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.

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.

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The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded$d$-dimensional cobordism category for all$d$. For$d$= 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW].

We investigate a generalization of Hopf algebra $\mathfrak{sl}_{q}\left( 2\right) $ by weakening the invertibility of the generator $K$, i.e. exchanging its invertibility $KK^{-1}=1$ to the regularity $K\overline{K}K=K$. This leads to a weak Hopf algebra $w\mathfrak{sl}_{q}\left( 2\right) $ and a $J$-weak Hopf algebra $v\mathfrak{sl}_{q}\left( 2\right) $ which are studied in detail. It is shown that the monoids of group-like elements of $w\mathfrak{sl}_{q}\left( 2\right) $ and $v\mathfrak{sl}_{q}\left( 2\right) $ are regular monoids, which supports the general conjucture on the connection betweek weak Hopf algebras and regular monoids. Moreover, from $w\mathfrak{sl}_{q}\left( 2\right) $ a quasi-braided weak Hopf algebra $\overline{U}_{q}^{w}$ is constructed and it is shown that the corresponding quasi-$R$-matrix is regular $R^{w}\hat{R}^{w}R^{w}=R^{w}$.