Given a real N by N matrix A, write p(A) for the maximum angle by which A rotates any unit vector. Suppose that A and B are positive definite symmetric (PDS) N by N matrices. Then their Jordan product {A, B} := AB + BA is also symmetric, but not necessarily positive definite. If p(A) + p(B) is obtuse, then there exists a special orthogonal matrix S such that {A, SBS^(-1)} is indefinite. Of course, if A and B commute, then {A, B} is positive definite. Our work grows from the following question: if A and B are commuting positive definite symmetric matrices such that p(A) + p(B) is obtuse, what is the minimal p(S) such that {A, SBS^(-1)} indefinite? In this dissertation we will describe the level curves of the angle function mapping a unit vector x to the angle between x and Ax for a 3 by 3 PDS matrix A, and discuss their interaction with those of a second such matrix.

The current state-of-the-art tyre models tend to be demanding in parameterisation terms, typically requiring extensive and expensive testing, and computational power. Consequently, an alternative parameterisation approach, which also allows for the separation of model fidelity from computational demand, is essential. Based on the above, a tyre model is introduced in this work. Tyre motion is separated into two components, the first being the non-linear global motion of the tyre as a rigid body and the second being the linear local deformation of each node. The resulting system of differential equations of motion consists of a reduced number of equations, depending on the number of rigid and elastic modes considered rather than the degrees of freedom. These equations are populated by the eigenvectors and the eigenvalues of the elastic tyre modes, the eigenvectors corresponding to the rigid tyre modes and the inertia properties of the tyre. The contact sub-model consists of bristles attached to each belt node. Shear forces generated in the contact area are calculated by a distributed LuGre friction model while vertical tread dynamics are obtained by the vertical motion of the contact nodes and the corresponding bristle stiffness and damping characteristics. To populate the abovementioned system of differential equations, the modal properties of the rigid and the elastic belt modes are required. In the context of the present work, rigid belt modes are calculated analytically, while in-plane and out-of-plane elastic belt modes are identified experimentally by performing modal testing on the physical tyre. To this end, the eigenvalue of any particular mode is obtained by fitting a rational fraction polynomial expression to frequency response data surrounding that mode. The eigenvector calculation requires a different approach as typically modes located in the vicinity of the examined mode have an effect on the apparent residue. Consequently, an alternative method has been developed which takes into account the out-of-band modes leading to identified residues representing only the modes of interest. The validation of the proposed modelling approach is performed by comparing simulation results to experimental data and trends found in the literature. In terms of vertical stiffness, correlation with experimental data is achieved for a limited vertical load range, due to the nature of the identified modal properties. Moreover, the tyre model response to transient lateral slip is investigated for a range of longitudinal speeds and vertical loads, and the resulting relaxation length trends are compared with the relevant literature.

The purpose of this investigation is to identify appropriate location of capacitor banks and sources of reactive power by studying power system stability in the vicinity of system equilibrium states. The locations for reactive power sources are determined by identifying those modes of the system that participate most in the system behavior in general and in dictating the final state of the system after experiencing faults or disturbances. To identify the relevant modes of the system that participate most in the system dynamic, we shall make use of modal and participation analysis for different system conditions. We also apply modal and participation analysis to a system in order to identify the components of greatest impact that result in the most efficient system control. The ideas developed in this study are used to analyze and identify weak boundaries of the IEEE 39- Bus system that contribute to the system’s instability.

Um novo critério, proposto por Jorge Navaza a partir de considerações teóricas para resolver o problema das fases, é avaliado numericamente. Este critério se baseia na propriedade de atomicidade da função densidade eletrônica, generalizando resultados obtidos por Goedkoop. O problema das fases é resolvido teoricamente pela minimização de uma função, R, que é formada pela soma dos menores autovalores de uma matriz, Q, construída a partir de todos os fatores de estrutura observados. O conjunto de fases procurado é aquele que minimiza R. Como a matriz Q depende em forma relativamente complexa do grupo de simetria espacial do cristal, teoria dos grupos é utilizada para reduzir a ordem desta matriz. O algoritmo e a implantação computacional do cálculo da função R, juntamente com testes numéricos que demonstram a utilidade do critério de Navaza, são descritos em detalhe. Como corolário, que pode talvez resultar de grande importância prática, é mostrado que a função R pode ser utilizada como uma nova figura de mérito nos métodos diretos por multissolução clássicos. Finalmente, é desenvolvida a álgebra correspondente ao cálculo do gradiente da função R, indicando a direção de trabalhos futuros / A new criterion, proposed by Jorge Navaza from theoretical considerations to solve the Phase Problem, is numerically tested. The criterion is based on the atomicity property of the electron density function, generalizing previous results by Goedkoop. The Phase Problem is theoretically solved by the minimization of a function, R, which is formed from the sum of the smallest eigenvalues of a matrix Q, constructed from the set of all observed structure factors. The sought set of phases is that which minimizes R. Because the matrix Q depends in a relatively complex fashion on the space group of the crystal, group theory is employed to reduce the order of Q. The algorithm and computational implementation for the calculation of R, together with numerical tests which demonstrate the usefulness of Navaza´s criterion, are described in detail. As a corollary, that might turn out to be of a practical importance, it is shown that the minimum value of the function R can be used as a novel Figure of Merit in the classical Multisolution Direct Methods. Finally, the rather complex algebra necessary for the calculation of the gradient of the function R is developed, indicating also the possible trends for future work

In general relativity, angular momentum of the gravitational field in some volume bounded by an axially symmetric sphere is well-defined as a boundary integral. The definition relies on the symmetry generating vector field, a Killing field, of the boundary. When no such symmetry exists, one defines angular momentum using an approximate Killing field. Contained in the literature are various approximations that capture certain properties of metric preserving vector fields. We explore the continuity of an angular momentum definition that employs an approximate Killing field that is an eigenvector of a particular second-order differential operator. We find that the eigenvector varies continuously in Hilbert space under smooth perturbations of a smooth boundary geometry. Furthermore, we find that not only is the approximate Killing field continuous but that the eigenvalue problem which defines it is stable in the sense that all of its eigenvalues and eigenvectors are continuous in Hilbert space. We conclude that the stability follows because the eigenvalue problem is strongly elliptic. Additionally, we provide a practical introduction to the mathematical theory of strongly elliptic operators and generalize the above stability results for a large class of such operators. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2014. / FAU Electronic Theses and Dissertations Collection

Denna avhandling är fokuserad på modellering av elkraftsystemkomponenter och deras representation vid simuleringar av elkraftsystem. Avhandlingen jämför olika linjäriseringstekniker. Dessa tekniker är såväl numeriska som analytiska och används vid linjärisering av ett dynamiskt system. Efter en linjärisering är det möjligt att beräkna egenvärdena av det linjäriserade systemet samt använda andra verktyg ämnade för studier av linjära system. I avhandlingen visas hur olika linjäriseringtekniker influerar egenvärdesberäkningen av det linjära systemet. I avhandlingen tas fram bandviddsreducerade linjära modeller av en kraftsystemkomponent med hjälp av två tekniker. Senare görs simuleringar med de linjära modellerna tillsammans med ett introducerat gränssnitt. Den studerade kraftsystemkomponenten är en tyristorstyrd seriekondensator (TCSC). En fördel med att använda en linjär representation av en kraftsystemkomponent är att det förenklar simuleringarna. Storleken på komplexiteten av en simulering vid lösandet av ekvationerna minskar och den konsumerade fysiska tiden att simulera minskar. En nackdel med en linjär modell är att dess giltighet kan vara begränsad. Behovet av att bygga linjära modeller av kraftsystemkomponenter torde även finnas i framtiden. Med dagens horisont (år 2006) finns behov av att bygga linjära modeller utgående från detaljerade modeller av bl a högspända likströmslänkar (HVDC-länkar), reaktiva effektkompensatorer (SVC) samt tyristorstyrda seriekondensatorer (TCSC). Hur skall dessa representeras när vi vill studera dynamiken av ett helt kraftsystem och det då är nödvändigt att reducera deras komplexitet? Denna frågeställning uppkommer när vi vill genomföra tidsdomänsimuleringar på en inte alltför detaljerad nivå av de individuella kraftsystemkomponenterna eller när vi vill linjärisera kraftsystemet för att studera dess stabilitet med hjälp av småsignalanalys. / This thesis is focused in modelling of power system components and their representation in simulations of power systems. The thesis compares different linearization techniques. These techniques are both numerical as well as analytical and are utilized when linearization of a dynamic system is desired. After a linearization it is possible to calculate the eigenvalues of the linearized system as well as to perform other applicable activities on a linear system. In the thesis it is shown how the linearization techniques influence the calculation of eigenvalues of the linear system. In the thesis bandwidth-reduced linear models of a power system component are developed using two techniques. The simulations with the linear models are done with an introduced interface system. The studied power system component is a Thyristor-Controlled Series Capacitor (TCSC). One advantage with using a linear representation of a power system component is that it simplifies the simulations. The size of the complexity of a simulation when solving the equations decreases and the consumed physical time to simulate becomes shorter. A disadvantage of a linear model is that its validity might be limited. The need of building linear models of power systems will continue to attract interest in the future. With the horizon of today (year 2006) there is a need of among other models to build linear models of detailed models of High Voltage Direct Current-links (HVDC-links), Static Var Compensators (SVCs), as well as Thyristor-Controlled Series Capacitors (TCSCs). How should these be represented when we want to study the dynamics of a whole power system and it is necessary to reduce their complexity? This question rises when we want to perform time-domain simulations with a not too detailed level of complexity of each individual power system component or if we want to linearize the power system and study it within small-signal stability analysis. / QC 20100915

This thesis develops ways of localizing the Ritz values of non-Hermitian matrices. The restarted Arnoldi method with exact shifts, useful for determining a few desired eigenvalues of a matrix, employs Ritz values to refine eigenvalue estimates. In the Hermitian case, using selected Ritz values produces convergence due to interlacing. No generalization of interlacing exists for non-Hermitian matrices, and as a consequence no satisfactory general convergence theory exists. To study Ritz values, I propose the inverse field of values problem for k Ritz values, which asks if a set of k complex numbers can be Ritz values of a matrix. This problem is always solvable for k = 1 for any complex number in the field of values; I provide an improved algorithm for finding a Ritz vector in this case. I show that majorization can be used to characterize, as well as localize, Ritz values. To illustrate the difficulties of characterizing Ritz values, this work provides a complete analysis of the Ritz values of two 3 × 3 matrices: a Jordan block and a normal matrix. By constructing conditions for localizing the Ritz values of a matrix with one simple, normal, sought-after eigenvalue, this work develops sufficient conditions that guarantee convergence of the restarted Arnoldi method with exact shifts. For general matrices, the conditions provide insight into the subspace dimensions that ensure that shifts do not cluster near the wanted eigenvalue. As Ritz values form the basis for many iterative methods for determining eigenvalues and solving linear systems, an understanding of Ritz value behavior for non-Hermitian matrices has the potential to inform a broad range of analysis.

We show that amongst all trees with a given degree sequence it is a ball where the vertex degrees decrease with increasing distance from its center that maximizes the spectral radius of the graph (i.e., its adjacency matrix). The resulting Perron vector is decreasing on every path starting from the center of this ball. This result it also connected to Faber-Krahn like theorems for Dirichlet matrices on trees. The above result is extended to connected graphs with given degree sequence. Here we give a necessary condition for a graph that has greatest maximum eigenvalue. Moreover, we show that the greatest maximum eigenvalue is monotone on degree sequences with respect to majorization. (author's abstract). Note: There is a more recent version of this paper available: "Graphs with Given Degree Sequence and Maximal Spectral Radius", Research Report Series / Department of Statistics and Mathematics, no. 72. / Series: Research Report Series / Department of Statistics and Mathematics

In this study, the mapped Galerkin method was applied to solve the vector wave equation based on H&amp / #8722 / field and to obtain the propagation constant in x &amp / #8722 / y space. The vector wave equation was solved by the transformation of the infinite x &amp / #8722 / y plane onto a unit square. Two-dimensional Fourier series expansions were used in the solutions. Modal fields and propagation constants of dielectric waveguides were calculated. In the first part of the study, all of the calculations were made in step index fibers. Transverse magnetic fields were obtained in the u &amp / #8722 / v and x &amp / #8722 / y space through the solution of the matrix eigenvalue equation. Some graphics were plotted in the light of the results obtained. The results are found to be in accord with the results of other numerical techniques and exact solutions. After that, the propagation constant in x&amp / #8722 / y space was calculated with ease using the solution of the modal field components. In the second part of the study, the similar calculations were made in graded index fibers.

Darbą sudaro: įvadas, analitinė ir praktinės dalys. Analitinėje dalyje trumpai aptariama su nagrinėjamu uždaviniu susijusi teorija ir pats uždavinys. Analitinėje dalyje yra trys poskyriai, juose pateikiama teorija, reikalinga nagrinėjamo uždavinio tyrimui. Praktinėje dalyje nagrinėjamas Šturmo ir Liuvilio kraštinis uždavinys su viena klasikine ir kita nelokalia dvitaške kraštine sąlyga. Ištirti keturių nelokalių kraštinių sąlygų atvejai. Kiekvienu kraštinių sąlygų atveju ieškomos kompleksinės tikrinės reikšmės ir tiriama jų kokybinė priklausomybė nuo uždavinio nelokaliosios sąlygos parametrų ir . Darbas iliustruotas charakteristinių funkcijų grafikais. Nustatyta, kad dviejų pirmųjų sąlygų atveju yra pakankamai nesudėtinga charakterisitnių funkcijų priklausomybė nuo parametro . Kitiems dviems atvejams ta priklausomybė yra žymiai sudėtingesnė. Tokios situacijos kruopščiai ištirtos ketvirtame atvejyje. Surasta tikrinių reikšmių elgsena bifurkacinių taškų aplinkoje. / This master thesis consists of introduction, analitical and practical parts. In the analitical part are considered the master thesis problem and theoretical studies, that are correlative with it. This part is rubricated to tree sections. They are designed for theoretical studies that are used to solve all problem of the master thesis. In practical part are examinated four problems: Šturm Louville problem with classical and nonlocal boundary condition. Here are investigated types of four nonlocal boundary conditions. In case of every nonlocal boundary conditions locking for complex eigenvalues and investigating thier quality that dependsupon nonlocal boundary condition parametere and . The work is pictorial of charakterical fukcion graphics. Is determinated, that in case of two condition charakteristical funkcion dependen of parameter is simple. In case of other two condition the dependen is convulated. This situation is examinated in the fourth case. Besides here resolved behaviour of eigenvalues in their points environment.