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.

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We develop a general theory of canonical bases for quantum symmetric pairs (U, U^i) with parameters of arbitrary finite type. We construct new canonical bases for the simple integrable U-modules and their tensor products regarded as U^i-modules. We also construct a canonical basis for the modified form of the i-quantum group U^i. To that end, we establish several new structural results on quantum symmetric pairs, such as bilinear forms, braid group actions, integral forms, Levi subalgebras (of real rank one), and integrality of the intertwiners.

Factor models are a class of powerful statistical models that have been widely used to deal with dependent measurements that arise frequently from various applications from genomics and neuroscience to economics and finance. As data are collected at an ever-growing scale, statistical machine learning faces some new challenges: high dimensionality, strong dependence among observed variables, heavy-tailed variables and heterogeneity. High-dimensional robust factor analysis serves as a powerful toolkit to conquer these challenges.

We prove pinching estimates for dual flows provided the curvature function used in the inverse flow in de Sitter space is convex.