In the first part of this thesis we discuss classes of new exact NUT-charged solutions in four dimensions and higher, while in the remainder of the thesis we make a study of their properties and their possible applications. <br /><br /> Specifically, in four dimensions we construct new families of axisymmetric vacuum solutions using a solution-generating technique based on the hidden <em>SL</em>(2,R) symmetry of the effective action. In particular, using the Schwarzschild solution as a seed we obtain the Zipoy-Voorhees generalisation of the Taub-NUT solution and of the Eguchi-Hanson soliton. Using the <em>C</em>-metric as a seed, we obtain and study the accelerating versions of all the above solutions. In higher dimensions we present new classes of NUT-charged spaces, generalizing the previously known even-dimensional solutions to odd and even dimensions, as well as to spaces with multiple NUT-parameters. We also find the most general form of the odd-dimensional Eguchi-Hanson solitons. We use such solutions to investigate the thermodynamic properties of NUT-charged spaces in (A)dS backgrounds. These have been shown to yield counter-examples to some of the conjectures advanced in the still elusive dS/CFT paradigm (such as the maximal mass conjecture and Bousso's entropic N-bound). One important application of NUT-charged spaces is to construct higher dimensional generalizations of Kaluza-Klein magnetic monopoles, generalizing the known 5-dimensional Kaluza-Klein soliton. Another interesting application involves a study of time-dependent higher-dimensional bubbles-of-nothing generated from NUT-charged solutions. We use them to test the AdS/CFT conjecture as well as to generate, by using stringy Hopf-dualities, new interesting time-dependent solutions in string theory. Finally, we construct and study new NUT-charged solutions in higher-dimensional Einstein-Maxwell theories, generalizing the known Reissner-Nordström solutions.

The first loudspeaker was invented almost 150 years ago and even though much has changed regarding the manufacturing, the main idea is still the same. To produce clean sound, modern loudspeaker consist of expensive materials that often need advanced manufacturing equipment. The relatively newly established company Actiwave AB uses digital signal processing to enhance the audio for loudspeakers with poor acoustic properties. Their algorithms concentrate on attenuating the linear distortion but there is no compensation for the loudspeakers' non-linear distortion, such as harmonic distortion. To attenuate the harmonic distortion, this thesis presents controllers based on exact input-output linearisation. This type of controller needs an accurate model of the system. A loudspeaker model has been derived based on the LR-2 model, an extension of the more common Thiele-Small model. A controller based on exact input-output linearisation also needs full state feedback, but since feedback risk being expensive, state estimators were used. The state estimators were based on feed-forward or observers using the extended Kalman filter or the unscented Kalman filter. A combination of feed-forward state estimation and a PID controller were designed as well. In simulations, the total harmonic distortion was attenuated for all controllers up to 180 Hz. The simulations also showed that the controllers are sensitive to inaccurate parameter values in the loudspeaker model. During real-life experiments, the controllers needed to be extended with a model of the used amplifier to function properly. The controllers that were able to attenuate the harmonic distortion were the two methods using feed-forward state estimation. Both controllers showed improvement compared to the uncontrolled case for frequencies up to 40 Hz.

Finding the maximum size of a matching in an undirected graph and finding the maximum size of branching in a directed graph can be formulated as matrix rank problems. The Tutte matrix, introduced by Tutte as a representation of an undirected graph, has rank equal to the maximum number of vertices covered by a matching in the associated graph. The branching matrix, a representation of a directed graph, has rank equal to the maximum number of vertices covered by a branching in the associated graph. A mixed graph has both undirected and directed edges, and the matching forest problem for mixed graphs, introduced by Giles, is a generalization of the matching problem and the branching problem. A mixed graph can be represented by the matching forest matrix, and the rank of the matching forest matrix is related to the size of a matching forest in the associated mixed graph. The Tutte matrix and the branching matrix have indeterminate entries, and we describe algorithms that evaluate the indeterminates as rationals in such a way that the rank of the evaluated matrix is equal to the rank of the indeterminate matrix. Matroids in the context of graphs are discussed, and matroid formulations for the matching, branching, and matching forest problems are given.

In the first part of this thesis we discuss classes of new exact NUT-charged solutions in four dimensions and higher, while in the remainder of the thesis we make a study of their properties and their possible applications. <br /><br /> Specifically, in four dimensions we construct new families of axisymmetric vacuum solutions using a solution-generating technique based on the hidden <em>SL</em>(2,R) symmetry of the effective action. In particular, using the Schwarzschild solution as a seed we obtain the Zipoy-Voorhees generalisation of the Taub-NUT solution and of the Eguchi-Hanson soliton. Using the <em>C</em>-metric as a seed, we obtain and study the accelerating versions of all the above solutions. In higher dimensions we present new classes of NUT-charged spaces, generalizing the previously known even-dimensional solutions to odd and even dimensions, as well as to spaces with multiple NUT-parameters. We also find the most general form of the odd-dimensional Eguchi-Hanson solitons. We use such solutions to investigate the thermodynamic properties of NUT-charged spaces in (A)dS backgrounds. These have been shown to yield counter-examples to some of the conjectures advanced in the still elusive dS/CFT paradigm (such as the maximal mass conjecture and Bousso's entropic N-bound). One important application of NUT-charged spaces is to construct higher dimensional generalizations of Kaluza-Klein magnetic monopoles, generalizing the known 5-dimensional Kaluza-Klein soliton. Another interesting application involves a study of time-dependent higher-dimensional bubbles-of-nothing generated from NUT-charged solutions. We use them to test the AdS/CFT conjecture as well as to generate, by using stringy Hopf-dualities, new interesting time-dependent solutions in string theory. Finally, we construct and study new NUT-charged solutions in higher-dimensional Einstein-Maxwell theories, generalizing the known Reissner-Nordström solutions.

Consider the multiresponse polynomial regression model with one control variable and arbitrary covariance matrix among responses. The present results complement solutions by Krafft and Schaefer (1992) and Imhof (2000), who obtained the n-point D-optimal designs for the multiresponse regression model with one linear and one quadratic. We will show that the D-optimal design is invariant under linear transformation of the control variable. Moreover, the most cases of the exact D-optimal designs on [-1,1] for responses consisting of linear and quadratic polynomials only are derived. The efficiency of the exact D-optimal designs for the univariate quadratic model to that for the above model are also discussed. Some conjectures based on intensively numerical results are also included.

The D-optimal design problems in polynomial regression models with a one-dimensional control variable and k-dimensional response variable Y=(Y_1,...,Y_k) where there are some common unknown parameters are discussed. The approximate D-optimal designs are shown to be independent of the covariance structure between the k responses when the degrees of the k responses are of the same order. Then, the exact n-point D-optimal designs are also discussed. Krafft and Schaefer (1992) and Imhof (2000) are useful in obtaining our results. We extend the proof of symmetric cases for k>= 2.

In this paper we consider the exact $D$-optimal design problem for linear trigonometric regression models with or without intercept on a partial circle. In a recent papper Dette, Melas and Pepelyshev (2001) found explicit solutions of approximate $D$-optimal designs for trigonometric regression models with intercept on a partial circle. The exact optimal designs are determined by means of moment sets of trigonometric functions. It is shown that the structure of the optimal designs depends on both the length of the design interval and the number of the design points.

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The topic of exact, exponential-time algorithms for NP-hard problems has received a lot of attention, particularly with the focus of producing algorithms with stronger theoretical guarantees, e.g. upper bounds on the running time on the form O(c^n) for some c. Better methods of analysis may have an impact not only on these bounds, but on the nature of the algorithms as well. The most classic method of analysis of the running time of DPLL-style ("branching" or "backtracking") recursive algorithms consists of counting the number of variables that the algorithm removes at every step. Notable improvements include Kullmann's work on complexity measures, and Eppstein's work on solving multivariate recurrences through quasiconvex analysis. Still, one limitation that remains in Eppstein's framework is that it is difficult to introduce (non-trivial) restrictions on the applicability of a possible recursion. We introduce two new kinds of complexity measures, representing two ways to add such restrictions on applicability to the analysis. In the first measure, the execution of the algorithm is viewed as moving between a finite set of states (such as the presence or absence of certain structures or properties), where the current state decides which branchings are applicable, and each branch of a branching contains information about the resultant state. In the second measure, it is instead the relative sizes of the modelled attributes (such as the average degree or other concepts of density) that controls the applicability of branchings. We adapt both measures to Eppstein's framework, and use these tools to provide algorithms with stronger bounds for a number of problems. The problems we treat are satisfiability for sparse formulae, exact 3-satisfiability, 3-hitting set, and counting models for 2- and 3-satisfiability formulae, and in every case the bound we prove is stronger than previously known bounds.

General relativity has a small number of known, exact solutions which model astronomically relevant systems. These models are highly idealized situations. Either perturbation theory or numerical simulations are typically needed to produce more realistic models. Numerical simulations are time-consuming and suffer from a difficulty in interpreting the results. In addition, global properties of numerical solutions are nearly impossible to uncover. On the other hand, standard perturbation methods are very difficult to implement beyond the second order, which means they barely scratch the surface of non-linear phenomena which distinguishes general relativity from Newtonian gravity. This work develops a method of finding exact solutions, inspired by perturbation theory, which have energy-momentum tensor components that approximately satisfy desired relationships. We find a spherical lump of matter which has a density profile $\mu \propto r^{-2}$ in a Robertson-Walker background; it looks like a galaxy in an expanding universe. We also find a plane-symmetric perturbation of a Bianchi type I metric with a density profile $\mu \propto z^{-2}$; it models a jet impacting a sheet-like structure. The former solution involves a wormhole while the latter involves a two dimensional singularity. These are both non-linear structures which perturbation theory can never produce.