In this article, we propose the notion of the general $p$-affine capacity and prove some basic properties for the general $p$-affine capacity, such as affine invariance and monotonicity. The newly proposed general $p$-affine capacity is compared with several classical geometric quantities, e.g., the volume, the $p$-variational capacity and the $p$-integral affine surface area. Consequently, several sharp geometric inequalities for the general $p$-affine capacity are obtained. These inequalities extend and strengthen many well-known (affine) isoperimetric and (affine) isocapacitary inequalities.

Prochlorococcus marinus, one of the most abundant marine cyanobacteria in the global ocean, is classified into low-light (LL) and high-light (HL) adapted ecotypes. These two adapted ecotypes differ in their ecophysiological characteristics, especially whether adapted for growth at high-light or low-light intensities. However, some evolutionary relationships of Prochlorococcus phylogeny remain to be resolved, such as whether the strains SS120 and MIT9211 form a monophyletic group. We use the Natural Vector (NV) method to represent the sequence in order to identify the phylogeny of the Prochlorococcus. The natural vector method is alignment free without any model assumptions. This study added the covariances of amino acids in protein sequence to the natural vector method. Based on these new natural vectors, we can compute the Hausdorff distance between the two clades which represents the dissimilarity. This method enables us to systematically analyze both the dataset of ribosomal proteomes and the dataset of 16s-23s rRNA sequences in order to reconstruct the phylogeny of Prochlorococcus. Furthermore, we apply classification to inspect the relationship of SS120 and MIT9211. From the reconstructed phylogenetic trees and classification results, we may conclude that the SS120 does not cluster with MIT9211. This study demonstrates a new method for performing phylogenetic analysis. The results confirm that these two strains do not form a monophyletic clade in the phylogeny of Prochlorococcus.

We prove that torsion-free G2 structures are (weakly) dynamically stable along the Laplacian flow for closed G2 structures. More precisely, given a torsion-free G2 structure φ on a compact 7-manifold, the Laplacian flow with initial value cohomologous and sufficiently close to φ will converge to a torsion-free G2 structure which is in the orbit of φ under diffeomorphisms isotopic to the identity. We deduce, from fundamental work of Joyce [18], that the Laplacian flow starting at any closed G2 structure with sufficiently small torsion will exist for all time and converge to a torsion-free G2 structure.

We present a new stabilization technique for multiscale convection diffusion problems. Stabilization for these problems has been a challenging task, especially for the case with high Peclet numbers. Our method is based on a constraint energy minimization idea and the discontinuous Petrov-Galerkin formulation. In particular, the test functions are constructed by minimizing an appropriate energy subject to certain orthogonality conditions, and are related to the trial space. The resulting test functions have a localization property, and can therefore be computed locally. We will prove the stability, and present several numerical results. Our numerical results confirm that our test space gives a good stability, in the sense that the solution error is close to the best approximation error.

In [8], Hitchin proved that a compact K/ihlerian twistor space is in fact a projective algebraic manifold. Moreover, it is the twistor space associated to either the Euclidean 4-sphere S 4 or the complex projective plane CF 2 with Fubini-Study metric. The twistor spaces are 3 and the flag of lines in~ p2 respectively. In [10, 11], the author proved the existence of self-dual metric with positive scalar curvature on the connected-sums of two or three copies of the complex projective planes. Their twistor spaces are Moish6zon spaces. In fact, they are the small resolution of the intersection of two quadrics in~ ps with four nodes and the double covering of CP 3 branched along a quartic with thirteen nodes. Recently, the joint work of Donaldson and Friedman [3] produced a general procedure to construct new twistor spaces and hence self-dual manifolds. In this article, we shall follow the spirit of Hitchin's work and prove the