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Hyper-Dual Numbers for Exact Second-Derivative Calculations
49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition AIAA 2011-886 4 - 7 January 2011, Orlando, Florida The Development of Hyper-Dual Numbers for Exact Second-Derivative Calculations Jeffrey A. Fike∗ andJuanJ.Alonso† Department of Aeronautics and Astronautics Stanford University, Stanford, CA 94305 The complex-step approximation for the calculation of derivative information has two significant advantages: the formulation does not suffer from subtractive cancellation er- rors and it can provide exact derivatives without the need to search for an optimal step size. However, when used for the calculation of second derivatives that may be required for approximation and optimization methods, these advantages vanish. In this work, we develop a novel calculation method that can be used to obtain first (gradient) and sec- ond (Hessian) derivatives and that retains all the advantages of the complex-step method (accuracy and step-size independence). In order to accomplish this task, a new number system which we have named hyper-dual numbers and the corresponding arithmetic have been developed. The properties of this number system are derived and explored, and the formulation for derivative calculations is presented. Hyper-dual number arithmetic can be applied to arbitrarily complex software and allows the derivative calculations to be free from both truncation and subtractive cancellation errors. A numerical implementation on an unstructured, parallel, unsteady Reynolds-Averaged Navier Stokes (URANS) solver, -
A Nestable Vectorized Templated Dual Number Library for C++11
cppduals: a nestable vectorized templated dual number library for C++11 Michael Tesch1 1 Department of Chemistry, Technische Universität München, 85747 Garching, Germany DOI: 10.21105/joss.01487 Software • Review Summary • Repository • Archive Mathematical algorithms in the field of optimization often require the simultaneous com- Submitted: 13 May 2019 putation of a function and its derivative. The derivative of many functions can be found Published: 05 November 2019 automatically, a process referred to as automatic differentiation. Dual numbers, close rela- License tives of the complex numbers, are of particular use in automatic differentiation. This library Authors of papers retain provides an extremely fast implementation of dual numbers for C++, duals::dual<>, which, copyright and release the work when replacing scalar types, can be used to automatically calculate a derivative. under a Creative Commons A real function’s value can be made to carry the derivative of the function with respect to Attribution 4.0 International License (CC-BY). a real argument by replacing the real argument with a dual number having a unit dual part. This property is recursive: replacing the real part of a dual number with more dual numbers results in the dual part’s dual part holding the function’s second derivative. The dual<> type in this library allows this nesting (although we note here that it may not be the fastest solution for calculating higher order derivatives.) There are a large number of automatic differentiation libraries and classes for C++: adolc (Walther, 2012), FAD (Aubert & Di Césaré, 2002), autodiff (Leal, 2019), ceres (Agarwal, Mierle, & others, n.d.), AuDi (Izzo, Biscani, Sánchez, Müller, & Heddes, 2019), to name a few, with another 30-some listed at (autodiff.org Bücker & Hovland, 2019). -
Application of Dual Quaternions on Selected Problems
APPLICATION OF DUAL QUATERNIONS ON SELECTED PROBLEMS Jitka Prošková Dissertation thesis Supervisor: Doc. RNDr. Miroslav Láviˇcka, Ph.D. Plze ˇn, 2017 APLIKACE DUÁLNÍCH KVATERNIONU˚ NA VYBRANÉ PROBLÉMY Jitka Prošková Dizertaˇcní práce Školitel: Doc. RNDr. Miroslav Láviˇcka, Ph.D. Plze ˇn, 2017 Acknowledgement I would like to thank all the people who have supported me during my studies. Especially many thanks belong to my family for their moral and material support and my advisor doc. RNDr. Miroslav Láviˇcka, Ph.D. for his guidance. I hereby declare that this Ph.D. thesis is completely my own work and that I used only the cited sources. Plzeˇn, July 20, 2017, ........................... i Annotation In recent years, the study of quaternions has become an active research area of applied geometry, mainly due to an elegant and efficient possibility to represent using them ro- tations in three dimensional space. Thanks to their distinguished properties, quaternions are often used in computer graphics, inverse kinematics robotics or physics. Furthermore, dual quaternions are ordered pairs of quaternions. They are especially suitable for de- scribing rigid transformations, i.e., compositions of rotations and translations. It means that this structure can be considered as a very efficient tool for solving mathematical problems originated for instance in kinematics, bioinformatics or geodesy, i.e., whenever the motion of a rigid body defined as a continuous set of displacements is investigated. The main goal of this thesis is to provide a theoretical analysis and practical applications of the dual quaternions on the selected problems originated in geometric modelling and other sciences or various branches of technical practise. -
Mathematicians Fleeing from Nazi Germany
Mathematicians Fleeing from Nazi Germany Mathematicians Fleeing from Nazi Germany Individual Fates and Global Impact Reinhard Siegmund-Schultze princeton university press princeton and oxford Copyright 2009 © by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW All Rights Reserved Library of Congress Cataloging-in-Publication Data Siegmund-Schultze, R. (Reinhard) Mathematicians fleeing from Nazi Germany: individual fates and global impact / Reinhard Siegmund-Schultze. p. cm. Includes bibliographical references and index. ISBN 978-0-691-12593-0 (cloth) — ISBN 978-0-691-14041-4 (pbk.) 1. Mathematicians—Germany—History—20th century. 2. Mathematicians— United States—History—20th century. 3. Mathematicians—Germany—Biography. 4. Mathematicians—United States—Biography. 5. World War, 1939–1945— Refuges—Germany. 6. Germany—Emigration and immigration—History—1933–1945. 7. Germans—United States—History—20th century. 8. Immigrants—United States—History—20th century. 9. Mathematics—Germany—History—20th century. 10. Mathematics—United States—History—20th century. I. Title. QA27.G4S53 2008 510.09'04—dc22 2008048855 British Library Cataloging-in-Publication Data is available This book has been composed in Sabon Printed on acid-free paper. ∞ press.princeton.edu Printed in the United States of America 10 987654321 Contents List of Figures and Tables xiii Preface xvii Chapter 1 The Terms “German-Speaking Mathematician,” “Forced,” and“Voluntary Emigration” 1 Chapter 2 The Notion of “Mathematician” Plus Quantitative Figures on Persecution 13 Chapter 3 Early Emigration 30 3.1. The Push-Factor 32 3.2. The Pull-Factor 36 3.D. -
Vector Bundles and Brill–Noether Theory
MSRI Series Volume 28, 1995 Vector Bundles and Brill{Noether Theory SHIGERU MUKAI Abstract. After a quick review of the Picard variety and Brill–Noether theory, we generalize them to holomorphic rank-two vector bundles of canonical determinant over a compact Riemann surface. We propose sev- eral problems of Brill–Noether type for such bundles and announce some of our results concerning the Brill–Noether loci and Fano threefolds. For example, the locus of rank-two bundles of canonical determinant with five linearly independent global sections on a non-tetragonal curve of genus 7 is a smooth Fano threefold of genus 7. As a natural generalization of line bundles, vector bundles have two important roles in algebraic geometry. One is the moduli space. The moduli of vector bun- dles gives connections among different types of varieties, and sometimes yields new varieties that are difficult to describe by other means. The other is the linear system. In the same way as the classical construction of a map to a pro- jective space, a vector bundle gives rise to a rational map to a Grassmannian if it is generically generated by its global sections. In this article, we shall de- scribe some results for which vector bundles play such roles. They are obtained from an attempt to generalize Brill–Noether theory of special divisors, reviewed in Section 2, to vector bundles. Our main subject is rank-two vector bundles with canonical determinant on a curve C with as many global sections as pos- sible: especially their moduli and the Grassmannian embeddings of C by them (Section 4). -
ON MATHEMATICAL TERMINOLOGY: CULTURE CROSSING in NINETEENTH-CENTURY CHINA the History of Mathematical Terminologies in Nineteent
ANDREA BRÉARD ON MATHEMATICAL TERMINOLOGY: CULTURE CROSSING IN NINETEENTH-CENTURY CHINA INTRODUCTORY REMARKS: ON MATHEMATICAL SYMBOLISM The history of mathematical terminologies in nineteenth-century China is a multi-layered issue. Their days of popularity and eventual decline are bound up with the complexities of the disputes between the Qian-Jia School who were concerned with textual criticism and the revival of ancient native Chinese mathematical texts, and those scholars that were interested in mathematical studies per se1 (and which we will discuss in a more detailed study elsewhere). The purpose of the following article is to portray the ‘introduction’ of Western mathematical notations into the Chinese consciousness of the late nineteenth century as a major signifying event, both (i) in its own right within the context of the translation of mathematical writ- ings from the West; and (ii) with respect to the way in which the tradi- tional mathematical symbolic system was interpreted by Qing commentators familiar with Western symbolical algebra. Although one might assume to find parallel movements in the commentatorial and translatory practices of one and the same person, nineteenth-cen- tury mathematical activities will be shown to be clearly compartmen- talized. In fact, for the period under consideration here, it would be more appropriate to speak of the ‘reintroduction’ of Western mathematical notations. Already in 1712 the French Jesuit Jean-François Foucquet S. J. (1665–1741) had tried to introduce algebraic symbols in his Aer- rebala xinfa ̅ (New method of algebra). This treatise, which was written for the Kangxi emperor, introduced symbolical 1 In particular, the ‘three friends discussing astronomy and mathematics’ (Tan tian san you dž ) Wang Lai (1768–1813), Li Rui (1763–1820) and Jiao Xun nj (1765–1814). -
Definitions and Nondefinability in Geometry 475 2
Definitions and Nondefinability in Geometry1 James T. Smith Abstract. Around 1900 some noted mathematicians published works developing geometry from its very beginning. They wanted to supplant approaches, based on Euclid’s, which han- dled some basic concepts awkwardly and imprecisely. They would introduce precision re- quired for generalization and application to new, delicate problems in higher mathematics. Their work was controversial: they departed from tradition, criticized standards of rigor, and addressed fundamental questions in philosophy. This paper follows the problem, Which geo- metric concepts are most elementary? It describes a false start, some successful solutions, and an argument that one of those is optimal. It’s about axioms, definitions, and definability, and emphasizes contributions of Mario Pieri (1860–1913) and Alfred Tarski (1901–1983). By fol- lowing this thread of ideas and personalities to the present, the author hopes to kindle interest in a fascinating research area and an exciting era in the history of mathematics. 1. INTRODUCTION. Around 1900 several noted mathematicians published major works on a subject familiar to us from school: developing geometry from the very beginning. They wanted to supplant the established approaches, which were based on Euclid’s, but which handled awkwardly and imprecisely some concepts that Euclid did not treat fully. They would present geometry with the precision required for general- ization and applications to new, delicate problems in higher mathematics—precision beyond the norm for most elementary classes. Work in this area was controversial: these mathematicians departed from tradition, criticized previous standards of rigor, and addressed fundamental questions in logic and philosophy of mathematics.2 After establishing background, this paper tells a story about research into the ques- tion, Which geometric concepts are most elementary? It describes a false start, some successful solutions, and a demonstration that one of those is in a sense optimal. -
An Inquiry Into Alfred Clebsch's Geschlecht
”Are the genre and the Geschlecht one and the same number?” An inquiry into Alfred Clebsch’s Geschlecht François Lê To cite this version: François Lê. ”Are the genre and the Geschlecht one and the same number?” An inquiry into Alfred Clebsch’s Geschlecht. Historia Mathematica, Elsevier, 2020, 53, pp.71-107. hal-02454084v2 HAL Id: hal-02454084 https://hal.archives-ouvertes.fr/hal-02454084v2 Submitted on 19 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. “Are the genre and the Geschlecht one and the same number?” An inquiry into Alfred Clebsch’s Geschlecht François Lê∗ Postprint version, April 2020 Abstract This article is aimed at throwing new light on the history of the notion of genus, whose paternity is usually attributed to Bernhard Riemann while its original name Geschlecht is often credited to Alfred Clebsch. By comparing the approaches of the two mathematicians, we show that Clebsch’s act of naming was rooted in a projective geometric reinterpretation of Riemann’s research, and that his Geschlecht was actually a different notion than that of Riemann. We also prove that until the beginning of the 1880s, mathematicians clearly distinguished between the notions of Clebsch and Riemann, the former being mainly associated with algebraic curves, and the latter with surfaces and Riemann surfaces. -
From Riemann Surfaces to Complex Spaces Reinhold Remmert∗
From Riemann Surfaces to Complex Spaces Reinhold Remmert∗ We must always have old memories and young hopes Abstract This paper analyzes the development of the theory of Riemann surfaces and complex spaces, with emphasis on the work of Rie- mann, Klein and Poincar´e in the nineteenth century and on the work of Behnke-Stein and Cartan-Serre in the middle of this cen- tury. R´esum´e Cet article analyse le d´eveloppement de la th´eorie des surfaces de Riemann et des espaces analytiques complexes, en ´etudiant notamment les travaux de Riemann, Klein et Poincar´eauXIXe si`ecle et ceux de Behnke-Stein et Cartan-Serre au milieu de ce si`ecle. Table of Contents 1. Riemann surfaces from 1851 to 1912 1.1. Georg Friedrich Bernhard Riemann and the covering principle 1.1∗. Riemann’s doctorate 1.2. Christian Felix Klein and the atlas principle 1.3. Karl Theodor Wilhelm Weierstrass and analytic configurations AMS 1991 Mathematics Subject Classification: 01A55, 01A60, 30-03, 32-03 ∗Westf¨alische Wilhelms–Universit¨at, Mathematisches Institut, D–48149 Munster,¨ Deutschland This expos´e is an enlarged version of my lecture given in Nice. Gratias ago to J.-P. Serre for critical comments. A detailed exposition of sections 1 and 2 will appear elsewhere. SOCIET´ EMATH´ EMATIQUE´ DE FRANCE 1998 204 R. REMMERT 1.4. The feud between G¨ottingen and Berlin 1.5. Jules Henri Poincar´e and automorphic functions 1.6. The competition between Klein and Poincar´e 1.7. Georg Ferdinand Ludwig Philipp Cantor and countability of the topology 1.8. -
Philosophia Scientić, 17-1
Philosophia Scientiæ Travaux d'histoire et de philosophie des sciences 17-1 | 2013 The Epistemological Thought of Otto Hölder Paola Cantù et Oliver Schlaudt (dir.) Édition électronique URL : http://journals.openedition.org/philosophiascientiae/811 DOI : 10.4000/philosophiascientiae.811 ISSN : 1775-4283 Éditeur Éditions Kimé Édition imprimée Date de publication : 1 mars 2013 ISBN : 978-2-84174-620-0 ISSN : 1281-2463 Référence électronique Paola Cantù et Oliver Schlaudt (dir.), Philosophia Scientiæ, 17-1 | 2013, « The Epistemological Thought of Otto Hölder » [En ligne], mis en ligne le 01 mars 2013, consulté le 04 décembre 2020. URL : http:// journals.openedition.org/philosophiascientiae/811 ; DOI : https://doi.org/10.4000/ philosophiascientiae.811 Ce document a été généré automatiquement le 4 décembre 2020. Tous droits réservés 1 SOMMAIRE General Introduction Paola Cantù et Oliver Schlaudt Intuition and Reasoning in Geometry Inaugural Academic Lecture held on July 22, 1899. With supplements and notes Otto Hölder Otto Hölder’s 1892 “Review of Robert Graßmann’s 1891 Theory of Number”. Introductory Note Mircea Radu Review of Graßmann, Robert, Theory of Number or Arithmetic in Strict Scientific Presentation by Strict Use of Formulas (1891) Otto Hölder Between Kantianism and Empiricism: Otto Hölder’s Philosophy of Geometry Francesca Biagioli Hölder, Mach, and the Law of the Lever: A Case of Well-founded Non-controversy Oliver Schlaudt Otto Hölder’s Interpretation of David Hilbert’s Axiomatic Method Mircea Radu Geometry and Measurement in Otto Hölder’s Epistemology Paola Cantù Varia Hat Kurt Gödel Thomas von Aquins Kommentar zu Aristoteles’ De anima rezipiert? Eva-Maria Engelen Philosophia Scientiæ, 17-1 | 2013 2 General Introduction Paola Cantù and Oliver Schlaudt 1 The epistemology of Otto Hölder 1 This special issue is devoted to the philosophical ideas developed by Otto Hölder (1859-1937), a mathematician who made important contributions to analytic functions and group theory. -
The Translation of Modern Western Science in Nineteenth-Century China, 1840-1895
The Translation of Modern Western Science in Nineteenth-Century China, 1840-1895 David Wright Isis, Vol. 89, No. 4. (Dec., 1998), pp. 653-673. Stable URL: http://links.jstor.org/sici?sici=0021-1753%28199812%2989%3A4%3C653%3ATTOMWS%3E2.0.CO%3B2-Q Isis is currently published by The University of Chicago Press. Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/journals/ucpress.html. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academic journals and scholarly literature from around the world. The Archive is supported by libraries, scholarly societies, publishers, and foundations. It is an initiative of JSTOR, a not-for-profit organization with a mission to help the scholarly community take advantage of advances in technology. For more information regarding JSTOR, please contact [email protected]. http://www.jstor.org Sun Mar 2 21:45:07 2008 The Translation of Modern Western Science in Nineteenth-Century China, 1840-1 895 By David Wright* ABSTRACT The translation of Western science texts into Chinese began with the Jesuits in the sixteenth century, but by 1800 their impact had waned. -
A Survey of the Development of Geometry up to 1870
A Survey of the Development of Geometry up to 1870∗ Eldar Straume Department of mathematical sciences Norwegian University of Science and Technology (NTNU) N-9471 Trondheim, Norway September 4, 2014 Abstract This is an expository treatise on the development of the classical ge- ometries, starting from the origins of Euclidean geometry a few centuries BC up to around 1870. At this time classical differential geometry came to an end, and the Riemannian geometric approach started to be developed. Moreover, the discovery of non-Euclidean geometry, about 40 years earlier, had just been demonstrated to be a ”true” geometry on the same footing as Euclidean geometry. These were radically new ideas, but henceforth the importance of the topic became gradually realized. As a consequence, the conventional attitude to the basic geometric questions, including the possible geometric structure of the physical space, was challenged, and foundational problems became an important issue during the following decades. Such a basic understanding of the status of geometry around 1870 enables one to study the geometric works of Sophus Lie and Felix Klein at the beginning of their career in the appropriate historical perspective. arXiv:1409.1140v1 [math.HO] 3 Sep 2014 Contents 1 Euclideangeometry,thesourceofallgeometries 3 1.1 Earlygeometryandtheroleoftherealnumbers . 4 1.1.1 Geometric algebra, constructivism, and the real numbers 7 1.1.2 Thedownfalloftheancientgeometry . 8 ∗This monograph was written up in 2008-2009, as a preparation to the further study of the early geometrical works of Sophus Lie and Felix Klein at the beginning of their career around 1870. The author apologizes for possible historiographic shortcomings, errors, and perhaps lack of updated information on certain topics from the history of mathematics.