We discuss a research-based theoretical framework based on affect as an internal representational system. Key ideas include the concepts of meta-affect and affective structures, and the constructs of mathematical intimacy and mathematical integrity. We understand these as fundamental to powerful mathematical problem solving, and deserving of closer attention by educators. In a study of elementary school children we characterize some features of emotional states inferred from individual problem solving behavior, including interactions between affect and cognition. We describe our methodology, illustrating theoretical ideas with brief qualitative examples from a longitudinal series of task-based interviews.

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In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations. Such nonlinear Schr odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge-invariant, and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin and by Bialynicki-Birula and Mycielski. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics.

The case of a noninteracting infinite Bose gas at zero temperature is studied in the formalism of local current algebras, using the representation theory of nuclear Lie groups. The class of representations describing such a system is obtained by taking an ``N / V limit'' of the finite case. These representations can also be determined uniquely from the solutions of a functional differential equation, which follows in turn from a condition on the ground state vector. Finally a system of functional differential equations is formulated for a theory with interactions, using a proposed definition of indefinite functional integration.

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