Root system of singular perturbations of the harmonic oscillator type operators

We analyze perturbations of the harmonic oscillator type operators in a Hilbert space H, i.e. of the self-adjoint operator with simple positive eigenvalues μ k satisfying μ k+1 − μ k ≥ Δ > 0. Perturbations are considered in the sense of quadratic forms. Under a local subordination assumption, the eigenvalues of the perturbed operator become eventually simple and the root system contains a Riesz basis.

Absolute polarity determination of teeth cementum by phase sensitive second harmonic generation microscopy

The absolute sign of local polarity in relation to the biological growth direction has been investigated for teeth cementum using phase sensitive second harmonic generation microscopy (PS-SHGM) and a crystal of 2-cyclooctylamino-5-nitropyridine (COANP) as a nonlinear optic (NLO) reference material. A second harmonic generation (SHG) response was found in two directions of cementum: radial (acellular extrinsic fibers that are oriented more or less perpendicular to the root surface) and circumferential (cellular intrinsic fibers that are oriented more or less parallel to the surface). A mono-polar state was demonstrated for acellular extrinsic cementum. However, along the different parts of cementum in circumferential direction, two corresponding domains were observed featuring an opposite sign of polarity indicative for a bi-polar microscopic state of cellular intrinsic cementum. The phase information showed that the orientation of radial collagen fibrils of cementum is regularly organized with the donor (D) groups pointing to the surface. Circumferential collagen molecules feature orientational disorder and are oriented up and down in random manner showing acceptor or donor groups at the surface of cementum. Considering that the cementum continues to grow in thickness throughout life, we can conclude that the cementum is growing circumferentially in two opposite directions and radially in one direction. A Markov chain type model for polarity formation in the direction of growth predicts D-groups preferably appearing at the fiber front.

Numerical Magnitude Skills in 6-Years-Old Children: Exploring Specific Associations with Components of Executive Function

Little is known about how children learn to associate numbers with their corresponding magnitude and about individual characteristics contributing to performance differences on the numerical magnitude tasks within a relatively homogenous sample of 6-year-olds. The present study investigated the relationships between components of executive function and two different numerical magnitude skills in a sample of 162 kindergartners. The Symbolic Number Line was predicted by verbal updating and switching, whereas the Symbolic Magnitude Comparison was predicted by inhibition. Both symbolic tasks were predicted by visuo-spatial updating. Current findings suggest that visuo-spatial updating underlies young children’s retrieval and processing of numbers’ magnitude.