Symmetry, Geometry, and Quantization with Hypercomplex
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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. -
Nilpotent Elements Control the Structure of a Module
Nilpotent elements control the structure of a module David Ssevviiri Department of Mathematics Makerere University, P.O BOX 7062, Kampala Uganda E-mail: [email protected], [email protected] Abstract A relationship between nilpotency and primeness in a module is investigated. Reduced modules are expressed as sums of prime modules. It is shown that presence of nilpotent module elements inhibits a module from possessing good structural properties. A general form is given of an example used in literature to distinguish: 1) completely prime modules from prime modules, 2) classical prime modules from classical completely prime modules, and 3) a module which satisfies the complete radical formula from one which is neither 2-primal nor satisfies the radical formula. Keywords: Semisimple module; Reduced module; Nil module; K¨othe conjecture; Com- pletely prime module; Prime module; and Reduced ring. MSC 2010 Mathematics Subject Classification: 16D70, 16D60, 16S90 1 Introduction Primeness and nilpotency are closely related and well studied notions for rings. We give instances that highlight this relationship. In a commutative ring, the set of all nilpotent elements coincides with the intersection of all its prime ideals - henceforth called the prime radical. A popular class of rings, called 2-primal rings (first defined in [8] and later studied in [23, 26, 27, 28] among others), is defined by requiring that in a not necessarily commutative ring, the set of all nilpotent elements coincides with the prime radical. In an arbitrary ring, Levitzki showed that the set of all strongly nilpotent elements coincides arXiv:1812.04320v1 [math.RA] 11 Dec 2018 with the intersection of all prime ideals, [29, Theorem 2.6]. -
Classifying the Representation Type of Infinitesimal Blocks of Category
Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt (Under the Direction of Brian D. Boe) Abstract Let g be a simple Lie algebra over the field C of complex numbers, with root system Φ relative to a fixed maximal toral subalgebra h. Let S be a subset of the simple roots ∆ of Φ, which determines a standard parabolic subalgebra of g. Fix an integral weight ∗ µ µ ∈ h , with singular set J ⊆ ∆. We determine when an infinitesimal block O(g,S,J) := OS of parabolic category OS is nonzero using the theory of nilpotent orbits. We extend work of Futorny-Nakano-Pollack, Br¨ustle-K¨onig-Mazorchuk, and Boe-Nakano toward classifying the representation type of the nonzero infinitesimal blocks of category OS by considering arbitrary sets S and J, and observe a strong connection between the theory of nilpotent orbits and the representation type of the infinitesimal blocks. We classify certain infinitesimal blocks of category OS including all the semisimple infinitesimal blocks in type An, and all of the infinitesimal blocks for F4 and G2. Index words: Category O; Representation type; Verma modules Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. Platt B.A., Utah State University, 1999 M.S., Utah State University, 2001 M.A, The University of Georgia, 2006 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Athens, Georgia 2008 c 2008 Kenyon J. Platt All Rights Reserved Classifying the Representation Type of Infinitesimal Blocks of Category OS by Kenyon J. -
Hyperbolicity of Hermitian Forms Over Biquaternion Algebras
HYPERBOLICITY OF HERMITIAN FORMS OVER BIQUATERNION ALGEBRAS NIKITA A. KARPENKO Abstract. We show that a non-hyperbolic hermitian form over a biquaternion algebra over a field of characteristic 6= 2 remains non-hyperbolic over a generic splitting field of the algebra. Contents 1. Introduction 1 2. Notation 2 3. Krull-Schmidt principle 3 4. Splitting off a motivic summand 5 5. Rost correspondences 7 6. Motivic decompositions of some isotropic varieties 12 7. Proof of Main Theorem 14 References 16 1. Introduction Throughout this note (besides of x3 and x4) F is a field of characteristic 6= 2. The basic reference for the staff related to involutions on central simple algebras is [12].p The degree deg A of a (finite dimensional) central simple F -algebra A is the integer dimF A; the index ind A of A is the degree of a central division algebra Brauer-equivalent to A. Conjecture 1.1. Let A be a central simple F -algebra endowed with an orthogonal invo- lution σ. If σ becomes hyperbolic over the function field of the Severi-Brauer variety of A, then σ is hyperbolic (over F ). In a stronger version of Conjecture 1.1, each of two words \hyperbolic" is replaced by \isotropic", cf. [10, Conjecture 5.2]. Here is the complete list of indices ind A and coindices coind A = deg A= ind A of A for which Conjecture 1.1 is known (over an arbitrary field of characteristic 6= 2), given in the chronological order: • ind A = 1 | trivial; Date: January 2008. Key words and phrases. -
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). -
Exploring Mathematical Objects from Custom-Tailored Mathematical Universes
Exploring mathematical objects from custom-tailored mathematical universes Ingo Blechschmidt Abstract Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are toposes in which the axiom of choice and the intermediate value theorem from undergraduate calculus fail. The purpose of this contribution is to give a glimpse of the toposophic landscape, presenting several specific toposes and exploring their peculiar properties, and to explicate how toposes provide distinct lenses through which the usual mathematical objects of the standard topos can be viewed. Key words: topos theory, realism debate, well-adapted language, constructive mathematics Toposes can be pictured as mathematical universes in which we can do mathematics. Most mathematicians spend all their professional life in just a single topos, the so-called standard topos. However, besides the standard topos, there is a colorful host of alternate toposes which are just as worthy of mathematical study and in which mathematics plays out slightly differently (Figure 1). For instance, there are toposes in which the axiom of choice and the intermediate value theorem from undergraduate calculus fail, toposes in which any function R ! R is continuous and toposes in which infinitesimal numbers exist. The purpose of this contribution is twofold. 1. We give a glimpse of the toposophic landscape, presenting several specific toposes and exploring their peculiar properties. 2. We explicate how toposes provide distinct lenses through which the usual mathe- matical objects of the standard topos can be viewed. -
Commutative Algebra
Commutative Algebra Andrew Kobin Spring 2016 / 2019 Contents Contents Contents 1 Preliminaries 1 1.1 Radicals . .1 1.2 Nakayama's Lemma and Consequences . .4 1.3 Localization . .5 1.4 Transcendence Degree . 10 2 Integral Dependence 14 2.1 Integral Extensions of Rings . 14 2.2 Integrality and Field Extensions . 18 2.3 Integrality, Ideals and Localization . 21 2.4 Normalization . 28 2.5 Valuation Rings . 32 2.6 Dimension and Transcendence Degree . 33 3 Noetherian and Artinian Rings 37 3.1 Ascending and Descending Chains . 37 3.2 Composition Series . 40 3.3 Noetherian Rings . 42 3.4 Primary Decomposition . 46 3.5 Artinian Rings . 53 3.6 Associated Primes . 56 4 Discrete Valuations and Dedekind Domains 60 4.1 Discrete Valuation Rings . 60 4.2 Dedekind Domains . 64 4.3 Fractional and Invertible Ideals . 65 4.4 The Class Group . 70 4.5 Dedekind Domains in Extensions . 72 5 Completion and Filtration 76 5.1 Topological Abelian Groups and Completion . 76 5.2 Inverse Limits . 78 5.3 Topological Rings and Module Filtrations . 82 5.4 Graded Rings and Modules . 84 6 Dimension Theory 89 6.1 Hilbert Functions . 89 6.2 Local Noetherian Rings . 94 6.3 Complete Local Rings . 98 7 Singularities 106 7.1 Derived Functors . 106 7.2 Regular Sequences and the Koszul Complex . 109 7.3 Projective Dimension . 114 i Contents Contents 7.4 Depth and Cohen-Macauley Rings . 118 7.5 Gorenstein Rings . 127 8 Algebraic Geometry 133 8.1 Affine Algebraic Varieties . 133 8.2 Morphisms of Affine Varieties . 142 8.3 Sheaves of Functions . -
Nilpotent Ideals in Polynomial and Power Series Rings 1609
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 138, Number 5, May 2010, Pages 1607–1619 S 0002-9939(10)10252-4 Article electronically published on January 13, 2010 NILPOTENT IDEALS IN POLYNOMIAL AND POWER SERIES RINGS VICTOR CAMILLO, CHAN YONG HONG, NAM KYUN KIM, YANG LEE, AND PACE P. NIELSEN (Communicated by Birge Huisgen-Zimmermann) Abstract. Given a ring R and polynomials f(x),g(x) ∈ R[x] satisfying f(x)Rg(x) = 0, we prove that the ideal generated by products of the coef- ficients of f(x)andg(x) is nilpotent. This result is generalized, and many well known facts, along with new ones, concerning nilpotent polynomials and power series are obtained. We also classify which of the standard nilpotence properties on ideals pass to polynomial rings or from ideals in polynomial rings to ideals of coefficients in base rings. In particular, we prove that if I ≤ R[x]is aleftT -nilpotent ideal, then the ideal formed by the coefficients of polynomials in I is also left T -nilpotent. 1. Introduction Throughout this paper, all rings are associative rings with 1. It is well known that a polynomial f(x) over a commutative ring is nilpotent if and only if each coefficient of f(x) is nilpotent. But this result is not true in general for noncommutative rings. For example, let R = Mn(k), the n × n full matrix ring over some ring k =0. 2 Consider the polynomial f(x)=e12 +(e11 − e22)x − e21x ,wheretheeij’s are 2 the matrix units. In this case f(x) = 0, but e11 − e22 is not nilpotent. -
Math Book from Wikipedia
Math book From Wikipedia PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. PDF generated at: Mon, 25 Jul 2011 10:39:12 UTC Contents Articles 0.999... 1 1 (number) 20 Portal:Mathematics 24 Signed zero 29 Integer 32 Real number 36 References Article Sources and Contributors 44 Image Sources, Licenses and Contributors 46 Article Licenses License 48 0.999... 1 0.999... In mathematics, the repeating decimal 0.999... (which may also be written as 0.9, , 0.(9), or as 0. followed by any number of 9s in the repeating decimal) denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all integer bases, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the only representation. The non-terminating form is more convenient for understanding the decimal expansions of certain fractions and, in base three, for the structure of the ternary Cantor set, a simple fractal. -
Generalisations of Singular Value Decomposition to Dual-Numbered
Generalisations of Singular Value Decomposition to dual-numbered matrices Ran Gutin Department of Computer Science Imperial College London June 10, 2021 Abstract We present two generalisations of Singular Value Decomposition from real-numbered matrices to dual-numbered matrices. We prove that every dual-numbered matrix has both types of SVD. Both of our generalisations are motivated by applications, either to geometry or to mechanics. Keywords: singular value decomposition; linear algebra; dual num- bers 1 Introduction In this paper, we consider two possible generalisations of Singular Value Decom- position (T -SVD and -SVD) to matrices over the ring of dual numbers. We prove that both generalisations∗ always exist. Both types of SVD are motivated by applications. A dual number is a number of the form a + bǫ where a,b R and ǫ2 = 0. The dual numbers form a commutative, assocative and unital∈ algebra over the real numbers. Let D denote the dual numbers, and Mn(D) denote the ring of n n dual-numbered matrices. × arXiv:2007.09693v7 [math.RA] 9 Jun 2021 Dual numbers have applications in automatic differentiation [4], mechanics (via screw theory, see [1]), computer graphics (via the dual quaternion algebra [5]), and geometry [10]. Our paper is motivated by the applications in geometry described in Section 1.1 and in mechanics given in Section 1.2. Our main results are the existence of the T -SVD and the existence of the -SVD. T -SVD is a generalisation of Singular Value Decomposition that resembles∗ that over real numbers, while -SVD is a generalisation of SVD that resembles that over complex numbers.