Given a compact Riemannian manifold $M$, we consider a warped product manifold $\bar M = I \times_h M$, where $I$ is an open interval in $\Bbb R$. For a positive function $\psi$ defined on $\bar M$, we generalize the arguments in \cite{GRW2015} and \cite{RW16}, to obtain the curvature estimates for Hessian equations $\sigma_k(\kappa)=\psi(V,\nu(V))$. We also obtain some existence results for the starshaped compact hypersurface $\Sigma$ satisfying the above equation with various assumptions.

Comparing DNA and protein sequence groups plays an important role in biological evolutionary relationship research. Despite many methods available for sequence comparison, only a few can be used for group comparison. In this study, we propose a novel approach using convex hulls. We use statistical information contained within the sequences to represent each sequence as a point in high dimensional space. We find that the points belonging to one biological group are located in a different region of space than points belonging to other biological groups. To be more precise, the convex hull of the points from one group are disjoint from the convex hulls of points from other groups. This finding allows us to do phylogenetic analysis for groups in an efficient way. Five different theorems are presented for checking whether two convex hulls intersect or are disjoint. Test results for datasets related to HRV, HPV, Ebolavirus, PKC and protein phosphatase domains demonstrate that our method performs well and provides a new tool for studying group phylogeny. More significantly, the convex analysis presents a new way to search for sequences belonging to a biological group by examining points within the group’s convex hull.

Suppose V^G is the fixed-point vertex operator subalgebra of a compact group G acting on a simple abelian intertwining algebra V. We show that if all irreducible V^G-modules contained in V live in some braided tensor category of V^G-modules, then they generate a tensor subcategory equivalent to the category Rep G of finite-dimensional representations of G, with associativity and braiding isomorphisms modified by the abelian 3-cocycle defining the abelian intertwining algebra structure on V. Additionally, we show that if the fusion rules for the irreducible V^G-modules contained in V agree with the dimensions of spaces of intertwiners among G-modules, then the irreducibles contained in V already generate a braided tensor category of V^G-modules. These results do not require rigidity on any tensor category of V^G-modules and thus apply to many examples where braided tensor category structure is known to exist but rigidity is not known; for example they apply when V^G is C_2-cofinite but not necessarily rational. When V^G is both C_2-cofinite and rational and V is a vertex operator algebra, we use the equivalence between Rep G and the corresponding subcategory of V^G-modules to show that V is also rational. As another application, we show that a certain category of modules for the Virasoro algebra at central charge 1 admits a braided tensor category structure equivalent to Rep SU(2), up to modification by an abelian 3-cocycle.

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Let M be a complete Riemannian manifold with Riemannian volume volg and f be a smooth function on M. A sharp upper bound estimate on the first eigenvalue of symmetric diffusion operator 􏰂 f = 􏰂 − ∇ f · ∇ was given by Wu (J Math Anal Appl 361:10–18, 2010) and Wang (Ann Glob Anal Geom 37:393–402, 2010) under a condition that finite dimensional Bakry–Émery Ricci curvature is bounded below, independently. They propounded an open problem is whether there is some rigidity on the estimate. In this note, we will solve this problem to obtain a splitting type theorem, which generalizes Li–Wang’s result in (J Differ Geom 58:501–534, 2001, J Differ Geom 62:143–162, 2002). For the case that infinite dimensional Bakry–Emery Ricci curvature of M is bounded below, we donotexpectanyupperboundestimateonthefirsteigenvalueof􏰂f withoutanyadditional assumption (see the example in Sect. 2). In this case, we will give a sharp upper bound estimate on the first eigenvalue of􏰂f under the additional assuption that |∇f| is bounded.We also obtain the rigidity result on this estimate, as another Li–Wang type splitting theorem.