Noncommutative Geometry and Gauge theories on AF algebras

Non-commutative geometry (NCG) is a mathematical discipline developed in the 1990s by Alain Connes. It is presented as a new generalization of usual geometry, both encompassing and going beyond the Riemannian framework, within a purely algebraic formalism. Like Riemannian geometry, NCG also has links with physics. Indeed, NCG provided a powerful framework for the reformulation of the Standard Model of Particle Physics (SMPP), taking into account General Relativity, in a single "geometric" representation, based on Non-Commutative Gauge Theories (NCGFT). Moreover, this accomplishment provides a convenient framework to study various possibilities to go beyond the SMPP, such as Grand Unified Theories (GUTs). This thesis intends to show an elegant method recently developed by Thierry Masson and myself, which proposes a general scheme to elaborate GUTs in the framework of NCGFTs. This concerns the study of NCGFTs based on approximately finite $C^*$-algebras (AF-algebras), using either derivations of the algebra or spectral triples to build up the underlying differential structure of the Gauge Theory. The inductive sequence defining the AF-algebra is used to allow the construction of a sequence of NCGFTs of Yang-Mills Higgs types, so that the rank $n+1$ can represent a grand unified theory of the rank $n$. The main advantage of this framework is that it controls, using appropriate conditions, the interaction of the degrees of freedom along the inductive sequence on the AF algebra. This suggests a way to obtain GUT-like models while offering many directions of theoretical investigation to go beyond the SMPP

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Quantum physics; Mathematical physics; Matrix theory; Algebr

Alternatives for Jet Engine Control. Volume 1: Modelling and Control Design with Jet Engine Data

This document compiles a comprehensive list of publications supported by, or related to, National Aeronautics and Space Administration Grant NSG-3048, entitled "Alternatives for Jet Engine Control". Dr. Kurt Seldner was the original Technical Officer for the grant, at Lewis Research Center. Dr. Bruce Lehtinen was the final Technical Officer. At the University of Notre Dame, Drs. Michael K. Sain and R. Jeffrey Leake were the original Project Directors, with Dr. Sain becoming the final Project Director. Publications cover work over a ten-year period. The Final Report is divided into two parts. Volume i, "Modelling and Control Design with Jet Engine Data", follows in this report. Volume 2, "Modelling and Control Design with Tensors", has been bound separately

Advances in Discrete Differential Geometry

Differential Geometr

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This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2

New Directions for Contact Integrators

Contact integrators are a family of geometric numerical schemes which guarantee the conservation of the contact structure. In this work we review the construction of both the variational and Hamiltonian versions of these methods. We illustrate some of the advantages of geometric integration in the dissipative setting by focusing on models inspired by recent studies in celestial mechanics and cosmology.Comment: To appear as Chapter 24 in GSI 2021, Springer LNCS 1282

Advances in Discrete Differential Geometry

Differential Geometr

Linear Algebra and Applications: An Inquiry-Based Approach

Linear Algebra and Applications: An Inquiry-Based Approach provides a novel open-source inquiry-based learning approach to linear algebra. The emphasis is on active learning and developing intuition through investigation of examples. The content is introduced through inquiry-based activities, starting with experimentation with hands-on concrete examples and continuing on to developing a deep understanding of the topics through working with conceptual questions. To provide motivation and context for the linear algebra content, the text includes 35 real-life applications projects. While working through all of this material in the text, readers are actively DOING mathematics instead of being passive learners. Although it is difficult to capture the essence of active learning in a textbook, this book is our attempt to do just that.https://scholarworks.gvsu.edu/books/1021/thumbnail.jp