We discuss the problem of optimal allocation of the design points of an experiment for the case where the trials may fail with non-zero probability. Numerical results for D-optimal designs are given for estimating the coefficients of a polynomial regression. For small sample sizes these designs may deviate substantially from the corresponding designs in the case of certain response. They can be less efficient, but are less affected by failing trials. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

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This thesis contains an exposition of the new strongly polynomial algorithm for the generalized flow problem by Laszlo Vegh (2013). It has been a long-standing open question whether such an algorithm exists, until it was resolved by Vegh in 2013. Generalized flows have many applications in economic problems, such as transportation of goods and foreign currency exchange. The main presentation follows Vegh's paper, but this exposition contains some simplifications and differences in the algorithm and its analysis. The main difference is that we consider the running time of the strongly polynomial algorithm up to one arc contraction before starting fresh on a smaller network. This increases the running time of the algorithm slightly, but the analysis becomes easier.

An examination of some properties that interrelate the computational complexities of evaluating multivariate polynomial functions is presented. The kind of relationship between polynomial functions that is studied takes the form of linear transformations of the arguments and results of a polynomial function that transform it into another such function. Such transformations are a generalisation of projection (a form of reduction in algebraic complexity first introduced by Valiant, whereby variables and constants are substituted for the arguments of a polynomial function in order to transform it into another polynomial function). In particular, two restricted forms of this generalised projection are considered: firstly, those that relate a polynomial function to itself, and secondly, those that are invertable. Call these symmetries and similarities, respectively. The structure of the set of symmetries of a polynomial function is explored, and the computationally useful members of the set identified; a technique for finding all such symmetries is presented. It is shown that polynomials related by similarity have "isomorphic" sets of symmetries, and this condition may be used as a criterion for similarity. Similarity of polynomial functions is shown to be an equivalence relation, and "similar polynomials" can be seen to possess closely comparable complexities. A fast probabilistic algorithm for finding the symmetries of a polynomial function is given. The symmetries of the determinant and of the permanent (which differs from the determinant only in that all of its monomials have coefficients of +1), and those of some other polynomials, are explicitly found using the above theory. Fast algorithms using linear algebra for evaluating the determinant are known, whereas evaluating the permanent is known to be a #p-complete problem, and is apparently intractable; the reasons for this are exposed. As an easy corollary it is shown that the permanent is not preserved by any bilinear product of matrices, in con'trast to the determinant which is preserved by matrix multiplication. The result of Marcus and Minc, that the determinant cannot be transformed into the permanent by substitution of linear combinations of variables for its arguments (i.e. the permanent and determinant are not similar), also follows as an easy corollary. The relationship between symmetries and ease of evaluation is discussed.

Black-Scholes Model is a widely used mathematical model for stock price behaviors, of which the return is assumed to be normally distributed. But this 'normally distributed' assumption is doubted and proved to be not true by realistric data. The main goal of this thesis is to explore polynomial-normal distribution, and use this distribution in the stock return, as a non-normal extension of the Black-Scholes Model. We will develop the properties of polynomial-normal distribtuion in the thesis, and also give the European call and put option price formulas under this model, and show how to use this model to estimate real stock returns.

Polynomial functors were introduced by Professors Eilenberg and Mac Lane in 1954, who used them to study certain homology rings. Strict polynomial functors were invented by Professors Friedlander and Suslin in 1997, in order to develop the theory of group schemes. The first real investigation of their intrinsic properties was performed in 1988, when Professor Pirashvili showed that polynomial functors are equivalent to modules over a certain ring. A similar study was conducted on strict polynomial functors in 2003 by Dr. Salomonsson in his doctoral thesis. A radically different method of attack was initiated by Dr. Dreckman and Professors Pirashvili, Franjou, and Baues in the year 2000. Their approach was to combinatorially encode polynomial functors, and utilised for this purpose the category of sets and surjections. Dr. Salomonsson would later repeat the feat for strict polynomial functors, employing instead the category of multi-sets. This thesis proposes the following: 1:o. To generalise the notion of polynomial functor to more general base rings than Z, so that it smoothly agree with the existing definition of strict polynomial functor, allowing for easy comparison. This results in the definition of numerical functors. 2:o. To make an extensive study of numerical maps of modules, to see how they fit into Professor Roby's framework of strict polynomial maps. 3:o. To conduct a survey of numerical rings. 4:o. To develop the theories of numerical and strict polynomial functors so that they run in parallel. 5:o. To show how also numerical functors may be interpreted as modules over a certain ring. 6:o. To expound the theory of mazes, which will be seen to vastly generalise the category of surjections employed by Professor Pirashvili et al., since they turn out to encode, not only polynomial or numerical functors, but all module functors over any base ring. 7:o. To simplify Dr. Salomonsson's construction involving multi-sets, making it more amenable to a comparison with mazes. 8:o. To prove comparison theorems interrelating numerical and strict polynomial functors. 9:o. And, finally, to indicate how polynomial functors may be used to extend the operad concept.

Thesis (Ph. D.)--University of Oregon, 2004. / Typescript. Includes vita and abstract. Includes bibliographical references (leaves 85-86). Also available for download via the World Wide Web; free to University of Oregon users.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1941. / Typescript. Includes abstract and vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaf [61]).

This thesis is devoted to a number of structural and behavioural problems in linear multivariable control system theory. The first problem addresses the subject of determination of the finite and infinite frequency structure of a rational matrix. A novel method is proposed that determines the finite and infinite frequency structure of any rational matrix. Some neat and numerically stable algorithms are developed to implement this method. The second problem concerns the resol vent decompositions of a regular polynomial matrix and solutions of regular polynomial matrix descriptions (PMDs). Regarding these fundamental is'sues, three contributions are made therein. Firstly, based on a general resolvent decomposition a complete solution of regular PMDs is presented that takes into account both the non-zero initial conditions of the pseudo state and the non-zero initial conditions of the input. Secondly, two special resolvent decompositions are proposed, both of which are applied to formulate the solution of the regular PMDs. The first one is formulated in terms of the finite, infinite, and the generalised infinite Jordan pairs, which is a refinement of the results given by Gohberg et al. [74] and Vardulakis [25]. The second resolvent decomposition is proposed on the Weierstrass canonical form of the generalised companion matrix of the polynomial matrix. Thirdly, a new characterization of the impulsive free initial conditions of regular PMDs is given and the relationship between the finite and infinite frequency structure of a regular polynomial matrix and its generalised companion matrix is determined. In the third problem a generalization of the chain-scattering representation for general plants is presented. Through the notion of input-output consistency, the conditions under which the generalised chain-scattering representation and the dual generalised chain-scattering representation exist are proposed. Some algebraic system properties of the GCSRs and DGCSRs are studied. The fourth problem is devoted to a new notion of realization of behaviour. We introduce a notion realization of behavior which is shown to be a generalization of the classical concept of a realization of transfer function. By using this approach, the input-output structures of the generalized chain-scattering representations and the dual generalized chain-scattering representations are investigated in a behavioral theory context. The last problem is devoted to the subjects of system wellposedness and internal stability. We present certain generalisations to the classical concepts of wellposedness and internal stability. The input consistency and output uniqueness of the closed-loop system in the standard control feedback configurations are discussed. Based on this, a number of notions are introduced such as fully internal wellposedness, externally internal wellposedness, and externally internal stability, which characterize the rich input-output and stability features of the general control systems in a general setting. On the basis of these notions the extended JL control problem is defined in a general setting.