In this thesis we find deep results on the structure of normal right n-Engel subgroups that are contained in some term of the upper central series of a group. We start with some known results, and one new result, on the structure of locally nilpotent n-Engel groups. These are closely related to the solution of the restricted Burnside problem. We also give specific details of the structure of 2-Engel and 3-Engel groups in the context of these results. The main idea of this thesis is to generalise these results to apply to normal upper central right n-Engel subgroups. We also consider the special case of locally finite p-groups and again generalise some deep results on the structure of n-Engel such groups to apply to right n-Engel subgroups. For each of the theorems on right n-Engel subgroups, complete details are given for the case n = 2. Right 3-Engel subgroups have a more complicated structure. For these we prove a Fitting result, for which we exclude the prime 3, and using this we also find a sharp bound on the upper central degree in the torsion-free case. In fact we only need to exclude the primes 2, 3 and 5 for this result. This gives some further information on the structure of right 3-Engel subgroups in the context of the main theorems.

This paper is a study of the properties of commutators, and deals exclusively with finite groups.

An (associative) algebra is a vector space over a field equipped with an associative, bilinear multiplication. By use of a new bilinear operation, any associative algebra morphs into a nonassociative abstract Lie algebra, where the new product in terms of the given associative product, is the commutator. The crux of this paper is to investigate the commutator as it pertains to the general linear group and its subalgebras. This forces us to examine properties of ring theory under the lens of linear algebra, as we determine subalgebras, ideals, and solvability as decomposed into an extension of abelian ideals, and nilpotency, as decomposed into the lower central series and eventual zero subspace. The map sending the Lie algebra L to a derivation of L is called the adjoint representation, where a given Lie algebra is nilpotent if and only if the adjoint is nilpotent. Our goal is to prove Engel's Theorem, which states that if all elements of L are ad-nilpotent, then L is nilpotent.

The subject of the thesis is the study of operator functions in the setting of symmetric operator spaces. In this latter setting, it is of great importance to analyze the properties of so-called operator functions A --> f(A), where the variable A is a self-adjoint operator and f is a complex-valued Borel function on the real line. The thesis study the question of differentiability of this type of operator functions. The latter question is intimately related to the study of commutators. Text not only extends existing results to the setting of unbounded self-adjoint linear operators, but it is also shown that this can be obtained via a unified approach utilizing the left regular representation of von Neumann algebras.

No description available.

Rotor Position information is essential in the operation of the Switched Reluctance Motor (SRM) for properly controlling its phase currents. This thesis uses Field Programmable Gate Array (FPGA) technology to implement a method to estimate the SRMs rotor position using the inverse inductance value of the SRMs phases. The estimated rotor position is given as input to the Commutator circuit, also implemented in the FPGA, to determine when torque-producing currents should be input in the SRM phase windings. The Estimator and Commutator design is coded using Verilog HDL and is simulated using Xilinx tools. This circuit is implemented on a Xilinx Virtex XCV800 FPGA system. The experimentally generated output is validated by comparing it with simulation results from a Simulink model of the Estimator. The performance of the FPGA based SRM rotor position estimator in terms of calculation time is compared to a digital signal processor (DSP) implementation of the same position estimator algorithm. It is found that the FPGA rotor position Estimator with a 5MHz clock can update its rotor position estimate every 7s compared to an update time of 50s for a TMS320C6701-150 DSP implementation using a commercial DSP board. This is a greater than 7 to one reduction in the update time.

This work is dedicated to small distance contact-less measurement. Especially the surface detection of the commutator or collector of the electrical machines was described. The probes utilising the induction principle of the distance measurement have been developed. The part of the work described the history of the induction distances sensing in FEEC BUT and the using of the induction principle in the industry small distances sensing nowadays. The paper concretely determines the possibilities of the small distances measurement using mentioned probes. The sensitivity for different materials was investigated and the error of the measurement for different measuring object was investigated, too. The effects, that could affected the measurement error, were described in the work. The contribution of the each effect for the total measuring error was specified. Next part is including the description of the measuring workstation and of the hardware and software. At the end the verification of the measuring principle and measuring workstation is presented. The verification has been made on three different commutator types. The verification results are in the conclusion.

Neste trabalho, apresentamos uma nova versão do injetor comutador, baseada na geometria retangular, mantendo a configuração de três peças, sendo a central móvel e as laterais fixas. Nesta condição, os parafusos de ajuste não funcionam como guia, tal como ocorre nos injetores da primeira geração. Este design facilita o deslocamento da parte móvel, contribuindo para a redução do atrito, o que permite o emprego de um pequeno motor de corrente contínua (12 V; 0,4 A), o qual foi usado para fazer o deslocamento da barra central da posição de amostragem para a de injeção e vice-versa. O funcionamento do injetor foi controlado pelo computador, usando apenas 2 bits de controle. A interface de controle projetada para esse fim, inclui a condição de reversão da direção de rotação do motor e lógica anti-duplicidade, então se os dois bits forem ativados ao mesmo tempo, a rotação do motor é mantida no sentido horário. Para testar a viabilidade deste injetor, o mesmo foi empregado para desenvolver um procedimento para determinação de etanol em bebidas destiladas. O procedimento fotométrico foi baseado na reação de etanol com dicromato de potássio. A detecção fotométrica foi realizada empregando um fotômetro de LED com máximo de emissão em 590 nm. O módulo de análise constituído pelo injetor e um conjunto de válvulas solenoide foi projetado para implementar o processo de reamostragem, adição intermitente de reagentes e transporte da zona de amostra com fluxo de ar. Após a definição das variáveis de controle, o sistema apresentou bom desempenho analítico, aliando vantagens tais como, boa velocidade, simplicidade de operação, versatilidade, proporcionando também um menor consumo de amostra e reagentes. Com relação às características analíticas, o sistema proposto apresentou faixa de resposta linear (r = 0,9972) para concentração etanol entre 10 - 50 % (v/v), limite de detecção de 2,0 % (v/v) etanol, desvio padrão relativo de 2,0 % (n =10) para uma amostra de etanol com concentração 40 % (v/v), um consumo de dicromato de 0,012 g por determinação e uma frequência analítica de 34 determinações por hora. / We present a new version of the commutator injector, based on the rectangular geometry, maintaining the three parts configuration with the central mobile and fixed side. In this condition, the adjusting screws do not functi as a guide, as occurs in the injector of first generation. This design facilitates the displacement of the movable part and contributes in reducing the friction, thus allowing the use of a small direct current motor (12 V, 0.4 A), which was used to perform the displacement of the central bar from the sampling position to the injection and vice-versa. The injector operation was controlled by a computer, which was done using only two control bits. The control interface designed for this purpose includes the condition for reversing the direction of the motor rotation and anti-duplication logic, so that both bits were activated at the same time, the motor rotation was maintained at clockwise direction. To test the feasibility of the injector, it was used to develop a procedure for determination of ethanol in spirit beverage. The procedure was based on the photometric reaction of ethanol with potassium dichromate. The photometric detection was performed using a LED based photometer with maximum of emission at 590 nm. The analytical module comprising the injector and a set of solenoid valves, was designed to implement the zone sampling process and Intermittent addition of reagents and sample zone displacement using an air stream. After definition of control variables, the system shown good analytical performance, combining advantages such as speed, simplicity of operation, versatility and also providing lower consumption of sample and reagents. With respect to analytical characteristics, the proposed system presented linear response range (r = 0.9972) between ethanol concentration to 10 - 50% (v/v), a detection limit of 2.0% (v/v) ethanol, relative standard deviation of 2.0% (n = 10) for a sample with an ethanol concentration of 40% (w/v), a dichromate consumption of 0.012 g by determination and a sampling rate of 34 determination per hour.

This thesis is devoted to use Lie group analysis to obtain all invariant solutions by constructing optimal system of one-dimensional subalgebras of the Lie algebra L5 for a nonlinear wave equation. I will show how the given symmetries ( Eq.2) are admitted by using partial differential equation (Eq.1), In addition to obtain the commutator table by using the same given symmetries. Subsequently, I calculate the transformations of the generators with the Lie algebra L5, which provide the 5-parameter group of linear transformations for the operators. Finally, I construct the invariant solutions for each member of the optimal system.

The main purpose of this thesis is to use modern goal-oriented adaptive methods of Lie group analysis to construct the optimal sys- tem of Black-Scholes equation. We will show in this thesis how to obtain all invariant solutions by constructing what has now become so popular, optimal system of sub-algebras, the main Lie algebra admit- ted by the Black-Scholes equation. First, we obtain the commutator table of already calculated symmetries of the Black-Scholes equation. We then followed with the calculations of transformation of the gen- erators with the Lie algebra L6 which provides one-parameter group of linear transformations for the operators. Here we make use of the method of Lie equations to solve the partial di®erential equations. Next, we consider the construction of optimal systems of the Black- Scholes equation where the method requires a simpli¯cation of a vector to a general form to each of the transformations of the generators. Further, we construct the invariant solutions for each of the op- timal system. This study is motivated by the analysis of Lie groups which is being taken to another level by ALGA here in Blekinge In- stitute Technology, Sweden. We give a practical and in-depth steps and explanation of how to construct the commutator table, the calcu- lation of the transformation of the generators and the construction of the optimal system as well as their invariant solutions. Keywords: Black-Scholes Equation, commutators, commutator table, Lie equa- tions, invariant solution, optimal system, generators, Airy equation, structure constant, / It was an accolade for us to work with Professor Nail.H. Ibrgimov. +46762600953