Smooth maps minimizing the energy and the calibrated geometry

We generalize the notion of calibrated submanifolds to smooth maps and show that the several examples of smooth maps appearing in the differential geometry become the examples of our situation. Moreover, we apply these notion to give the lower bound to the energy of smooth maps in the given homotopy class between Riemannian manifolds, and consider the energy functional which is minimized by the identity maps on the Riemannian manifolds with special holonomy groups

Spectral convergence in geometric quantization --- the case of non-singular Langrangian fibrations

We develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of K\"ahler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of $\bar{\partial}$-Laplacians, as well as the convergence result of quantum Hilbert spaces. We also consider the case of almost K\"ahler quantization for compatible almost complex structures, and show the analogous convergence results

Spectrocolorimetric evaluation of repaired articular cartilage after a microfracture

<p>Abstract</p> <p>Background</p> <p>In clinical practice, surgeons differentiate color changes in repaired cartilage compared with surrounding intact cartilage, but cannot quantify these color changes. Objective assessments are required. A spectrocolorimeter was used to evaluate whether intact and repaired cartilage can be quantified.</p> <p>Findings</p> <p>We investigated the use of a spectrocolorimeter and the application of two color models (L* a* b* colorimetric system and spectral reflectance distribution) to describe and quantify articular cartilage. In this study, we measured the colors of intact and repaired cartilage after a microfracture. Histologically, the repaired cartilage was a mixture of fibrocartilage and hyaline cartilage. In the L* a* b* colorimetric system, the L* and a* values recovered to close to the values of intact cartilage, whereas the b* value decreased over time after the operation. Regarding the spectral reflectance distribution at 12 weeks after the operation, the repaired cartilage had a higher spectral reflectance ratio than intact cartilage between wavelengths of 400 to 470 nm.</p> <p>Conclusion</p> <p>This study reports the first results regarding the relationship between spectrocolorimetric evaluation and the histological findings of repair cartilage after a microfracture. Our findings demonstrate the ability of spectrocolorimetric measurement to judge the repair cartilage after treatment on the basis of objective data such as the L*, a* and b* values and the SRP as a coincidence index of the spectral reflectance curve.</p