We study the heat kernel expansion of the Laplacian on n-forms defined on a subgraph of a directed complete graph. We derive two expressions for the subgraph heat kernel on 0-forms and compute the coefficients of the expansion. We also obtain the subgraph heat kernel of the Laplacian on 1-forms.

Let $W$ be an extended affine Weyl group. We prove that the minimal length elements $w_{\co}$ of any conjugacy class $\co$ of $W$ satisfy some nice properties, generalizing results of Geck and Pfeiffer \cite{GP} on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some $p$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra $H$. We prove that $T_{w_\co}$, where $\co$ ranges over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H, H]$. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne-Lusztig varieties \cite{H99}.

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With the advance in virtual reality applications, garment industry has strived for new developments. This paper reviews state-of-the-art CAD methods in 3D garment design. A large range of techniques are selected and organized into several key modules which form the core of a 3D garment design technology platform. In each module, basic techniques are presented first. Then advanced developments are systematically discussed and commented. The selected key modules – digital human modeling, 3D garment design and modification, numerical integration of draping, 2D pattern generation, geometric details modeling, parallel computation and GPU acceleration – are discussed in turn. Major challenges and solutions that have been addressed over the years are discussed. Finally, some of the ensuing challenges in 3D garment CAD technologies are outlined.