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.

As a continuation of\lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the recent work of Kachru-Vafa, we degenerate a three-moduli family of Calabi-Yau toric varieties along a codimension one subfamily which can be described by the vanishing of certain Mori coordinate, corresponding to going to the``large volume limit''in a certain direction. Then we see that the deformation space of the subfamily is the same as a certain family of K3 toric surfaces. This family can in turn be studied by further degeneration along a subfamily which in the end is described by a family of elliptic curves. The periods of the K3 family (and hence the original Calabi-Yau family) can be described by the squares of the periods of the elliptic curves. The consequences include:(1) proofs of various conjectural formulas of physicists\vk\lkm involving mirror maps and modular functions;(2) new identities involving multi-variable hypergeometric series and modular functions--generalizing\lianyaufour. In Part B, we study for two-moduli families the perturbation series of the mirror map and the type A Yukawa couplings near certain large volume limits. Our main tool is a new class of polynomial PDEs associated with Fuchsian PDE systems. We derive the first few terms in the perturbation series. For the case of degree 12 hypersurfaces in\P^ 4 [6, 2, 2, 1, 1], in one limit the series of the couplings are expressed in terms of the j function. In another limit, they are expressed in terms of rational functions. The latter give explicit formulas for infinite sequences of``instanton

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In this paper we present a rigorous derivation of the material parameters for both the cylinder and rectangle cloaking structures. Numerical results using these material parameters are presented to demonstrate the cloaking effect.