<p>This report consider a system describing three competing species with populations <em>x</em>, <em>y</em> and <em>z</em>. Sufficient conditions for every positive equilibrium to be asymptotically stable have been found. First it is shown that conditions on the pairwise competitive interaction between the populations are needed. Actually, these conditions are equivalent to asymptotic stability for any two-dimensional competing system of the three species. It is also shown that these alone are not enough, and that a condition on the competitive interaction between all three populations is also needed. If all conditions are fulfilled, each population will survive on a long-term basis and there will be a stable coexistence.</p>

<p>In this thesis we analyze the phase of the heavy symmetric top and the tippe top. These tops are two examples of physical systems for which the usefulness of integrals of motion and invariant manifolds, in phase space picture analysis, can be illustrated</p><p>In the case of the heavy symmetric top, simplified proofs of stability of the vertical rotation have been perpetuated by successive textbooks during the last century. In these proofs correct perturbations of integrals of motion are missing. This may seem harmless since the deduced threshold value for stability is correct. However, perturbations of first integrals are essential in rigorous proofs of stability of motions for both tops.</p><p>The tippe top is a toy that has the form of a truncated sphere equipped with a little peg. When spun fast on the spherical bottom its center of mass rises above its geometrical center and after a few seconds the top is spinning vertically on the peg. We study the tippe top through a sequence of embedded invariant manifolds to unveil the structure of the top's phase space. The last manifold, consisting of the asymptotic trajectories, is analyzed completely. We prove that trajectories in this manifold attract solutions in contact with the plane of support at all times and we give a complete description of their stability/instability properties for all admissible choices of model parameters and of the initial conditions.</p> / Report code: LiU-TEK-LIC-2004:35.

Dans ce travail, nous proposons des procédures de test paramétriques et nonparamétriques localement et asymptotiquement optimales au sens de Hajek et Le Cam, pour deux modèles de données de panel. Notre approche est fondée sur la théorie de Le Cam d'une part, pour obtenir les propriétés de normalité asymptotique, bases de la construction des tests paramétriques optimaux, et la théorie de Hajek d'autre part, qui, via un principe d'invariance, permet d'obtenir les procédures nonparamétriques. Dans le premier chapitre, nous considérons un modèle à erreurs composées et nous nous intéressons au problème qui consiste à tester l'absence de l'effet individuel aléatoire. Nous établissons la propriété de normalité locale asymptotique (LAN), ce qui nous permet de construire des procédures paramétriques localement et asymptotiquement optimales (“les plus stringentes”) pour le problème considéré. L'optimalité de ces procédures est liée à la densité-cible f1. Ces propriétés d'optimalité sont hautement paramétriques puisqu'elles requièrent que la densité sous-jacente soit f1. De plus, ces procédures ne seront valides que si la densité-cible f1 et la densité sous-jacent g1 coincïdent. Or, en pratique, une spécification correcte de la densité sous-jacente g1 est non réaliste, et g1 doit être considérée comme un paramètre de nuissance. Pour éliminer cette nuisance, nous adoptons l'argument d'invariance et nous nous restreignons aux procédures fondées sur des statistiques qui sont mesurables par rapport au vecteur des rangs. Les tests que nous obtenons restent valide quelle que soit la densité sous-jacente et sont localement et asymptotiquement les plus stringents. Afin d'avoir des renseignements sur l'efficacité des tests fondés sur les rangs sous différentes lois, nous calculons les efficacités asymptotiques relatives de ces tests par rapport aux tests pseudo-gaussiens, sous des densités g1 quelconques. Enfin, nous proposons quelques simulations pour comparer les performances des procédures proposées. Dans le deuxième chapitre, nous considérons un modèle à erreurs composées avec autocorrélation d'ordre 1 et nous montrons que ce modèle jouit de la propriété LAN. A partir de ce résultat, nous construisons des tests optimaux, au sens local et asymptotique, pour trois problèmes de tests importants dans ce contexte : (a) test de l'absence d'effet individuel et d'autocorrélation; (b) test de l'absence d'effet individuel en présence d'une autocorrélation non spécifiée; et (c) test de l'absence d'autocorrélation en présence d'un effet individuel non spécifié. Enfin, nous proposons quelques simulations pour comparer les performances des tests pseudogaussiens et des tests classiques.

It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.

Axicons are optical elements that produce Bessel beams,i.e., long and narrow focal lines along the optical axis. Thenarrow focus makes them useful ine.g. alignment, harmonicgeneration, and atom trapping, and they are also used toincrease the longitudinal range of applications such astriangulation, light sectioning, and optical coherencetomography. In this thesis, axicons are designed andcharacterized for different kinds of illumination, using thestationary-phase and the communication-modes methods. The inverse problem of axicon design for partially coherentlight is addressed. A design relation, applicable toSchell-model sources, is derived from the Fresnel diffractionintegral, simplified by the method of stationary phase. Thisapproach both clarifies the old design method for coherentlight, which was derived using energy conservation in raybundles, and extends it to the domain of partial coherence. Thedesign rule applies to light from such multimode emitters aslight-emitting diodes, excimer lasers and some laser diodes,which can be represented as Gaussian Schell-model sources. Characterization of axicons in coherent, obliqueillumination is performed using the method of stationary phase.It is shown that in inclined illumination the focal shapechanges from the narrow Bessel distribution to a broadasteroid-shaped focus. It is proven that an axicon ofelliptical shape will compensate for this deformation. Theseresults, which are all confirmed both numerically andexperimentally, open possibilities for using axicons inscanning optical systems to increase resolution and depthrange. Axicons are normally manufactured as refractive cones or ascircular diffractive gratings. They can also be constructedfrom ordinary spherical surfaces, using the sphericalaberration to create the long focal line. In this dissertation,a simple lens axicon consisting of a cemented doublet isdesigned, manufactured, and tested. The advantage of the lensaxicon is that it is easily manufactured. The longitudinal resolution of the axicon varies. The methodof communication modes, earlier used for analysis ofinformation content for e.g. line or square apertures, isapplied to the axicon geometry and yields an expression for thelongitudinal resolution. The method, which is based on abi-orthogonal expansion of the Green function in the Fresneldiffraction integral, also gives the number of degrees offreedom, or the number of information channels available, forthe axicon geometry. Keywords:axicons, diffractive optics, coherence,asymptotic methods, communication modes, information content,inverse problems

No description available.

In the thesis we consider two types of problems. In Paper 1, we study small solutions to a time-independent nonlinear elliptic partial differential equation of Emden-Fowler type in a semi-infnite cylinder. The asymptotic behaviour of these solutions at infnity is determined. First, the equation under the Neumann boundary condition is studied. We show that any solution small enough either vanishes at infnity or tends to a nonzero periodic solution to a nonlinear ordinary differential equation. Thereafter, the same equation under the Dirichlet boundary condition is studied, the non-linear term and right-hand side now being slightly more general than in the Neumann problem. Here, an estimate of the solution in terms of the right-hand side of the equation is given. If the equation is homogeneous, then every solution small enough tends to zero. Moreover, if the cross-section is star-shaped and the nonlinear term in the equation is subject to some additional constraints, then every bounded solution to the homogeneous Dirichlet problem vanishes at infnity. In Paper 2, we study asymptotics as t → ∞ of solutions to a linear, parabolic system of equations with time-dependent coefficients in Ωx(0,∞), where Ω is a bounded domain. On δΩ(0,∞) we prescribe the homogeneous Dirichlet boundary condition. For large values of t, the coefficients in the elliptic part are close to time-independent coefficients in an integral sense which is described by a certain function κ(t). This includes in particular situations when the coefficients may take different values on different parts of Ω and the boundaries between them can move with t but stabilize as t → ∞. The main result is an asymptotic representation of solutions for large t. As a corollary, it is proved that if κєL1(0,∞), then the solution behaves asymptotically as the solution to a parabolic system with time-independent coefficients.

Estimating the integrated volatility of high frequency realized prices is an important issue in microstructure literature. Bandi and Russell (2006) derived the optimal-sampling frequency, and Zhang et al. (2005) proposed a "two-scales estimator" to solve the problem. In this study, we propose a new estimator based on a signal to noise ratio statistic with convergence rate of Op (n^(−1/ 4) ). The method is applicable to both constant and stochastic volatility models and modi¡Âes the Op (n^(−1/ 6) ) convergence rate of Zhang et al. (2005). The proposed estimator is shown to be asymptotic e¡Ócient as the maximum likelihood estimate for the constant volatility case. Furthermore, unbiased estimators of the two elements, the variance of the microstructure noise and the fourth moment of the realized log returns, are also proposed to facilitate the estimation of integrated volatility. The asymptotic prop- erties and e&#x00AE;ectiveness of the proposed estimators are investigated both theoretically and via simulation study.

The k-sample problem, i.e., testing whether two or more data sets come from the same population, is a classic one in statistics. Instead of having a small number of k groups of samples, this dissertation works on a large number of p groups of samples, where within each group, the sample size, n, is a fixed, small number. We call this as a "Large p, but Small n" setting. The primary goal of the research is to provide a test statistic based on kernel density estimation (KDE) that has an asymptotic normal distribution when p goes to infinity with n fixed. In this dissertation, we propose a test statistic called Tp(S) and its standardized version, T(S). By using T(S), we conduct our test based on the critical values of the standard normal distribution. Theoretically, we show that our test is invariant to a location and scale transformation of the data. We also find conditions under which our test is consistent. Simulation studies show that our test has good power against a variety of alternatives. The real data analyses show that our test finds differences between gene distributions that are not due simply to location.

In this dissertation, the classical problems of testing goodness-of-fit of uniformity and parametric families are reconsidered. A new omnibus test for these problems is proposed and investigated. The new test statistics are a combination of Bayesian and score test ideas. More precisely, singletons that contain only one more parameter than the null describing departures from the null model are introduced. A Laplace approximation to the posterior probability of the null hypothesis is used, leading to test statistics that are weighted sums of exponentiated squared Fourier coefficients. The weights depend on prior probabilities and the Fourier coefficients are estimated based on score tests. Exponentiation of Fourier components leads to tests that can be exceptionally powerful against high frequency alternatives. Comprehensive simulations show that the new tests have good power against high frequency alternatives and perform comparably to some other well-known omnibus tests at low frequency alternatives. Asymptotic distributions of the proposed test are derived under null and alternative hypotheses. An application of the proposed test to an interesting real problem is also presented.