Nous étudions la mesure spectrale des transformations stationnaires, puis nous l’utilisons pour étudier le théorème ergodique et le théorème limite central. Nous étudions également les martingales avec une nouvelle preuve du théorème central limite, sans analyse de Fourier. Pour le théorème limite central pour marches aléatoires dans un environnement aléatoire sur la dimension 1, on donne deux méthodes pour l’obtenir: approximation pour une martingale et méthode des moments. La méthode des martingales fait résoudre l’équation de Dirichlet (I - P)h = 0, alors que celle des moments résoudre l’équation de Poisson (I - P)h = f. Enfin, nous pouvons utiliser la deuxième méthode pour prouver la relation d’Einstein pour des diffusions réversibles dans un environnement aléatoire dans une dimension. / We study the spectral measure for stationary transformations, and then apply to Ergodic theorem and Central limit theorem. We study also martingale process with a new proof of the central limit theorem without Fourier analysis. For the central limit theorem for random walks in random environment, we give two methods to obtain it: martingale approximation and moments. The method of martingales solves Dirichlet’s equation (I - P)h = 0, and the method of moments solves Poisson’s equation (I - P)h = f. Finally, we can use the second method to prove the Einstein relation for reversible diffusions in random environment in one dimension.

In this thesis we investigate Robinson-Trautman solutions of Einstein's gravity cou- pled to a matter field in higher dimensions, specifically a conformally invariant and non- linear electromagnetic field. The latter possesses in general a non-zero energy-momentum tensor, which provides a source term in Einstein's equations. We focus concretely on an electromagnetic field aligned with the null vector field generating the expanding con- gruence of Robinson-Trautman spacetimes. At the beginning, we review the concept of optical scalars for a null vector field in higher dimensions and we use those to define the higher-dimensional Robinson-Trautman class of spacetimes. Next, we solve the corre- sponding Einstein's equations and present the complete family of exact solutions of the theory under consideration. We then contrast the obtained results with the known ones for the linear Maxwell theory in higher dimensions. As a check, we also compare our results to the well-known results in D = 4, since in this case our matter theory reduces to the standard linear Maxwell theory. Finally, we study properties of a subfamily of solutions which represent the static black holes within our class. In particular, we ana- lyze the asymptotic behaviour, we show that a curvature singularity is always present for r → 0 and the...

Cosmology is the study of the universe as a whole, and attempts to describe the behaviour of the universe mathematically. The simplest relativistic cosmological models are derived from Einstein's field equations with the assumptions of isotropy and homogeneity. In this thesis, a few simple cosmological models will be derived and evaluated with respect to their description of our universe i.e., how well they match observational data from e.g., the cosmic background radiation and redshift from distant supernovae. The models are derived from Einstein's field equations, which is why a large portion of the thesis will lay the ground work for the field equations by introducing and explaining the language of tensors. / Kosmologi är läran om universum i stort samt dess matematiska beskrivning. De enklaste relativistiska kosmologiska modellerna kan härledas från Einsteins fältekvationer med hjälp av antaganden om isotropi och homogenitet. I denna rapport kommer ett par av de enklaste modellerna att härledas, samt evalueras baserat på hur väl de beskriver vårt universum, det vill säga hur bra de passar de observationer som gjorts på exempelvis den kosmiska bakgrundsstrålningen och rödskifte från avlägsna supernovor. Modellerna härleds utifrån Einsteins fältekvationer, varför en stor del av rapporten består av en introduktion till tensoranalys.

Nous nous intéressons dans cette thèse à un aspect assez peu documenté de l'histoire de la théorie de la relativité restreinte, la recherche d'une vérification expérimentale de ses prédictions sur la variation de l'inertie en fonction de leur vitesse. Nous complétons les études historiques antérieures sur les expériences de Kaufmann (1906) et de Bucherer (1908), et montrons que la vérification de la formule de Lorentz-Einstein constitue encore un enjeu expérimental après 1911. Nous étudions plus particulièrement les recherches dirigées par Charles-Eugène Guye en collaboration avec ses étudiants Simon Ratnowsky (1907-1910) et Charles Lavanchy (1913-1915). Nous montrons que la seconde phase de ce travail est très largement considérée dans les années 1920 comme la vérifiation expérimentale la plus précise de la formule de Lorentz-Einstein. Nous utilisons la méthode de réplication, appliquée à l'expérience de Guye et Lavanchy. La très grande maîtrise de l'émission cathodique, associée à une méthode d'investigation spécifique, leur permet de surmonter toutes les difficultés identifiées alors comme préjudiciables au succès de ce type d'expérience. / We focus on an aspect of the history of special relativity theory that has not received much attention yet, namely the search for an experimental verification of the relativistic velocity dependency of inertia. Former historical studies on Kaufmann's 1906 and Bucherer's 1908 experiments are pursued. It is shown that verifying Lorentz-Einstein's formula is still a challenge after 1911. We concentrate here on Charles-Eugène Guye's works with students Simon Ratnowsky (1907-1910) and Charles Lavanchy (1913-1915). The second experiment is shown to be considered as the most precise verification of Lorentz-Einstein's formula by a large number of 1920s physicists. The replication method is used to probe Guye and Lavanchy's experiment. They are able to solve every then known difficulty in successfully performing such an experiment, through a great mastership of the crucial issue of cathode ray emission, together with a specific investigation method.

The study investigated the impact of computer simulations on the teaching and learning of photoelectric effect in Grade 12. The Grade 12 Physical Sciences curriculum has components of physics and chemistry. The photoelectric effect is a section in the physics curriculum and examination in the National Senior Certificate. In this case study, thirty learners were randomly divided into three groups in one rural school in the Frances Baard district in the Northern Cape Province. A randomised pre-test - post-test control group design was implemented. Data were collected through pre and post tests, by observation of the lessons and learner interviews. An analysis of variance performed showed that there was no significant difference on pre-test scores for the three groups. A paired -sample t-test on the post-test scores discovered that the Teacher-Centred Experimental Group (TCEG) performed better than the Learner-Centred Experimental Group (LCEG); (t statics, t (9) = -6.135, p < 0.05). In addition, the Control Group (CG) where the teacher used the traditional method of teaching performed even better than the Learner-Centred Experimental group. An analysis of covariance on the post-test scores with learners' pre-test scores as the covariate showed a significant effect on the instructional group favouring the TCEG (F (2,29) = 52.763, p < 0.05). The Hake's normalised gain, <g> was used to measure the effectiveness of the intervention. The normalised gain showed a high-g (0.794) for the TCEG, a medium-g (0.405) for the CG and a low-g (0.134) for the LCEG. The interview data also confirms that the TCEG learners benefited more than the LCEG learners. It is, therefore, suggested that the TCEG approach is a better method for the effective teaching of photoelectric effect. / Science and Technology Education / M. Sc. (MSTE)