In this thesis, we study the existence of random periodic solutions of random dynamical systems (RDS) by geometric and topological approach. We employed an extension of ergodic theory to random setting to prove that a random invariant set with some kind of dissipative structure, can be expressed as union of random periodic curves. We extensively characterize the dissipative structure by random invariant measures and Lyapunov exponents. For stochastic flows induced by stochastic differential equations (SDEs), we studied the dissipative structure by two point motion of the SDE and prove the existence exponential stable random periodic solutions.

Neste trabalho analisamos a positividade definida estrita de núcleos contínuos sobre um espaço compacto 2-homogêneo. R. Gangolli (1967) apresentou uma caracterização completa para os núcleos que são contínuos, isotrópicos e positivos definidos sobre um espaço compacto 2-homogêneo Md: a parte isotrópica do núcleo é uma série de Fourier uniformemente convergente, com coeficientes não negativos, em relação a certos polinômios de Jacobi atrelados a Md. Uma das contribuições de nosso trabalho é uma caracterização para a positividade definida estrita de tais núcleos, complementando a caracterização apresentada por Chen et al. (2003) no caso em que Md é uma esfera unitária de dimensão maior ou igual a 2. Outra contribuição do trabalho é uma extensão do resultado de Gangolli para núcleos sobre produtos cartesianos de espaços compactos 2-homogêneos, e a consequente caracterização para núcleos estritamente positivos definidos neste mesmo contexto. Por fim, a última contribuição do trabalho envolve a análise do grau de diferenciabilidade da parte isotrópica de um núcleo contínuo, isotrópico e positivo definido sobre Md e a aplicabilidade de tal análise em resultados envolvendo a positividade definida estrita. / In this work we analyze the strict positive definiteness of continuous kernels on compact two-point homogeneous spaces Md. R. Gangolli (1967) presented a complete characterization for continuous, isotropic and positive definite kernels on Md: the isotropic part of the kernel is a uniformly convergent Fourier series of certain Jacobi polynomials associated to Md, with nonnegative coefficients. One of the contributions of our work is a characterization for the strict positive definiteness of such kernels, completing that one presented by Chen et al. (2003) in the case Md is the unit sphere of dimension at least 2. Another contribuition of this work is an extension of Gangolli\'s result for kernels on a product of compact two-point homogeneous spaces, and the subsequent characterization of strict positive definiteness in this same context. Finally, the last contribution in this work involves the analysis of the differentiability of the isotropic part of a continuous, isotropic and positive definite kernel on Md and the applicability of such analysis in results involving the strict positive definiteness.

<p>The oral sensory ability in children can be assessed through test of intraoral stereognosis and two-point discrimination. Currently, these two tests are not frequently used in clinical settings, and normative data are not always available. The aim of this study was to establish normative data for these two tests in children aged 5;6 to 7;0 years. Further, intention was to compare the results with respect to age and gender. In total 103 children participated, 49 boys and 54 girls. The participants were divided into three age-groups. The results regarding intraoral stereognosis showed significant differences between genders, where boys performed poorer than girls. Differences between two of the three age-groups could also be established. The results show that an improvement can be expected in children six years and older.</p><p>Regarding two-point discrimination 73 of the 103 children performed perfectly. The remaining children’s scores were also consistently high. No differences between gender and age were found.</p><p>There was no correlation between the test results of intraoral stereognosis and two-point discrimination. Therefore it would be necessary to perform both these tests in assessment, since they probably examine two separate aspects of the oral sensory ability.</p><p> </p> / <p>Oral sensorisk förmåga hos barn kan bland annat utredas genom test av intraoral stereognosi och tvåpunktsdiskrimination. Dessa två test är i nuläget inte särskilt kliniskt utbredda, och normvärden saknas ofta. Föreliggande studie syftade till att fastställa normvärden för dessa två test hos barn mellan 5;6 och 7;0 år. Vidare jämfördes även testresultaten med avseende på åldersgrupp och kön. 103 barn deltog i studien, varav 49 var pojkar och 54 var flickor. Deltagarna delades in i tre åldersgrupper. Resultatet visade signifikanta könsskillnader på test av intraoral stereognosi, där pojkarnas resultat var något sämre än flickornas. Även skillnader mellan två av åldersgrupperna konstaterades gällande intraoral stereognosi. Resultatet visar att en förbättring på testet kan förväntas då barnen har fyllt sex år.</p><p>Angående test av tvåpunktsdiskrimination klarade 73 av de 103 barnen testet felfritt, och överlag presterade barnen mycket bra på testet. Inga skillnader mellan kön eller åldersgrupper noterades beträffande test av tvåpunktsdiskrimination.</p><p>Testresultaten av intraoral stereognosi och tvåpunktsdiskrimination korrelerade inte. Därför är det sannolikt nödvändigt att i utredning utföra båda testen, eftersom det kan röra sig om två olika aspekter av intraoral sensorik.</p><p> </p>

The oral sensory ability in children can be assessed through test of intraoral stereognosis and two-point discrimination. Currently, these two tests are not frequently used in clinical settings, and normative data are not always available. The aim of this study was to establish normative data for these two tests in children aged 5;6 to 7;0 years. Further, intention was to compare the results with respect to age and gender. In total 103 children participated, 49 boys and 54 girls. The participants were divided into three age-groups. The results regarding intraoral stereognosis showed significant differences between genders, where boys performed poorer than girls. Differences between two of the three age-groups could also be established. The results show that an improvement can be expected in children six years and older. Regarding two-point discrimination 73 of the 103 children performed perfectly. The remaining children’s scores were also consistently high. No differences between gender and age were found. There was no correlation between the test results of intraoral stereognosis and two-point discrimination. Therefore it would be necessary to perform both these tests in assessment, since they probably examine two separate aspects of the oral sensory ability. / Oral sensorisk förmåga hos barn kan bland annat utredas genom test av intraoral stereognosi och tvåpunktsdiskrimination. Dessa två test är i nuläget inte särskilt kliniskt utbredda, och normvärden saknas ofta. Föreliggande studie syftade till att fastställa normvärden för dessa två test hos barn mellan 5;6 och 7;0 år. Vidare jämfördes även testresultaten med avseende på åldersgrupp och kön. 103 barn deltog i studien, varav 49 var pojkar och 54 var flickor. Deltagarna delades in i tre åldersgrupper. Resultatet visade signifikanta könsskillnader på test av intraoral stereognosi, där pojkarnas resultat var något sämre än flickornas. Även skillnader mellan två av åldersgrupperna konstaterades gällande intraoral stereognosi. Resultatet visar att en förbättring på testet kan förväntas då barnen har fyllt sex år. Angående test av tvåpunktsdiskrimination klarade 73 av de 103 barnen testet felfritt, och överlag presterade barnen mycket bra på testet. Inga skillnader mellan kön eller åldersgrupper noterades beträffande test av tvåpunktsdiskrimination. Testresultaten av intraoral stereognosi och tvåpunktsdiskrimination korrelerade inte. Därför är det sannolikt nödvändigt att i utredning utföra båda testen, eftersom det kan röra sig om två olika aspekter av intraoral sensorik.

In this research, the homogenization relations for elastic properties in isotropic and anisotropic materials are studied by applying two-point statistical functions to composite and polycrystalline materials. The validity of the results is investigated by direct comparison with experimental results. In todays technology, where advanced processing methods can provide materials with a variety of morphologies and features in different scales, a methodology to link property to microstructure is necessary to develop a framework for material design. Statistical distribution functions are commonly used for the representation of microstructures and also for homogenization of materials properties. The use of two-point statistics allows the materials designer to consider morphology and distribution in addition to properties of individual phases and components in the design space. This work is focused on studying the effect of anisotropy on the homogenization technique based on two-point statistics. The contribution of one-point and two-point statistics in the calculation of elastic properties of isotropic and anisotropic composites and textured polycrystalline materials will be investigated. For this purpose, an isotropic and anisotropic composite is simulated and an empirical form of the two-point probability functions are used which allows the construction of a composite Hull. The homogenization technique is also applied to two samples of Al-SiC composite that were fabricated through extrusion with two different particle size ratios (PSR). To validate the applied methodology, the elastic properties of the composites are measured by Ultrasonic methods. This methodology is then extended to completely random and textured polycrystalline materials with hexagonal crystal symmetry and the effect of cold rolling on the annealing texture of near- Titanium alloy are presented.

We connect the Mimetic Finite Difference method (MFD) with the finite-volume two-point flux scheme (TPFA) for Voronoi meshes. The main effect is reducing the saddle-point system to a much smaller symmetric-positive definite matrix. In addition, the generalization allows MFD to seamlessly integrate with existing porous media modeling technology. The generalization also imparts the monotonicity property of the TPFA method on MFD. The connection is achieved by altering the consistency condition of the velocity bilinear operator. First-order convergence theory is presented as well as numerical results that support the claims. We demonstrate a methodology for using MFD in modeling fluid flow in fractures coupled with a reservoir. The method can be used for nonplanar fractures. We use the method to demonstrate the effects of fracture curvature on single-phase and multi-phase flows. Standard benchmarks are used to demonstrate the accuracy of the method. The approach is coupled with existing reservoir simulation technology. / text

Neste trabalho analisamos a positividade definida estrita de núcleos contínuos sobre um espaço compacto 2-homogêneo. R. Gangolli (1967) apresentou uma caracterização completa para os núcleos que são contínuos, isotrópicos e positivos definidos sobre um espaço compacto 2-homogêneo Md: a parte isotrópica do núcleo é uma série de Fourier uniformemente convergente, com coeficientes não negativos, em relação a certos polinômios de Jacobi atrelados a Md. Uma das contribuições de nosso trabalho é uma caracterização para a positividade definida estrita de tais núcleos, complementando a caracterização apresentada por Chen et al. (2003) no caso em que Md é uma esfera unitária de dimensão maior ou igual a 2. Outra contribuição do trabalho é uma extensão do resultado de Gangolli para núcleos sobre produtos cartesianos de espaços compactos 2-homogêneos, e a consequente caracterização para núcleos estritamente positivos definidos neste mesmo contexto. Por fim, a última contribuição do trabalho envolve a análise do grau de diferenciabilidade da parte isotrópica de um núcleo contínuo, isotrópico e positivo definido sobre Md e a aplicabilidade de tal análise em resultados envolvendo a positividade definida estrita. / In this work we analyze the strict positive definiteness of continuous kernels on compact two-point homogeneous spaces Md. R. Gangolli (1967) presented a complete characterization for continuous, isotropic and positive definite kernels on Md: the isotropic part of the kernel is a uniformly convergent Fourier series of certain Jacobi polynomials associated to Md, with nonnegative coefficients. One of the contributions of our work is a characterization for the strict positive definiteness of such kernels, completing that one presented by Chen et al. (2003) in the case Md is the unit sphere of dimension at least 2. Another contribuition of this work is an extension of Gangolli\'s result for kernels on a product of compact two-point homogeneous spaces, and the subsequent characterization of strict positive definiteness in this same context. Finally, the last contribution in this work involves the analysis of the differentiability of the isotropic part of a continuous, isotropic and positive definite kernel on Md and the applicability of such analysis in results involving the strict positive definiteness.

Damage to the nervous system may cause sensorimotor impairment, often resulting in chronic neuropathic pain. Phantom limb pain affects multiple amputees and the treatment options are limited. A promising treatment option to reduce chronic pain is by training sensory discrimination. At the Center for Bionics and Pain Research, a sensory training device has been developed. Transcranial direct current stimulation (tDCS) is a technique to stimulate different regions of the cerebral cortex. In studies, anodal tDCS of the sensorimotor cortex has shown to improve tactile acuity. Until now, the effect of the sensory discrimination training, performed with the sensory training device, in combination with tDCS has not been tested. The purpose of this master’s thesis was to determine the effect of tDCS applied on the sensorimotor cortex on the outcomes of sensory discrimination training. The purpose was also to compare the effect of two different methods of stimulation, namely conventional and High Definition tDCS. 16 able-bodied participants underwent three single sessions with 40 minutes of sensory discrimination training: one session combined with conventional tDCS, one session combined with High Definition tDCS and one session without tDCS. The tactile acuity was determined by the two-point discrimination test and the Semmes-Weinstein monofilament test, prior to and one hour after each session. This study showed that 40 minutes of sensory discrimination training was sufficient to improve the two-point discrimination in the sensory trained areas, compared to the sensory untrained areas (p=0.02). However, the improvement in two-point discrimination was not statistically significant between the interventions, i.e. the improvement in two-point discrimination for the sessions with brain modulation was not statistically significant compared to the session without brain modulation. The monofilament assessments showed an improvement in monofilament score for the sensory untrained skin patches (p=0.053). This study concluded that single sessions of training was enough to improve two-point discrimination but not monofilament score at the site of stimulation. This study lays a foundation for what parameters to include in future studies.

<p><strong>ABSTRACT </strong> Approximating the kinetic energy as a functional of the electron density is a daunting, but important, task. For molecules in equilibrium geometries, the kinetic energy is equal in magnitude to the total electronic energy, so achieving the exquisite accuracy in the total energy that is needed for chemical applications requires similar accuracy for the kinetic energy functional. For this reason, most density functional theory (DFT) calculations use the Kohn-Sham method, which provides a good estimate for the kinetic energy. But the computational cost of Kohn-Sham DFT calculations has a direct dependence on the total number of electrons because the Kohn-Sham method is based on the orbital picture, with one orbital per electron. Explicit density functionals, where the kinetic energy is written explicitly in terms of the density, and not in terms of orbitals, are much faster to compute. Unfortunately, the explicit density functionals in the literature had disappointing accuracy. This dissertation introduces several new approaches for orbital-free density functional methods. One can try to include information about the Pauli principle using the exchange hole. In the weighted density approximation (WDA), a model for the exchange hole is used to approximate the one-electron density matrix, which is then used to compute the kinetic energy. This thesis introduces a symmetric, normalized, weighted density approximation using the exchange hole of the uniform electron gas. Though the key results on kinetic energy are not accurate enough, an efficient algorithm is introduced which, with a more sophisticated hole model, might give better results. The effects of electron correlation on the kinetic energy can be modeled by moving beyond the one-electron distribution function (the electron density) to higherorder electron distributions (k-electron DFT). For example, one can model electron correlation directly using the pair electron density. In this thesis, we investigated two different functionals of the pair density, the Weizsäcker functional and the March-Santamaria functional. The Weizsäcker functional badly fails to describe the accurate kinetic energy due to the N-representability problem. The March-Santamaria functional is exact for a single Slater determinant, but fails to adequately model the effects of electron correlation on the kinetic energy. Finally, we established a relation between Fisher information and Weizsäcker kinetic energy functional. This allowed us to propose generalisations of the Weizsäcker kinetic energy density functional. It is hoped that the link between information theory and kinetic energy might provide a new approach to deriving improved kinetic energy functionals. <strong> Keywords: </strong><em>Kinetic energy functional, Density functional theory (DFT), von-Weizsäcker</em> <em> functional, March-Santamaria functional, Thomas-Fermi model, density matrix, Twopoint normalization, Pair-density functional theory (PDFT). </em></p> / Doctor of Science (PhD)

Turbulent wakes generated from an aircraft or submarine vehicles has been of main interest to researchers due to the broad band noise associated with such wakes. One such case is the noise generated by spiral vortices shed of from one blade interacting with another oncoming blade of helicopter rotor. Consequently, researchers have been trying to understand the basic physics and evolution of such wakes. Although there has been numerous studies done on plane wakes, there has been little research being done on laterally curved wakes. Single and two-point velocity measurements were taken on a plane and laterally curved turbulent wakes to understand the evolution and effect of lateral curvature into the far wake region. The analyses provide useful information in modeling curved or spiral wakes such as turbulence field surrounding tip vortices shed from a wing. In order to achieve our objectives, the Virginia Tech 3’ x 2’ subsonic wind tunnel was used to take velocity measurements of toroidal ring model and a straight cylinder as a control case. Velocity measurements were done using four sensor hot-wire anemometers, to obtain all mean velocity, Reynolds stress, triple product components of the turbulence field. Single point, spectra and two-point measurements of the wakes were performed throughout the development into the far wake region. The single point results reveal the universality of the mean axial velocity, however the Reynolds stresses and triple products were not universal illustrating that the turbulence field has its own length and velocity scales different from that of the mean flow. The effect of lateral curvature is mainly evidenced in the early development of the curved ring wake. The turbulent energy budget reveals similar trend for both wakes and plane wake achieves approximate equilibrium. The spectra result reveals for the plane wake that self-preservation is achieved for all scales of motion, while the ring wake does not achieve such a state. While the longitudinal correlations of both wakes are similar in form, in general difference in form and orientation prevailed over all indicating the difference in the turbulent structure of both wakes. Linear stochastic estimation reveals the presence of spanwise and double-roller eddy structures in the plane wake and only spanwise eddies were detected for the ring wake. / Master of Science