Mathematical Education for Teachers

We study boundary-value problems with mixed boundary conditions in weakly Lipschitz domains of compact boundaryless Riemannian Lipschitz manifolds. These include the case of the Maxwell system, the Hodge–Dirac operator, and the Hodge–Laplacian. Our approach brings to bear tools from functional analysis, differential geometry, harmonic analysis, and topology. Bibliography: 45 titles.

Hodge decompositions with mixed boundary conditions and applications to partial differential equations on lipschitz manifolds

Abstract: Let Ω C R be a bounded domain with a smooth boundary Γ . The \"tangential trace\" 7 Έ of a differential form E on Ω is defined with the aid of the pullback i ' where t denotes the embedding of Γ into Ω . It is well known that the \"trace theorem\" for differential forms assures that Τ defines a topological isomorphism from R ' ( n ) (the space of L-forms E with exterior derivative d E in L) onto Λ ^ , (Γ) (the space of differential forms E on Γ such that both E and d E belong to a fractional order Sobolev space i f 1 / 2 ). We generalize and extend this and related results to domains Ω with a Lipschitz boundary Γ where even the definition of the spaces Η/(Γ) is not obvious. (For the special case of classical vector analysis corresponding results may be found in [3] and its bibliography.)

Traces of differential forms on lipschitz boundaries

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem, this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl, D) equipped with a symplectic pairing arising from the ${\wedge}$-product of 1-forms on ${\partial D}$. Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extensions. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.

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Self-adjoint curl operators

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain $D$. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl) equipped with a symplectic pairing arising from the $\wedge$-product of 1-forms on $\partial D$. Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extension. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.

There has been much debate in recent years as to the amount and type of mathematical knowledge that teachers need to acquire. One set of recommendations is provided by the Mathematical Education of Teachers (MET) report, produced jointly by the American Mathematical Society and the Mathematical Association of America. This paper reports on efforts at the University of Wisconsin-Milwaukee to implement the MET report recommendations for pre-service elementary and middle grades teachers, in the contexts of teacher education programmes and the teacher licensing structure of the state of Wisconsin. Data is also presented on the results of teaching the resulting courses to in-service teachers.

University of Wisconsin-Milwaukee mathematics focus courses: mathematics content for elementary and middle grades teachers

In [14], Teleman developed a theory of signature operators on Lipschitz manifolds. The theory includes both the classical index operator _D + and "twisted" operators with values in vector bundles. The main result of 1,-14] is Theorem 13.1 which states, in part, that for any closed, oriented, Lipschitz Riemannian manifold M" and for any Lipschitz complex vector bundle ~ over M, there is a natural signature operator _D[; the index of this operator is a Lipschitz invariant of the pair (M, ~). Using this result, Sullivan and Teleman 1-13] were able to show that (except in dimension 4) the index of _D~is actually a topological invariant of(M, 4). The computation of the index of _D~requires three tools: Hodge theory, a compact imbedding result and an excision theorem. The proof of the imbedding result in particular is quite complicated, using ideas from the theory of pseudodifferential operators. The main purpose of the present paper is to provide a simpler proof of the compact imbedding, based on ideas in 1-11], and to show that Hodge theory may be derived easily as a consequence of the imbedding result. (In the smooth case, this approach to Hodge theory may be traced back to Gaffney 15] and Friedrichs I-4].) Our techniques use only elementary facts from functional analysis. (For a concise summary of these see Appendix A of 1,8].) Before proceeding with the rigorous development, we would like to explain briefly the essential new idea in our approach. (The precise definitions of all spaces and operators may be found in the next section.) We denote the Hilbert space of square-integrable differential forms on M by L2(M), and define

A compact imbedding result on Lipschitz manifolds

We prove that the quasilinear parabolic initial-boundary value problem (1.1) below is globally well-posed in a class of high order Sobolev solutions, and that these solutions possess compact, regular attractors ast→+∞.

On some global well-posedness and asymptotic results for quasilinear parabolic equations

A Global Existence Result For Quasi–Linear Parabolic Equations

Administrative processes are ubiquitous in modern life and have been identified as a particular burden to those with accessibility needs. Students who have accessibility needs often have to understand guidance, fill in complex forms, and communicate with multiple parties to disclose disabilities and access appropriate support. Conversational user interfaces (CUIs) could allow us to reimagine such processes, yet there is currently limited understanding of how to design these to be accessible, or whether such an approach would be preferred. In the ADMINS (Assistants for the Disclosure and Management of Information about Needs and Support) project, we implemented a virtual assistant (VA) which is designed to enable students to disclose disabilities and to provide guidance and suggestions about appropriate support. ADMINS explores the potential of CUIs to reduce administrative burden and improve the experience of arranging support by replacing a static form with written or spoken dialogue. This article reports the results of two trials conducted during the project. A beta trial using an early version of the VA provided understanding of accessibility challenges and issues in user experience. The beta trial sample included 22 students who had already disclosed disabilities and 3 disability support advisors. After improvements to the design, a larger main trial was conducted with 134 students who disclosed their disabilities to the university using both the VA and the existing form-based process. The results show that the VA was preferred by most participants to completing the form (64.9% vs 23.9%). Qualitative and quantitative feedback from the trials also identified accessibility and user experience barriers for improving CUI design, and an understanding of benefits and preferences that can inform further development of accessible CUIs for this design space.

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Kevin McLeod

Papers

University of Wisconsin-Milwaukee mathematics focus courses: mathematics content for elementary and middle grades teachers

A compact imbedding result on Lipschitz manifolds

On some global well-posedness and asymptotic results for quasilinear parabolic equations

A Global Existence Result For Quasi–Linear Parabolic Equations

Creating ‘a Simple Conversation’: Designing a Conversational User Interface to Improve the Experience of Accessing Support for Study