A finite EI category is a small category with finitely many morphisms such that every endomorphism is an isomorphism. They include finite groups, finite posets and free categories of finite quivers as special cases. In this paper we consider the representation types of finite EI categories, describe some criteria for finite representation type, and use them to classify the representation types of several classes of finite EI categories with extra properties.

In this survey paper, we will present the recent work on the study of the compressible fluids with vacuum states by illustrating its interesting and singular behavior through some systems of fluid dynamics, that is, Euler equations, EulerPoisson equations and NavierStokes equations. The main concern is the well-posedness of the problem when vacuum presents and the singular behavior of the solution near the interface separating the vacuum and the gas. Furthermore, the relation of the solutions for the gas dynamics with vacuum to those of the Boltzmann equation will also be discussed. In fact, the results obtained so far for vacuum states are far from being complete and satisfactory. Therefore, this paper can only be served as an introduction to this interesting field which has many open and challenging mathematical problems. Moreover, the problems considered here are limited to the author's interest and

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This paper presents an algorithm for extending B-spline curves and surfaces. Based on the unclamping algorithm for B-spline curves, we propose a new algorithm for extending B-spline curves that extrapolates using the recurrence property of the de Boor algorithm. This algorithm provides a nice extension, with maximum continuity, to the original curve segment. Moreover, it can be applied to the extension of B-spline surfaces. Extension to both single and multiple target points/curves are considered in this paper.

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