The aim of this work is to study the interaction between two classical objects of mathematics: the modular group, and the absolute Galois group. The latter acts on the category of finite index subgroups of the modular group. However, it is a task out of reach do understand this action in this generality. We propose a lattice which parametrizes a certain system of &rdquo / geometric&rdquo / elements in this category. This system is setwise invariant under the Galois action, and there is a hope that one can explicitly understand the pointwise action on the elements of this system. These elements admit moreover a combinatorial description as quadrangulations of the sphere, satisfying a natural nonnegative curvature condition. Furthermore, their connections with hypergeometric functions allow us to realize these quadrangulations as points in the moduli space of rational curves with 8 punctures. These points are conjecturally defined over a number field and our ultimate wish is to compare the Galois action on the lattice elements in the category and the corresponding points in the moduli space.

The research described in this dissertation deals with an open problem concerning the synthesis of controllers of xed order and structure. This problem is encountered in a variety of applications. Simply put, the problem may be put as the determination of the set, S of controller parameter vectors, K = (k1; k2,...,kl), that render Hurwitz a family (indexed by F) of complex polynomials of the form {P0(s.a) + [summation] i=1 Pi(s,a)ki, a [set membership] F}, where the polynomials Pj(s,a), j = 0,...,l are given data. They are specied by the plant to be controlled, the structure of the controller desired and the performance that the controllers are expected to achieve. Simple examples indicate that the set S can be non-convex and even be disconnected. While the determination of the non-emptiness of S is decidable and amenable to methods such as the quantier elimination scheme, such methods have not been computationally tractable and more importantly, do not provide a reasonable approximation for the set of controllers. Practical applications require the construction of a set of controllers that will enable a control engineer to check the satisfaction of performance criteria that may not be mathematically well characterized. The transient performance criteria often fall into this category. From the practical viewpoint of the construction of approximations for S, this dissertation is dierent from earlier work in the literature on this problem. A novel feature of the proposed algorithm is the exploitation of the interlacing property of Hurwitz polynomials to provide arbitrarily tight outer and inner approximation to S. The approximation is given in terms of the union of polyhedral sets which are constructed systematically using the Hermite-Biehler theorem and the generalizations of the Descartes' rule of signs.

A spectacular instability has been observed in low density round jets when the density ratio of jet fluid to ambient fluid falls below a threshold of approximately 0.6. This phenomenon has been observed in non-buoyant jets of helium in air, heated air jets and heated buoyant jets. The oscillation of the flow near the nozzle is extremely regular and periodic and consists of ring vortices. Even the smaller scale structures that appear downstream exhibit similar regularity. A theory for predicting the onset of this oscillation is based on finding regions of absolute instability from linear stability analysis of parallel flow. However, experiments suggest that the theory is at least incomplete and fortuitous as the oscillation is not a linear process. The present work is to observe and understand the process of regeneration of these oscillations by conducting numerical simulations. Here, two-dimensional, plane jets were simulated because they undergo a qualitatively similar process. A spatial and temporal picture of a heated jet has been obtained numerically. A perturbation expansion was used to obtain a system of conservation laws for compressible flows which is valid for low Mach numbers. The low Mach number approximation removes the high frequency acoustic waves from the flow field. This enables a larger time step to be taken without making the calculation unstable. To ensure that all the scales of motion are properly resolved, calculations were done at a low Reynolds number. The governing equations were discretized in space using second-order finite difference formulas on a staggered grid. Velocity fields were advanced using a second-order Adams-Bashforth explicit scheme and then corrected by solving for pressure such that continuity is satisfied at every time step. The Poisson problem for pressure requires the time derivative of the density which was approximated by a third-order backward difference formula. Gauss-Siedel iteration was used to find the pressure. Several numerical tests were conducted prior to simulations of variable density jets to check the stability and accuracy of the code. Two dimensional driven cavity flow calculations were done as a first test. Then a calculation of a forced, spatially developing, incompressible, plane mixing layer was done to check the time accuracy of the code. After obtaining satisfactory performance of the code for the different test cases, two-dimensional, variable density jets were simulated. Since the plane jet extends ad infinitum in the streamwise direction, a sufficiently large domain was used to capture all the relevant physics in the downstream regions of the jet. An advective boundary condition was imposed at the exit plane. Rigid, slipwall conditions were employed to prescribe lateral boundary conditions. A 2-D, incompressible plane jet was simulated first. The jet profile was approximated by two hyperbolic tangent shear layers. The most unstable mode of the inviscid shear layer for this profile, along with its first and second harmonics, was imposed on the velocity profile at the inlet plane. The amplitude of oscillation of the harmonics was chosen so as to provide sufficient energy in the perturbation to accelerate the growth of the layer. No explicit phase lag was introduced in the perturbation. The flow was allowed to develop long enough to wash out the effect of the initial condition. The results obtained for this case indicate that experimentally realized phenomena such as vortex pairing were captured in this simulation. Furthermore, to check the convective nature of instability of the incompressible jet, the forcing at the inlet plane was turned off. The disturbances were gradually convected downstream, out of the computational domain. Next, two-dimensional heated, non-buoyant jets were studied numerically. The effects of the ratio of jet density to ambient density S, the velocity ratio R, and jet width W, on the near field behavior of an initial laminar jet and the regeneration mechanism of the self-sustaining vortices were explored. The theory based on domain of absolute/convective instability identifies these three parameters. No initial perturbation was necessary to start roll-up of the shear layer. For certain choices, e.g., S= 0.75, R = 20, W =10.5, self-sustaining oscillations appeared spontaneously, and these cycles repeated for very long simulation intervals. Waviness on the jet shear layers grow and roll-up into vortices as in constant density shear layers. But unlike the incompressible plane jet, these vortices grow much larger and mixes more with the surrounding fluid. As these vortices evolve, packets of fluid break away as trailing legs similar to side jet expulsions observed in round jets and plumes. The growing vortices disturb the upstream shear layer. Consistently with linear theory, which predicts absolute instability for these parameters, these disturbances are able to grow and roll up. If these disturbances travelled faster than the downstream vortices, it would not be possible for the cycle to repeat. With sufficient shear between the co-flowing streams (R not too small), the entire regeneration process was found to begin from roughly the same streamwise location. Furthermore, it is the symmetric, varicose mode which occurs. At a slightly larger density ratio (S = 0.8, R = 10), self-sustaining oscillations appeared, but each new cycle began slightly farther downstream. It seems likely that these values are close to the boundary in parameter space between self-sustained oscillatory and convectively unstable behaviors. Jet width also influences the selection of these two behaviors. When jet width was reduced, W = 6, even for S = 0.75,R = 20, each new cycle began to shift downstream. For larger jet width (W = 12.3), self-sustaining oscillations occur but the response is now as an asymmetric sinuous mode after a short initial varicose mode. The detailed processes that have now been revealed in plane jets should serve as guidelines for the study of such processes in the technologically more important round jets.

<p>This corpus based investigation deals with the present-day usage of and the semantic relation between the two verbs rise and arise. Concordance lines containing various forms of the two verbs in question have been taken from six different (sub)corpora and were examined in view of their collocational and semantic characteristics. The basic aims were to investigate the nowadays status of the verbs rise and arise and whether they (still) can be regarded as synonyms. The results show that both verbs can sometimes be used synonymously. Their general semantic relation appeared to be near-synonymy. Furthermore, both verbs seem to have developed a semantic specialisation, which is regarded a counterargument for the thesis that the verb arise is on the verge of dying out.</p>

Im Niedertemperaturplasma einer Stickstoff-Hochfrequenzgasentladung werden mit Hilfe der plasmadiagnostischen Methoden LIF und OES räumliche Konzentrationsprofile der N2-Triplett-Zustände A, B und C in verschiedenen Schwingungsniveaus studiert. Dabei wird die Druckabhängigkeit im Bereich 30 bis 100 Pa und die Abhängigkeit von der Brennspannung bis 250 V untersucht. Die deutlich unterschiedlichen räumlichen Verteilungen der verschiedenen Spezies werden mit Hilfe einer Modellierung beschrieben, die sowohl Elektronenstoßanregung in der Plasmarandschicht als auch Diffusion, Stoßabregung, Kaskadenprozesse sowie Deaktivierung durch Wandstöße berücksichtigt. Dabei zeigt sich, daß vor allem Wandstöße für die räumliche Verteilung der langlebigen Spezies A(v=0,1) von zentraler Bedeutung sind. Die kurzlebigen C und B finden sich hingegen hauptsächlich in der Plasmarandschicht. Durch die Kombination der LIF mit Rayleighstreulichtexperimenten werden absolute Teilchendichten der drei Triplett-Zustände bestimmt.

The study of microstructures in brightfield microscopy using unbiased stereology plays a large and growing role in bioscience research. Stereology enables objective quantitative analysis of biological structures within a tissue sample. A first step in the stereology process is to calculate the thickness of a tissue sample by locating the top and bottom surfaces of the sample. The aim of this project is to fully automate this location process by using the relative optical focus measure as an indicator of tissue surface boundary. The current method for identification of focus bounding planes requires a trained user to manually select the top and bottom of the tissue at each sample position examined. To automate finding the correct focal planes, i.e. the "just out of focus" planes at the top and bottom surfaces of the tissue sections, a novel approach was developed. Several gray scale focusing functions were analyzed, but while the traditional emphasis of microscopy focus functions is to find global maximums on the focus curve, in this project the aim was to find the sharp "knees" on the focus curve. Starting with a low focus measure value when the focal plane of the objective lens is out of focus above the tissue sample, the objective focal plane is moved downward through the tissue. The ideal focus measure should increase sharply as the upper surface of the tissue passes into the depth of field of the objective lens. As the focal plane is moved through the tissue, the focus measure value rises and falls as objects within the tissue come in and out of focus. As the bottom tissue surface passes into the depth of field the ideal focus measure should reflect some level of focus, dropping precipitously as the surface passes out of the depth of field into the unfocused region below the tissue.

A key proposition of Gottfredson and Hirschi's (1990) self-control theory is the stability hypothesis which suggests that an individual's level of self-control, once established between the ages of 8-10, is stable over the life course. Empirical results from examinations of the stability hypothesis have been mixed. Prior tests of the stability hypothesis have employed aggregate assessment methods (e.g., mean-level and correlational analyses). Such approaches fail to take into account the possibility that individual developmental pathways may differ. This study employs individual longitudinal data over a four year period for 3,249 7th to 10th grade subjects to assess the stability hypothesis using both traditional stability estimation techniques (e.g., ANOVAs and zero-order correlations), as well as heterogeneity assessment methods - semiparametric group-based trajectory modeling (SPGM). Multinomial logistic regression (MLOGIT) of theoretically and empirically relevant risk factors (i.e., parenting, parental criminality, deviant peers, bonds to school) was employed to distinguish between developmental trajectories. SPGM results suggest that self-control is stable for a majority of the sample; however, a sizeable portion of the sample evinced trajectories for which self-control was marked by considerable change. Specifically, 6 unique trajectories in the development of self-control were identified - two groups were identified with high stable trajectories of self-control and four groups were identified that had lower, less stable trajectories of self-control. Additionally, several risk factors differentiated these groups. The results indicate that those with lower, less stable trajectories have more deviant peer association, higher rates of parental criminality, less intense bonds to school, and lower levels of parenting. These results indicate that self control is not stable nor is it consistent across groups, leading to a rejection of Hirschi and Gottfredson's explanation.

Muzikinės klausos ugdymas – tiek absoliučios, tiek reliatyvios solmizacijos sistemų objektai. Jis abiem solmizacijos sistemoms yra vienodas: pastovūs ir nepastovūs laipsniai, gama (kaip mažoro, minoro dermės išraiška), intervalai, akordai ir t.t. Esminis skirtumas tarp absoliučios ir realiatyvios sistemos yra mokomosios medžiagos, mokymo turinio apimtyje. Šiuo darbu nesiūloma taikyti vieną iš sistemų. Tai kiekvieno muzikos pedagogo apsisprendimas. Tačiau jis turi būti apgalvotas, motyvuotas ir sąmoningas. Svarbus vaidmuo, lemiantis sėkmingą vaikų muzikinę veiklą, tenka šiam darbui tinkamai pasirengusiems muzikos mokytojams. Kiekvienas pedagogas turi gerai susipažinti tiek su reliatyvia, tiek su absoliučia solmizacijos sistema, nes nėra vieningos nuomonės dėl abiejų sistemų privalumų ir trūkumų. / Development of musical hearing is the object of both absolute and relative systems of solmisation. It is the same for both systems of solmisation: constant and unconstant degrees, scale (as the expression of major and minor codes), intervals, chords and so on. The main difference between absolute and relative systems is in the extent of teaching material and teaching content. This paper does not aim at the proposal of one of these systems. It is the decision of each teacher of music. However, it must be thought over, motivated and made deliberately. Teachers well-prepared for this work play a very important, role in the successful musical activity of children. Each teacher must be well acquainted both with relative and absolute systems of solmisation as there is no unanimuos opinion concerning advantages and drawbacks of each of them.

Liudytojų anonimiškumas - procesinė apsaugos priemonė, kurią taikant siekiama apsaugoti liudytoją nuo nusikalstamo poveikio bei gauti teisingus parodymus, minimaliai ribojant fundamentalią kaltinamojo teisę į gynybą ir teisingą teismą. Tiek anonimiškumo taikymas, tiek jo modelio pasirinkimas turi būti atliekami laikantis konstitucinio proporcingumo principo. Liudytojui apsaugoti turi būti sudaryta procesinių apsaugos priemonių sistema, sudaryta iš anonimiškumo modelių bei formų, garantuojančių veiksmingą tiek liudytojo, tiek kaltinamojo teisių užtikrinimą. Lietuvos baudžiamajame procese įtvirtinta liudytojo apsaugos sistema susidedanti iš absoliutaus ir dalinio anonimiškumo taikymo. Absoliutaus anonimiškumo modelis taikomas vykstant baudžiamajam procesui dėl labai sunkių, sunkių ar apysunkių nusikaltimų, taikant turi būti įsitikinta liudytojo patikimumu. Dalinio anonimiškumo modelis, kurio metu įslaptinama tik dalis asmenį identifikuojančių duomenų, gali būti taikomas ir tiriant nesunkius nusikaltimus. Taikant šį anonimiškumo modelį yra mažiau ribojamos kaltinamojo teisės, racionaliau panaudojamas pareigūnų darbas, procesas yra operatyvesnis ir greitesnis nei absoliutaus anonimiškumo taikymo atveju. / Witness anonymity is a procedural safeguard. It‘s aim is to protect a witness from criminal influence and to get truthful testimony. The fundamental rights of the accused to defence and to fair trial must be minimally restricted when applying anonymity. The application of anonymity and the choice of anonymity model shall be made in accordance with constitutional principle of proportionality. There shall be a system of procedural safeguards containing models and forms of anonymity created for witness protection. The rights of both accused and of witness shall be ensured by this system. Witness protection system containing absolute and limited anonymity models is enshrined in the Lithuanian code of criminal proceedings. Absolute anonymity is applied when serious offences are investigated. Before applying absolute anonymity the credibility of a witness must be confirmed. Limited anonymity can be applied also when less serious offences are investigated; there is only a part of identifying data classified. When applying model of limited anonymity the rights of the accused are less restricted, the operation of officers is used more rationally, the criminal proceedings are faster than when absolute anonymity is used.

A composition algebra is a non-zero algebra endowed with a strictly non-degenerate, multiplicative quadratic form. Finite-dimensional composition algebras exist only in dimension 1, 2, 4 and 8 and are in general not associative or unital. Over the real numbers, such algebras are division algebras if and only if they are absolute valued, i.e. equipped with a multiplicative norm. The problem of classifying all absolute valued algebras and, more generally, all composition algebras of finite dimension remains unsolved. In dimension eight, this is related to the triality phenomenon. We approach this problem using a categorical language and tools from representation theory and the theory of algebraic groups. We begin by considering the category of absolute valued algebras of dimension at most four. In Paper I we determine the morphisms of this category completely, and describe their irreducibility and behaviour under the actions of the automorphism groups of the algebras. We then consider the category of eight-dimensional absolute valued algebras, for which we provide a description in Paper II in terms of a group action involving triality. Then we establish general criteria for subcategories of group action groupoids to be full, and applying this to the present setting, we obtain hitherto unstudied subcategories determined by reflections. The reflection approach is further systematized in Paper III, where we obtain a coproduct decomposition of the category of finite-dimensional absolute valued algebras into blocks, for several of which the classification problem does not involve triality. We study these in detail, reducing the problem to that of certain group actions, which we express geometrically. In Paper IV, we use representation theory of Lie algebras to completely classify all finite-dimensional absolute valued algebras having a non-abelian derivation algebra. Introducing the notion of quasi-descriptions, we reduce the problem to the study of actions of rotation groups on products of spheres. We conclude by considering composition algebras over arbitrary fields of characteristic not two in Paper V. We establish an equivalence of categories between the category of eight-dimensional composition algebras with a given quadratic form and a groupoid arising from a group action on certain pairs of outer automorphisms of affine group schemes