Privileged Words and Sturmian Words

This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.

Effect of fertilizer treatment on the biomass production and nutrient uptake of short-rotation willow on cut-away peatlands

Summary

Biomass production and nutrient uptake of short-rotation plantations

Summary

ISBN (international standard book number) -an international perspective

Esitelmä Suomen ISBN-keskus 30 vuotta -juhlassa Helsingin yliopistossa 7.11.2002

Effects of rotation period on biomass production and atmospheric CO‚́‚ emissions from broadleaved stands growing on abandoned farmland

Abstract

Linear prediction application for modelling the relationships between a large number of stand characteristics of Norway spruce stands

Abstract

Performance Evaluation of Six-Sectored Configuration in Hexagonal WCDMA (UMTS) Cellular Network Layout

The objective of this master's thesis is to evaluate the optimum performance of sixsectored hexagonal layout of WCDMA (UMTS) network and analyze the performance at the optimum point. The maximum coverage and the maximum capacity are the main concern of service providers and it is always a challenging task for them to achieve economically. Because the optimum configuration of a network corresponds to a configuration which minimizes the number of sites required to provide a target service probability in the planning area which in turn reduces the deployment cost. The optimum performance means the maximum cell area and themaximum cell capacity the network can provide at the maximum antenna height satisfying the target service probability. Hexagon layout has been proven as the best layout for the cell deployment. In this thesis work, two different configurations using six-sectored sites have been considered for the performance comparison. In first configuration, each antenna is directed towards each corner of hexagon, whereas in second configurationeach antenna is directed towards each side of hexagon. The net difference in the configurations is the 30 degree rotation of antenna direction. The only indoor users in a flat and smooth semi-urban environment area have been considered for the simulation purpose where the traffic distribution is 100 Erl/km2 with 12.2 kbps speech service having maximum mobile speed of 3 km/hr. The simulation results indicate that a similar performance can be achieved in both the configurations, that is, a maximum of 947 m cellrange at antenna height of 49.5 m can be achieved when the antennas are directed towards the corner of hexagon, whereas 943.3 m cell range atantenna height of 54 m can be achieved when the antennas are directed towards the side of hexagon. However, from the interference point of view the first configuration provides better results. The simulation results also show that the network is coverage limited in both the uplink and downlink direction at the optimum point.

Torque vibration model of axial-flux surface-mounted permanent magnet synchronous machine

In order that the radius and thus ununiform structure of the teeth and otherelectrical and magnetic parts of the machine may be taken into consideration the calculation of an axial flux permanent magnet machine is, conventionally, doneby means of 3D FEM-methods. This calculation procedure, however, requires a lotof time and computer recourses. This study proves that also analytical methods can be applied to perform the calculation successfully. The procedure of the analytical calculation can be summarized into following steps: first the magnet is divided into slices, which makes the calculation for each section individually, and then the parts are submitted to calculation of the final results. It is obvious that using this method can save a lot of designing and calculating time. Thecalculation program is designed to model the magnetic and electrical circuits of surface mounted axial flux permanent magnet synchronous machines in such a way, that it takes into account possible magnetic saturation of the iron parts. Theresult of the calculation is the torque of the motor including the vibrations. The motor geometry and the materials and either the torque or pole angle are defined and the motor can be fed with an arbitrary shape and amplitude of three-phase currents. There are no limits for the size and number of the pole pairs nor for many other factors. The calculation steps and the number of different sections of the magnet are selectable, but the calculation time is strongly depending on this. The results are compared to the measurements of real prototypes. The permanent magnet creates part of the flux in the magnetic circuit. The form and amplitude of the flux density in the air-gap depends on the geometry and material of the magnetic circuit, on the length of the air-gap and remanence flux density of the magnet. Slotting is taken into account by using the Carter factor in the slot opening area. The calculation is simple and fast if the shape of the magnetis a square and has no skew in relation to the stator slots. With a more complicated magnet shape the calculation has to be done in several sections. It is clear that according to the increasing number of sections also the result will become more accurate. In a radial flux motor all sections of the magnets create force with a same radius. In the case of an axial flux motor, each radial section creates force with a different radius and the torque is the sum of these. The magnetic circuit of the motor, consisting of the stator iron, rotor iron, air-gap, magnet and the slot, is modelled with a reluctance net, which considers the saturation of the iron. This means, that several iterations, in which the permeability is updated, has to be done in order to get final results. The motor torque is calculated using the instantaneous linkage flux and stator currents. Flux linkage is called the part of the flux that is created by the permanent magnets and the stator currents passing through the coils in stator teeth. The angle between this flux and the phase currents define the torque created by the magnetic circuit. Due to the winding structure of the stator and in order to limit the leakage flux the slot openings of the stator are normally not made of ferromagnetic material even though, in some cases, semimagnetic slot wedges are used. In the slot opening faces the flux enters the iron almost normally (tangentially with respect to the rotor flux) creating tangential forces in the rotor. This phenomenon iscalled cogging. The flux in the slot opening area on the different sides of theopening and in the different slot openings is not equal and so these forces do not compensate each other. In the calculation it is assumed that the flux entering the left side of the opening is the component left from the geometrical centre of the slot. This torque component together with the torque component calculated using the Lorenz force make the total torque of the motor. It is easy to assume that when all the magnet edges, where the derivative component of the magnet flux density is at its highest, enter the slot openings at the same time, this will have as a result a considerable cogging torque. To reduce the cogging torquethe magnet edges can be shaped so that they are not parallel to the stator slots, which is the common way to solve the problem. In doing so, the edge may be spread along the whole slot pitch and thus also the high derivative component willbe spread to occur equally along the rotation. Besides forming the magnets theymay also be placed somewhat asymmetric on the rotor surface. The asymmetric distribution can be made in many different ways. All the magnets may have a different deflection of the symmetrical centre point or they can be for example shiftedin pairs. There are some factors that limit the deflection. The first is that the magnets cannot overlap. The magnet shape and the relative width compared to the pole define the deflection in this case. The other factor is that a shifting of the poles limits the maximum torque of the motor. If the edges of adjacent magnets are very close to each other the leakage flux from one pole to the other increases reducing thus the air-gap magnetization. The asymmetric model needs some assumptions and simplifications in order to limit the size of the model and calculation time. The reluctance net is made for symmetric distribution. If the magnets are distributed asymmetrically the flux in the different pole pairs will not be exactly the same. Therefore, the assumption that the flux flows from the edges of the model to the next pole pairs, in the calculation model from one edgeto the other, is not correct. If it were wished for that this fact should be considered in multi-pole pair machines, this would mean that all the poles, in other words the whole machine, should be modelled in reluctance net. The error resulting from this wrong assumption is, nevertheless, irrelevant.

Design of axial-flux permanent-magnet low-speed machines and performance comparison between radial-flux and axial-flux machines

This thesis presents an alternative approach to the analytical design of surface-mounted axialflux permanent-magnet machines. Emphasis has been placed on the design of axial-flux machines with a one-rotor-two-stators configuration. The design model developed in this study incorporates facilities to include both the electromagnetic design and thermal design of the machine as well as to take into consideration the complexity of the permanent-magnet shapes, which is a typical requirement for the design of high-performance permanent-magnet motors. A prototype machine with rated 5 kW output power at 300 min-1 rotation speed has been designed and constructed for the purposesof ascertaining the results obtained from the analytical design model. A comparative study of low-speed axial-flux and low-speed radial-flux permanent-magnet machines is presented. The comparative study concentrates on 55 kW machines with rotation speeds 150 min-1, 300 min-1 and 600 min-1 and is based on calculated designs. A novel comparison method is introduced. The method takes into account the mechanical constraints of the machine and enables comparison of the designed machines, with respect to the volume, efficiency and cost aspects of each machine. It is shown that an axial-flux permanent-magnet machine with one-rotor-two-stators configuration has generally a weaker efficiency than a radial-flux permanent-magnet machine if for all designs the same electric loading, air-gap flux density and current density have been applied. On the other hand, axial-flux machines are usually smaller in volume, especially when compared to radial-flux machines for which the length ratio (axial length of stator stack vs. air-gap diameter)is below 0.5. The comparison results show also that radial-flux machines with alow number of pole pairs, p < 4, outperform the corresponding axial-flux machines.

Experimental insights into deformation dynamics and intermittency in rapid granular shear flows

This work concerns the experimental study of rapid granular shear flows in annular Couette geometry. The flow is induced by continuous driving of the horizontal plate at the top of the granular bed in an annulus. The compressive pressure, driving torque, instantaneous bed height and rotational speed of the shearing plate are measured. Moreover, local stress fluctuations are measured in a medium made of steel spheres 2 and 3 mm in diameter. Both monodisperse packing and bidisperse packing are investigated to reveal the influence of size diversity in intermittent features of granular materials. Experiments are conducted in an annulus that can contain up to 15 kg of spherical steel balls. The shearing granular medium takes place via the rotation of the upper plate which compresses the material loaded inside the annulus. Fluctuations of compressive force are locally measured at the bottom of the annulus using a piezoelectric sensor. Rapid shear flow experiments are pursued at different compressive forces and shear rates and the sensitivity of fluctuations are then investigated by different means through monodisperse and bidisperse packings. Another important feature of rapid granular shear flows is the formation of ordered structures upon shearing. It requires a certain range for the amount of granular material (uniform size distribution) loaded in the system in order to obtain stable flows. This is studied more deeply in this thesis. The results of the current work bring some new insights into deformation dynamics and intermittency in rapid granular shear flows. The experimental apparatus is modified in comparison to earlier investigations. The measurements produce data for various quantities continuously sampled from the start of shearing to the end. Static failure and dynamic shearing ofa granular medium is investigated. The results of this work revealed some important features of failure dynamics and structure formation in the system. Furthermore, some computer simulations are performed in a 2D annulus to examine the nature of kinetic energy dissipation. It is found that turbulent flow models can statistically represent rapid granular flows with high accuracy. In addition to academic outcomes and scientific publications our results have a number of technological applications associated with grinding, mining and massive grain storages.