Open Math Notes

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Welcome to AMS Open Math Notes, a repository of freely downloadable mathematical works hosted by the American Mathematical Society as a service to researchers, faculty and students. Open Math Notes includes:

  • Draft works including course notes, textbooks, and research expositions. These have not been published elsewhere and are subject to revision.
  • Items previously published in the Journal of Inquiry-Based Learning in Mathematics, a refereed journal
  • Refereed publications at the AMS

Visitors are encouraged to download and use any of these materials as teaching and research aids, and to send constructive comments and suggestions to the authors.

Open Math Notes Advisory Board:

  • Karen Vogtmann, Chair | University of Warwick
  • Tom Halverson | Macalester College
  • Andrew Hwang | College of the Holy Cross
  • Robert Lazarsfeld | Stony Brook University
  • Mary Pugh | University of Toronto

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Showing 1 - 5 of 195 result(s)

Linear Algebra I

First part of two-semester course on Linear Algebra: Starting with the basics of propositional calculus and set theory, we then introduce functions, relations, and proofs by induction. After proving necessary facts regarding groups, rings, and fields, we study Linear Algebra's core
subjects, i.e. vector spaces, linear maps, matrices,
and linear systems. Appendix includes an introduction to axiomatic set theory and the axiom of choice.

Peter Philip · LMU Munich · Date posted: October 20, 2021 · Date revised: October 21, 2021

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Linear Algebra II

Second part of two-semester course on Linear Algebra: Affine geometry is introduced as well as duality, multilinear maps, determinants, and eigenvalues. Via characteristic and minimal polynomials, we obtain the Jordan normal form. We study vector spaces with inner products, including orthogonality, adjoint maps, as well as Hermitian, unitary, and normal maps. Appendix includes an existence proof for algebraic closures.

Peter Philip · LMU Munich · Date posted: October 20, 2021 · Date revised: October 21, 2021

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The Dirichlet-to-Neumann Map and Fractal Variants

These notes develop the study of the Dirichlet-to-Neumann map N, associated to a domain in a compact Riemannian manifold, in a variety of situations, starting with the classical case of smoothly bounded domains, then Lipschitz domains, then rougher finite perimeter domains, and finally more exotic cases, where the boundary has a fractal character. We consider the semigroup generated by -N, as a Markov semigroup, with novel properties in the fractal case.

Michael Taylor · University of North Carolina · Date posted: October 13, 2021

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Many Models for Water Waves: A unified theoretical approach

In this monograph we derive, discuss, and justify as much as possible a large class of models describing in an approximate manner the propagation of waves at the surface of water, at the interface between two homogeneous fluids, or in the bulk of a continuously density-stratified fluid. Our aim is to present standard and less-standard models in a unified framework, together with robust mathematical tools involved in their rigorous justification.

Vincent Duchene · Institut de Recherche Mathématique de Rennes · Date posted: October 11, 2021

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Analysis I: Calculus of One Real Variable

1st part of 3-semester course on Calculus and Real Analysis: Starting with the basics of propositional calculus and set theory, we then introduce functions, relations, and proofs by induction. We study convergence of real and complex sequences and series, as well as continuity, differentiability, and Riemann integrability of functions of one real variable.
Appendix includes an introduction to axiomatic set theory and the axiom of choice as well as a construction of the real numbers.

Peter Philip · LMU Munich · Date posted: September 13, 2021 · Date revised: October 21, 2021

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Showing 1 - 5 of 195 result(s)