6 research outputs found

    Intrinsic Riemannian Functional Data Analysis for Sparse Longitudinal Observations

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    A novel framework is developed to intrinsically analyze sparsely observed Riemannian functional data. It features four innovative components: a frame-independent covariance function, a smooth vector bundle termed covariance vector bundle, a parallel transport and a smooth bundle metric on the covariance vector bundle. The introduced intrinsic covariance function links estimation of covariance structure to smoothing problems that involve raw covariance observations derived from sparsely observed Riemannian functional data, while the covariance vector bundle provides a rigorous mathematical foundation for formulating the smoothing problems. The parallel transport and the bundle metric together make it possible to measure fidelity of fit to the covariance function. They also plays a critical role in quantifying the quality of estimators for the covariance function. As an illustration, based on the proposed framework, we develop a local linear smoothing estimator for the covariance function, analyze its theoretical properties, and provide numerical demonstration via simulated and real datasets. The intrinsic feature of the framework makes it applicable to not only Euclidean submanifolds but also manifolds without a canonical ambient space.Comment: 36 pages, 8 figure

    Spin fragmentation of Bose-Einstein condensates with antiferromagnetic interactions

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    We study spin fragmentation of an antiferromagnetic spin 1 condensate in the presence of a quadratic Zeeman (QZ) effect breaking spin rotational symmetry. We describe how the QZ effect turns a fragmented spin state, with large fluctuations of the Zeemans populations, into a regular polar condensate, where atoms all condense in the m=0m=0 state along the field direction. We calculate the average value and variance of the Zeeman state m=0m=0 to illustrate clearly the crossover from a fragmented to an unfragmented state. The typical width of this crossover is q∼kBT/Nq \sim k_B T/N, where qq is the QZ energy, TT the spin temperature and NN the atom number. This shows that spin fluctuations are a mesoscopic effect that will not survive in the thermodynamic limit N→∞N\rightarrow \infty, but are observable for sufficiently small atom number.Comment: submitted to NJ

    Etudes théoriques et expérimentales de condensats de spin 1 antiferromagnétiques

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    In this thesis, we study theoretically and in experimentally the properties of spin-1 Bose-Einstein condensates with antiferromagnetic interactions realized in the ultra-cold sodium gases confined in optical dipole traps. We present in detail how to realize, diagnose and control the spinor Bose-Einstein condensates in our experiment. In order to describe the condensate, a mean-field theory is first adopted. This theory predicts a phase transition when changing the magnetic field BB. The predictions of the mean-field theory agree very well with most of our experimental results, including the values of the critical magnetic field BcB_c, asymptotic values of n0n_0 (relative atom number in mF=0m_F=0 state) at large BB and the phase diagram of n0n_0. However, for small magnetization and small magnetic fields, we find abnormally large fluctuations (super-Poissonian) of n0n_0. In order to describe these large fluctuations, we develop a full quantum statistical description for the spinor condensate at finite temperature. To describe uncondensed thermal atoms (also present in the same trap), we use the ``semi-ideal" Hartree-Fock approximation to deal with the interactions between the condensate atoms and the thermal ones. The experimental results lead us to introduce two kinds of temperatures, the ``spin temperature" TsT_s and the ``kinetic temperature" TkT_k with Ts≪TkT_s\ll T_k, characterizing respectively the fluctuaitons of the condensate spin and the thermal gas. We conclude that the system is reaching a quasi-equilibrium states, where different degrees of freedom reach equilibrium by separate mechanisms but where the mutual thermalization does not occur over the lifetime of the cloud.Nous présentons en détail comment réaliser, analyser et contrôler de tels condensats spineurs. Afin de décrire le condensat, nous adoptons d'abord une théorie du champ moyen. Cette théorie prédit une transition de phase quand on change le champ magnétique BB. Dans la plupart des cas, les prédictions par cette théorie du champ moyen s'accordent très bien avec la plupart des nos résultats expérimentaux, incluant les valeurs de champ magnétique critique Bc, les valeurs asymptotiques de n0n_0 (nombre d'atome relative en état mF=0m_F=0) en grand B et le diagramme de phase de n0. Cependant, pour des magnétisations faibles et des champs magnétiques petits, nous constatons des fluctuations anormalement grandes (super-Poissoniennes) des populations des états Zeeman individuels. Pour comprendre leur origine et décrire ces systèmes fluctuants, nous développons une description statistique quantique des condensat spineur à température finie. Pour décrire le nuage thermique présent également dans le piège, nous utilisons l'approximation ``Hartree-Fock semi-idéale" pour traiter les interactions entre les condensats et les nuages thermiques. Les résultats expérimentaux nous amènent à introduire deux types des températures: la ``température de spin" T_s et la ``température cinétique" T_k,avec, avec T_s\ll T_k$, qui caractérisent les fluctuations du spin du condensat et le nuage thermique, respectivement. On en conclut que le système se trouve dans un état de quasi-équilibre , ou différents degrés de liberté s'équilibrent séparément mais leur thermalisation mutuelle ne s'opère pas sur la durée de vie du système

    Embryological observations on seed abortion in Hibiscus syriacus L. and physiological studies on nutrients, enzyme activity and endogenous hormones

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    Abstract Under natural conditions, most Hibiscus syriacus L. individuals form very few mature seeds or the mature seeds that do form are of poor quality. As a result, seed yield is poor and seeds have low natural germinability. These phenomena strongly hinder utilization of the excellent germplasm resources of H. syriacus. The study has shown that pollen activity and stigma receptivity were high on the day of anthesis, and the pistils and stamens were fertile. Pollen release and stigma receptivity were synchronous. But in styles following self and cross-pollination, pollen tube abnormalities (distortion and twisting of the pollen tubes) and callose deposition were observed. Cross-pollinated pollen tubes elongated faster and fewer pollen tube abnormalities were observed compared with self-pollinated pollen tubes. And during embryo development, abnormalities during the heart-shaped embryo stage led to embryo abortion. Imbalance in antioxidant enzyme activities and low contents of auxin and cytokinin during early stages of embryo development may affect embryo development. Therefore, a low frequency of outcrossing and mid-development embryo abortion may be important developmental causes of H. syriacus seed abortion. Nutrient deficiencies, imbalance in antioxidant enzyme activities, and a high content of abscisic acid at advanced stages of seed development may be physiological causes of seed abortion
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