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 193 result(s)

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: September 17, 2021

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Analysis II: Topology and Differential Calculus of Several Variables

2nd part of 3-semester course on Calculus and Real Analysis:
Convergence and function continuity are studied in normed
and metric spaces as well as in abstract topological spaces
(where net convergence is used). Topological notions of interest
include separation, compactness, and connectedness.
The part on advanced differential calculus includes the
chain rule, Taylor's theorem, necessary and sufficient
conditions for extrema, as well as constrained optimization.

Peter Philip · LMU Munich · Date posted: September 13, 2021 · Date revised: September 17, 2021

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Analysis III: Measure and Integration Theory of Several Variables

3rd part of 3-semester course on Calculus and Real Analysis:
A detailed introduction to abstract measure theory is followed
by the study of abstract integration of real- and complex-valued measurable maps.
Noteworthy theorems include dominated convergence,
Fubini, and change of variables. L^p spaces are studied,
including an introduction to convolution and Fourier transform.
An introductory section on the integration over submanifolds of R^n
culminates in the Gauss-Green Theorem.

Peter Philip · LMU Munich · Date posted: September 13, 2021 · Date revised: September 17, 2021

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Showing 1 - 5 of 193 result(s)