1-3 of 3 results
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Matrix Analysis and Operator Theory
PI Edward Poon
Matrices and operators are ubiquitous throughout science, engineering, and mathematics; they are the transformations that arise whenever one studies a linear system (or approximates a nonlinear system by a linear one). Examples include rotations and reflections (rigid motions of space), spin operators (quantum mechanics and quantum computing), stress tensors (mechanics), regression and curve fitting (statistics and data analysis), derivatives and linear differential operators (dynamical systems), to name just a few. By studying various properties, relations, and transformations of matrices and operators one may obtain insight into a wide range of phenomena.
One particular class of problems of interest is the study of preservers. For example, if M_n denotes the space of n x n matrices, one might ask for a complete classification of the isometries preserving a fixed norm. More generally, given any (possibly multi-valued) function f of a matrix (such as its determinant, rank, eigenvalues, singular values, numerical range, etc) one can ask for a description of the maps T:M_n -> M_n satisfying f(T(A)) = f(A) for all A in M_n; in this case one says that T preserves f. Usually one imposes some additional structure on T, requiring that it be linear, or simply additive, or multiplicative, and so on. One might also wish to describe those maps T leaving certain special subsets of matrices invariant (such as projections, unitaries, rank one matrices, etc.). A broad range of tools and concepts are used in solving such preserver problems; for example, consideration of the dual norm, coupled with convexity arguments, can be handy in classifying isometries, while majorization may appear in problems involving eigenvalues, singular values, and unitarily invariant norms. Currently, investigation is being conducted on isometries of certain matrix subalgebras, as well as preservers of certain collections of projections.
Categories: Faculty-Staff
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Analyticity and kernel stabilization of unbounded derivations on C*-algebras
We first show that a derivation studied recently by E. Christensen has a set of analytic elements which is strong operator topology-dense in the algebra of bounded operators on a Hilbert space, which strengthens a result of Christensen. Our second main result shows that this derivation has kernel stabilization, that is, no elements have derivative eventually equal to 0 unless their first derivative is 0. As applications, we (1) show that a family of derivations on C*-algebras studied by Bratteli and Robinson has kernel stabilization, and (2) we provide sufficient conditions for when two operators which satisfy the Heisenberg Commutation Relation must both be unbounded.
Categories: Faculty-Staff
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Simulation Based Inquiry Oriented Linear Algebra
CO-I Ashish Amresh
Games that teach introductory concepts in linear algebra such as vectors, span and dependence are created to be used by instructors in an undergraduate class.
A well-established National workforce need and critical challenge is to recruit and train students in Science, Technology, Engineering and Mathematics (STEM) fields. Since mathematics is a fundamental part of all STEM disciplines, success of undergraduate students in mathematics is a crucial ingredient to address this challenge. Linear algebra is a vital transition course for students in the STEM disciplines because of its unifying power within mathematics and its applicability to areas outside of mathematics. Accordingly, effective instruction at this stage in students' development is paramount. The focus of this project will be to improve teaching, learning, and student success in linear algebra by incorporating a blending of technology and several learning theories and applications to lead to new research results and production of curriculum resources. This project will leverage the investigators' previous research and curriculum development in Inquiry-Oriented Linear Algebra (IOLA) and expertise in Technology Based Learning to explore the unification of curriculum design and technology design theories and practices.
The goals of the project are to: (1) create a digital platform that will equip students with a virtual experience of a version of the IOLA curriculum; (2) document the affordances and constraints for learning using a game platform (IOLA-G) in comparison to face-to-face instruction by experienced IOLA instructors; (3) compare different digital gaming formats to determine which are most conducive to inquiry-oriented learning; and (4) use the knowledge gained from (1), (2), and (3) to improve student learning through the developed technology, and, reflexively, to enhance the existing IOLA curriculum and teacher support resources. The project team will investigate students' mathematical activity and learning while the students are engaged with the digital platform and will use this insight to inform further refinement of design. Building on prior research efforts in the learning and teaching of linear algebra and expertise in Game Based Learning (GBL), the team will design IOLA-G to mimic the problem-centered approach of the existing IOLA curriculum and will iteratively refine this platform through teaching experiments with students throughout the project. The project also will explore the extent to which GBL can provide a dynamic approach to addressing the constraints that larger class sizes place on instructors' implementation of inquiry-oriented curricula. In addition to, and as part of the process of, creating the resource technology, the investigators will incorporate a mixed methods approach with a blending of game-based learning design, curriculum design theory, and research from inquiry-based learning to explore the following research questions: What are the mathematical practices that students engage in and the conceptual understandings students develop using IOLA-G compared to when using only the face-to-face IOLA curriculum? What are the affordances and constraints of different game environments in terms of enacting an inquiry-oriented curriculum? The impact of the project will include the positive effects on STEM discipline student learning, knowledge, abilities, and overall success, which will lead to strengthening United States workforce needs in STEM areas.Categories: Faculty-Staff
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