteria of dual feasibility. The objective then becomes to obtain primal feasibility. Once this is attained, the revised simplex algorithm is applied to obtain optimality, if this has not been reached through use of the dual. This experimental change redirects the goal of the dual simplex method from obtaining objective function optimality to obtaining primal feasibility. Program testing has shown the experimental design to produce correct results for a variety of linear programming problems. The program is written for an IBM PC using PASCAL for coding. Spreadsheet format and menus provide ease in problem input and output. Devices for output of problem and solution are printer, screen and/or disk. A problem can be saved and retrieved at a later time for editing.

Methods for allocating redundant controls for systems with four or more objectives are studied. Previous research into aircraft control allocation has focused on allocating control effectors to provide commands for three rotational degrees of freedom. Redundant control systems have the capability to allocate commands for a larger number of objectives. For aircraft, direct force commands can be applied in addition to moment commands. When controls are limited, constraints must be placed on the objectives which can be achieved. Methods for meeting commands in the entire set of of achievable objectives have been developed. The Bisecting Edge Search Algorithm has been presented as a computationally efficient method for allocating controls in the three objective problem. Linear programming techniques are also frequently presented. This research focuses on an effort to extend the Bisecting Edge Search Algorithm to handle higher numbers of objectives. A recursive algorithm for allocating controls for four or more objectives is proposed. The recursive algorithm is designed to be similar to the three objective allocator and to require computational effort which scales linearly with the controls. The control allocation problem can be formulated as a linear program. Some background on linear programming is presented. Methods based on five formulations are presented. The recursive allocator and linear programming solutions are implemented. Numerical results illustrate how the average and worst case performance scales with the problem size. The recursive allocator is found to scale linearly with the number of controls. As the number of objectives increases, the computational time grows much faster. The linear programming solutions are also seen to scale linearly in the controls for problems with many more controls than objectives. In online applications, computational resources are limited. Even if an allocator performs well in the average case, there still may not be sufficient time to find the worst case solution. If the optimal solution cannot be guaranteed within the available time, some method for early termination should be provided. Estimation of solutions from current information in the allocators is discussed. For the recursive implementation, this estimation is seen to provide nearly optimal performance. / Ph. D.