The regularity of harmonic maps into spheres and applications to Bernstein problems

We show the regularity of, and derive a-priori estimates for (weakly) harmonic maps from a Riemannian manifold into a Euclidean sphere under the assumption that the image avoids some neighborhood of a half-equator. The proofs combine constructions of strictly convex functions and the regularity theory of quasi-linear elliptic systems. We apply these results to the spherical and Euclidean Bernstein problems for minimal hypersurfaces, obtaining new conditions under which compact minimal hypersurfaces in spheres or complete minimal hypersurfaces in Euclidean spaces are trivial

The geometry of Grassmannian manifolds and Bernstein type theorems for higher codimension

We identify a region \Bbb{W}_{\f{1}{3}} in a Grassmann manifold \grs{n}{m}, not covered by a usual matrix coordinate chart, with the following important property. For a complete $n-$submanifold in \ir{n+m} \, (n\ge 3, m\ge2) with parallel mean curvature whose image under the Gauss map is contained in a compact subset K\subset\Bbb{W}_{\f{1}{3}}\subset\grs{n}{m}, we can construct strongly subharmonic functions and derive a priori estimates for the harmonic Gauss map. While we do not know yet how close our region is to being optimal in this respect, it is substantially larger than what could be achieved previously with other methods. Consequently, this enables us to obtain substantially stronger Bernstein type theorems in higher codimension than previously known.Comment: 36 page

The Gauss image of entire graphs of higher codimension and Bernstein type theorems

Under suitable conditions on the range of the Gauss map of a complete submanifold of Euclidean space with parallel mean curvature, we construct a strongly subharmonic function and derive a-priori estimates for the harmonic Gauss map. The required conditions here are more general than in previous work and they therefore enable us to improve substantially previous results for the Lawson-Osseman problem concerning the regularity of minimal submanifolds in higher codimension and to derive Bernstein type results.Comment: 28 page

An improved classification of G-protein-coupled receptors using sequence-derived features

<p>Abstract</p> <p>Background</p> <p>G-protein-coupled receptors (GPCRs) play a key role in diverse physiological processes and are the targets of almost two-thirds of the marketed drugs. The 3 D structures of GPCRs are largely unavailable; however, a large number of GPCR primary sequences are known. To facilitate the identification and characterization of novel receptors, it is therefore very valuable to develop a computational method to accurately predict GPCRs from the protein primary sequences.</p> <p>Results</p> <p>We propose a new method called PCA-GPCR, to predict GPCRs using a comprehensive set of 1497 sequence-derived features. The <it>principal component analysis </it>is first employed to reduce the dimension of the feature space to 32. Then, the resulting 32-dimensional feature vectors are fed into a simple yet powerful classification algorithm, called intimate sorting, to predict GPCRs at <it>five </it>levels. The prediction at the first level determines whether a protein is a GPCR or a non-GPCR. If it is predicted to be a GPCR, then it will be further predicted into certain <it>family</it>, <it>subfamily</it>, <it>sub-subfamily </it>and <it>subtype </it>by the classifiers at the second, third, fourth, and fifth levels, respectively. To train the classifiers applied at five levels, a non-redundant dataset is carefully constructed, which contains 3178, 1589, 4772, 4924, and 2741 protein sequences at the respective levels. Jackknife tests on this training dataset show that the overall accuracies of PCA-GPCR at five levels (from the first to the fifth) can achieve up to 99.5%, 88.8%, 80.47%, 80.3%, and 92.34%, respectively. We further perform predictions on a dataset of 1238 GPCRs at the second level, and on another two datasets of 167 and 566 GPCRs respectively at the fourth level. The overall prediction accuracies of our method are consistently higher than those of the existing methods to be compared.</p> <p>Conclusions</p> <p>The comprehensive set of 1497 features is believed to be capable of capturing information about amino acid composition, sequence order as well as various physicochemical properties of proteins. Therefore, high accuracies are achieved when predicting GPCRs at all the five levels with our proposed method.</p