The present study examined the influence of category representations on exemplar generation, which has been neglected in previous category research. An experiment on college students manipulated the category representation of insects in three conditions (prototypes, exemplars, and the hybrid of prototypes and exemplars). Participants were asked to generate as many exemplars as possible. The results demonstrate that category representations affect and constrain exemplar generation. The new findings are as follows. In the prototype and hybrid conditions with the prototype representation, people tend to generate more valid exemplars by using the prototype mutation mechanism, and exemplar generation conforms to the family resemblance structure. Exemplar generation in the hybrid condition is additionally constrained by known exemplars. In the exemplar condition, people tend to generate fewer valid exemplars by using miscellaneous strategies, and their exemplar generation may not conform to the family resemblance structure.
In this research, we investigate an optimal control problem governed by elliptic PDEs with Dirichlet boundary conditions on complex connected domains, which can be utilized to model the cooling process of concrete dam pouring. A new convergence result for two-dimensional Dirichlet boundary control is proven with the Fourier finite volume element method. The Lagrange multiplier approach is employed to find the optimality systems of the Dirichlet boundary optimal control problem. The discrete optimal control problem is then obtained by applying the Fourier finite volume element method based on Galerkin variational formulation for optimality systems, that is, using Fourier expansion in the azimuthal direction and the finite volume element method in the radial direction, respectively. In this way, the original two-dimensional problem is reduced to a sequence of one-dimensional problems, with the Dirichlet boundary acting as an interval endpoint at which a quadratic interpolation scheme can be implemented. The convergence order of state, adjoint state, and Dirichlet boundary control are therefore proved. The effectiveness of the method is demonstrated numerically, and numerical data is provided to support the theoretical analysis.
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