DOCSLIB.ORG

Geometric Algebra (GA)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometric Algebra (GA) Geometric Algebra (GA) Werner Benger, 2007 CCT@LSU SciViz 1 Abstract • Geometric Algebra (GA) denotes the re-discovery and geometrical interpretation of the Clifford algebra applied to real fields. Hereby the so-called „geometrical product“ allows to expand linear algebra (as used in vector calculus in 3D) by an invertible operation to multiply and divide vectors. In two dimenions, the geometric algebra can be interpreted as the algebra of complex numbers. In extends in a natural way into three dimensions and corresponds to the well-known quaternions there, which are widely used to describe rotations in 3D as an alternative superior to matrix calculus. However, in contrast to quaternions, GA comes with a direct geometrical interpretation of the respective operations and allows a much finer differentation among the involved objects than is achieveable via quaternions. Moreover, the formalism of GA is independent from the dimension of space. For instance, rotations and reflections of objects of arbitrary dimensions can be easily described intuitively and generic in spaces of arbitrary higher dimensions. • Due to the elegance of the GA and its wide applicabililty it is sometimes denoted as a new „fundamental language of mathematics“. Its unified formalism covers domains such as differential geometry (relativity theory), quantum mechanics, robotics and last but not least computer graphics in a natural way. • This talk will present the basics of Geometric Algebra and specifically emphasizes on the visualization of its elementary operations. Furthermore, the potential of GA will be demonstrated via usage in various application domains. 2 Motivation of GA • Unification of many domains: quantum mechanics, computer graphics, general relativity, robotics… • Completing algebraic operations on vectors • Unified concept for geometry and algebra • Superior formalism for rotations in arbitrary dimensions • Explicit geometrical interpretation of the involved objects and operations on them 3 Definition: “Algebra” • Vector space V over field K with multiplication “ ” • Null-element, One-element, Inverse • Commutative? a b = b a • Associative? (a b) c = a (b c) • Division algebra? • a≠0 a-1 such that a a-1 = 1 = a-1 a • Alternatively: a b=0 a=0 or b=0 4 Historical Roots • Complex Plane (Gauss ~1800) • Real/Imaginary part: a+ib where i2= -1 • Associative, commutative division-algebra • Polar representation: r ei = r ( cos + i sin ) • Multiplication corresponds to rotation in the plane i sin cos 5 Historical Roots, II William Rowan Hamilton (1805-65) invents Quaternions (1844): – Generalization of complex numbers: • 4 components, non commutative: ab ba (in general) • Basic idea: ii=jj=kk= ijk = -1 • Alternative to younger vector- and matrix algebra (Josiah Willard Gibbs, 1839-1903) • p=(p,p), q=(q,q), p q=(pq - p q , pq + pq + p q) • rotation in R3 are around axis of the vector component: v’ = q v q-1 6 Historical Roots, III • Construction by Cayley-Dickson (a,b)(c,d) = (ac-d *b, *a d+cb) – hypercomplex numbers: • octaves/octonions (8 components) • sedenions/hexadekanions (16 components) • … – incremental loss of • commutativity (quaternions,…) • associativity (octonions,…) • division algebra (sedenions,…) 7 Renaissance of the GA 1878: Clifford introduces “geometric algebra”, but dies at age 34 superseded by Gibb’s vector calculus 1920er: Renaissance in quantum mechanics (Pauli, Dirac) algebra on complex fields no geometrical interpretation 1966-2005 David Hestenes (Arizona State University) revives the geometrical interpretation 1997: Gravitation theory using GA (Lasenby, Doran, Gull; Cambridge) 2001: Geometric Algebra at SIGGRAPH (L. Dorst, S. Mann) 8 9 Geometry and Vectors • Geometric interpretation of a vector – Directed line segment or tangent • Vector-algebra in Euclidean Geometry or Tp(M) • Addition / subtraction of vectors a+b • Multiplication / division by scalars a • Multiplication / Division of vectors?? Multiplication of vectors 10 Complete Vector-algebra? • Invertible product of vectors? • What means vector-division “a/b” ? • ab=C b=a-1C • Note: C not necessarily a vector! • Inner product (not associative): a b Skalar – Not invertible e.g. a b =0 with a≠0, b≠0 but orthogonal • Outer product (associative): a b Bivektor – Generalized cross-product from 3D: a b – Not invertible e.g. a b =0 with a≠0, b≠0 but parallel Multiplikation von Vectoren 11 Bivector a b Describes the plane spun by a and b, sign is orientation a b b a = -a b Defined in arbitrary dimensions, anti-symmetric ( not commutative), associative, distributive, spans a vector space, does not require additional structures Multiplikation von Vektoren 12 Constructing Bivectors No unique determination of the generating vectors possible = = a b = (a+λb) b b b b =0 Basis-element |a| |b| sin a+λb Multiplikation von Vektoren 13 Bivectors in R3 • 3 Basis-elements ex ey, ey ez, ez ex • Generalization: ex ey ez is a volume Multiplikation von Vektoren 14 Vectorspace of Bivectors Linear combinations possible e.g.: ex ey, ez ex Multiplikation von Vektoren 15 Coordinate representation of “ ”-product in R3 • Generic Bivector: A = Axy ex ey + Ayz ey ez + Azx ez ex • (axex + ayey + azez) (bxex + byey + bzez)= axex bxex + axex byey + axex bzez + ayey bxex + ayey byey + ayey bzez + azez bxex + azez byey + azez bzez = (axby - aybx)exy+(aybz-azby)eyz+(axbz- azbx)exz Multiplikation von Vektoren 16 Inner product a b • Describes projections a b = |a| |b| cos = b a Symmetric (commutative), requires quadratic form (Metric) as additional structure, not associative (a b) c a (b c) Multiplikation von Vektoren 17 Comparing the products • Inner product • Outer product – Not associative – Associative • (a b) c ≠ a (b c) • (a b) c= a (b c) – Commutative – Not commutative • a b = b a • a b ≠ b a – Not invertible – Not invertible – Yields a scalar – Yields a bivector 18 Geometric Product 1. Requirements and definition 2. Structure of the operands 3. Calculus using GP 4. Rotations using GP Das Geometrische Produkt 19 Requirements to GP • For elements A,B,C of a vector space with quadratic form Q(v) [i.e. a metric g(u,v) = Q(u+v) - Q(u) – Q(v)] we require: 1. Associative: (AB)C = A(BC) 2. Left-distributive: A(B+C) = AB+AC 3. Right-distributive: (B+C)A= BA+CA 4. Scalar product: A2 = Q(A) = |A|2 Das Geometrische Produkt 20 Properties of the GP • Right-angled triangle |a+b|2 = |a|2+|b|2 (A+B)(A+B) = AA+BA+AB+BB = A2 + B2 AB = -BA for A B = 0 anti-symm if orthogonal • However: not purely anti-symmetric |AB|2 =|A|2 |B|2 for A B = 0 (i.e. A,B parallel: B= A) Das Geometrische Produkt 21 Geometric Product • William Kingdon Clifford (1845-79): • Combine inner and outer product to defined the geometric product AB (1878): AB := A B A B • Result is not a vector, but the sum of a scalar + bivector! • Operates on “multivectors” • Subset of the tensoralgebra • Geometric Product is invertible! Das Geometrische Produkt 22 Multi-vector components 2 • R : A = A0 + A1 e0 + A2 e1 + A3 e0 e1 2.7819… + • R3: A = A0 + + + + A1 e0 + A2 e1 + A3 e2 + + + A4 e0 e1+A5 e1 e2+A6 e0 e2 + A e e e 7 0 1 2 + Struktur von Multivektoren 23 Structure of Multi-vectors Linear combination of anti-symmetric basis elements 2n components 0D 1 Scalar 1D 1 Scalar, 1 Vector 2D 1 Scalar, 2 Vectors, 1 Bivector 3D 1 Scalar, 3 Vectors, 3 Bivectors, 1 Volume 4D 1 Scalar, 4 Vectors, 6 Bivectors, 4 Volume, 1 Hyper-volume 5D … Struktur von Multivektoren 24 Inversion • Given vectors a,b: a b = ½ (ab + ba) symmetric part a b = ½ (ab - ba) anti-symmetric part a b = -(a b) (ex ey ez) Dual in 3D Rechnen mit Multivektoren 25 Reflection at a Vector • Unit vector n, arbitrary vector v Vector v projected to n: v║=(v n) n Reflected vector w = v┴ – v║ = v – 2v║ thus w = v – 2(v n) n with GP w = v – 2[½(vn+nv) ] n = v – vnn – nvn w = -nvn Rechnen mit Multivektoren 26 Rotations 1. Identification with Quaternions 2. Rotation in 2D 3. Rotation in nD 4. Rotation of arbitrary Multivectors in nD Rotation 27 Geometrical Quadrate Consider (AB)2 of Bivector-basis element where |A|=1, |B|=1, A B = 0 AB=A B=-BA Basiselement (AB)2 = (AB) (AB) = -(AB) (BA)=-A(BB) A= -1 Rotation 28 Quaternion Algebra • 2D: complex numbers 2 • i:= exey, i = -1 • 3D: quaternions • i:= ex ey= exey, j:= ey ez = eyez, k:=ex ez=exez • i2 = -1, j2 = -1 , k2 = -1 • ijk = (exey)(eyez)(exez) = -1 • 4D: Biquaternions (complex quaternions, spacetime algebra) Rotation 29 Rotation and GA Right-multiplication of Vectors by Bivectors ex i = ex (exey) = (exex ) ey= ey = ey i = ey(exey)=-ey(eyex)= -ex = Rotation 30 Generic Rotation in 2D • Multiple Rotation ex i i = (ex i) i = ey i = -ex = -1 ex • Arbitrary vector A = Ax ex + Ay ey A i = Ax ex i + Ay ey i = Ax ey - Ay ex • Rotation by arbitrary angle: A cos + A i sin ≡ “A e i ” rotates vector A by angle in plane i Inverse rotation: Ai = -iA : - A ei = e-i A Rotation 31 Rotor in 2D • Rotor R := e i = cos + i sin mit i² = -1 A ei = e-i A = e-i /2 A e i/2 = R A R-1 With R=e-i /2 “Rotor” R-1=ei /2 “inverse Rotor” A R-2 = R2 A = R A R-1 • Product of rotors is multiple rotation R=ABCD, R-1=DCBA is “reverse” R Rotation 32 Rotor in nD • Rotor in plane U, Vektor v: R = cos + sin U U² = -1 Expect: Rv or vR-1 or R v R-1 • Problem: With arbitrary vector v there would be a tri-vector component: Rv = v cos + sin (U v + U v ) iff U v ≠ 0 ( v not coplanar with U) Rotation 33 Rotation in nD -1 Consider: R v R mit v =v┴ + v║ : – We have: U v┴ = 0 d.h.
Load more

Recommended publications
  • Geometric Algebra for Vector Fields Analysis and Visualization: Mathematical Settings, Overview and Applications Chantal Oberson Ausoni, Pascal Frey
    Geometric algebra for vector fields analysis and visualization: mathematical settings, overview and applications Chantal Oberson Ausoni, Pascal Frey To cite this version: Chantal Oberson Ausoni, Pascal Frey. Geometric algebra for vector fields analysis and visualization: mathematical settings, overview and applications. 2014. hal-00920544v2 HAL Id: hal-00920544 https://hal.sorbonne-universite.fr/hal-00920544v2 Preprint submitted on 18 Sep 2014 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. Geometric algebra for vector field analysis and visualization: mathematical settings, overview and applications Chantal Oberson Ausoni and Pascal Frey Abstract The formal language of Clifford’s algebras is attracting an increasingly large community of mathematicians, physicists and software developers seduced by the conciseness and the efficiency of this compelling system of mathematics. This contribution will suggest how these concepts can be used to serve the purpose of scientific visualization and more specifically to reveal the general structure of complex vector fields. We will emphasize the elegance and the ubiquitous nature of the geometric algebra approach, as well as point out the computational issues at stake. 1 Introduction Nowadays, complex numerical simulations (e.g. in climate modelling, weather fore- cast, aeronautics, genomics, etc.) produce very large data sets, often several ter- abytes, that become almost impossible to process in a reasonable amount of time.
    [Show full text]
  • Quaternions and Cli Ord Geometric Algebras
    Quaternions and Cliord Geometric Algebras Robert Benjamin Easter First Draft Edition (v1) (c) copyright 2015, Robert Benjamin Easter, all rights reserved. Preface As a rst rough draft that has been put together very quickly, this book is likely to contain errata and disorganization. The references list and inline citations are very incompete, so the reader should search around for more references. I do not claim to be the inventor of any of the mathematics found here. However, some parts of this book may be considered new in some sense and were in small parts my own original research. Much of the contents was originally written by me as contributions to a web encyclopedia project just for fun, but for various reasons was inappropriate in an encyclopedic volume. I did not originally intend to write this book. This is not a dissertation, nor did its development receive any funding or proper peer review. I oer this free book to the public, such as it is, in the hope it could be helpful to an interested reader. June 19, 2015 - Robert B. Easter. (v1) [email protected] 3 Table of contents Preface . 3 List of gures . 9 1 Quaternion Algebra . 11 1.1 The Quaternion Formula . 11 1.2 The Scalar and Vector Parts . 15 1.3 The Quaternion Product . 16 1.4 The Dot Product . 16 1.5 The Cross Product . 17 1.6 Conjugates . 18 1.7 Tensor or Magnitude . 20 1.8 Versors . 20 1.9 Biradials . 22 1.10 Quaternion Identities . 23 1.11 The Biradial b/a .
    [Show full text]
  • Exploring Physics with Geometric Algebra, Book II., , C December 2016 COPYRIGHT
    peeter joot [email protected] EXPLORINGPHYSICSWITHGEOMETRICALGEBRA,BOOKII. EXPLORINGPHYSICSWITHGEOMETRICALGEBRA,BOOKII. peeter joot [email protected] December 2016 – version v.1.3 Peeter Joot [email protected]: Exploring physics with Geometric Algebra, Book II., , c December 2016 COPYRIGHT Copyright c 2016 Peeter Joot All Rights Reserved This book may be reproduced and distributed in whole or in part, without fee, subject to the following conditions: • The copyright notice above and this permission notice must be preserved complete on all complete or partial copies. • Any translation or derived work must be approved by the author in writing before distri- bution. • If you distribute this work in part, instructions for obtaining the complete version of this document must be included, and a means for obtaining a complete version provided. • Small portions may be reproduced as illustrations for reviews or quotes in other works without this permission notice if proper citation is given. Exceptions to these rules may be granted for academic purposes: Write to the author and ask. Disclaimer: I confess to violating somebody’s copyright when I copied this copyright state- ment. v DOCUMENTVERSION Version 0.6465 Sources for this notes compilation can be found in the github repository https://github.com/peeterjoot/physicsplay The last commit (Dec/5/2016), associated with this pdf was 595cc0ba1748328b765c9dea0767b85311a26b3d vii Dedicated to: Aurora and Lance, my awesome kids, and Sofia, who not only tolerates and encourages my studies, but is also awesome enough to think that math is sexy. PREFACE This is an exploratory collection of notes containing worked examples of more advanced appli- cations of Geometric Algebra (GA), also known as Clifford Algebra.
    [Show full text]
  • Geometric Algebra Rotors for Skinned Character Animation Blending
    Geometric algebra rotors for skinned character animation blending briefs_0080* QLB: 320 FPS GA: 381 FPS QLB: 301 FPS GA: 341 FPS DQB: 290 FPS GA: 325 FPS Figure 1: Comparison between animation blending techniques for skinned characters with variable complexity: a) quaternion linear blending (QLB) and dual-quaternion slerp-based interpolation (DQB) during real-time rigged animation, and b) our faster geometric algebra (GA) rotors in Euclidean 3D space as a first step for further character-simulation related operations and transformations. We employ geometric algebra as a single algebraic framework unifying previous separate linear and (dual) quaternion algebras. Abstract 2 Previous work The main goal and contribution of this work is to show that [McCarthy 1990] has already illustrated the connection between (automatically generated) computer implementations of geometric quaternions and GA bivectors as well as the different Clifford algebra (GA) can perform at a faster level compared to standard algebras with degenerate scalar products that can be used to (dual) quaternion geometry implementations for real-time describe dual quaternions. Even simple quaternions are identified character animation blending. By this we mean that if some piece as 3-D Euclidean taken out of their propert geometric algebra of geometry (e.g. Quaternions) is implemented through geometric context. algebra, the result is as efficient in terms of visual quality and even faster (in terms of computation time and memory usage) as [Fontijne and Dorst 2003] and [Dorst et al. 2007] have illustrated the traditional quaternion and dual quaternion algebra the use of all three GA models with applications from computer implementation. This should be so even without taking into vision, animation as well as a basic recursive ray-tracer.
    [Show full text]
  • Determinant Notes
    68 UNIT FIVE DETERMINANTS 5.1 INTRODUCTION In unit one the determinant of a 2× 2 matrix was introduced and used in the evaluation of a cross product. In this chapter we extend the definition of a determinant to any size square matrix. The determinant has a variety of applications. The value of the determinant of a square matrix A can be used to determine whether A is invertible or noninvertible. An explicit formula for A–1 exists that involves the determinant of A. Some systems of linear equations have solutions that can be expressed in terms of determinants. 5.2 DEFINITION OF THE DETERMINANT a11 a12 Recall that in chapter one the determinant of the 2× 2 matrix A = was a21 a22 defined to be the number a11a22 − a12 a21 and that the notation det (A) or A was used to represent the determinant of A. For any given n × n matrix A = a , the notation A [ ij ]n×n ij will be used to denote the (n −1)× (n −1) submatrix obtained from A by deleting the ith row and the jth column of A. The determinant of any size square matrix A = a is [ ij ]n×n defined recursively as follows. Definition of the Determinant Let A = a be an n × n matrix. [ ij ]n×n (1) If n = 1, that is A = [a11], then we define det (A) = a11 . n 1k+ (2) If na>=1, we define det(A) ∑(-1)11kk det(A ) k=1 Example If A = []5 , then by part (1) of the definition of the determinant, det (A) = 5.
    [Show full text]
  • Vectors, Matrices and Coordinate Transformations
    S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors. Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = |A|. In handwritten text, vectors are often expressed using the −→ arrow, or underbar notation, e.g. A , A. Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α, the result is a vector B = αA, which has magnitude B = |α|A. The vector B, is parallel to A and points in the same direction if α > 0.
    [Show full text]
  • International Centre for Theoretical Physics
    :L^ -^ . ::, i^C IC/89/126 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS NEW SPACETIME SUPERALGEBRAS AND THEIR KAC-MOODY EXTENSION Eric Bergshoeff and Ergin Sezgin INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL. SCIENTIFIC AND CULTURAL ORGANIZATION 1989 MIRAMARE - TRIESTE IC/89/I2G It is well known that supcrslrings admit two different kinds of formulations as far as International Atomic Energy Agency llieir siipcrsytnmclry properties arc concerned. One of them is the Ncveu-Sdwarz-tUmond and (NSIt) formulation [I] in which the world-sheet supcrsymmclry is manifest but spacetimn United Nations Educational Scientific and Cultural Organization supcrsymmclry is not. In this formulation, spacctimc supcrsymmctry arises as a consequence of Glbzzi-Schcrk-Olive (GSO) projections (2J in the Hilbcrt space. The field equations of INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS the world-sheet graviton and gravilino arc the constraints which obey the super-Virasoro algebra. In the other formulation, due to Green and Schwarz (GS) [3|, the spacrtimc super- symmetry is manifest but the world-sheet supersymmclry is not. In this case, the theory lias an interesting local world-sheet symmetry, known as K-symmetry, first discovered by NEW SPACETIME SUPERALGEBRAS AND THEIR KAC-MOODY EXTENSION Sicgcl [4,5) for the superparticlc, and later generalized by Green and Schwarz [3] for super- strings. In a light-cone gauge, a combination of the residual ic-symmetry and rigid spacetime supcrsyininetry fuse together to give rise to an (N=8 or 1G) rigid world-sheet supersymmclry. Eric BergflliocIT The two formulations are essentially equivalent in the light-cone gauge [6,7]. How- Theory Division, CERN, 1211 Geneva 23, Switzerland ever, for many purposes it is desirable to have a covariant quantization scheme.
    [Show full text]
  • Geometric-Algebra Adaptive Filters Wilder B
    1 Geometric-Algebra Adaptive Filters Wilder B. Lopes∗, Member, IEEE, Cassio G. Lopesy, Senior Member, IEEE Abstract—This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are Faces generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well- Edges suited for the description of geometric transformations. Also, (directed lines) differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows Fig. 1. A polyhedron (3-dimensional polytope) can be completely described for applying the same derivation techniques regardless of the by the geometric multiplication of its edges (oriented lines, vectors), which type (subalgebra) of the data, i.e., real, complex numbers, generate the faces and hypersurfaces (in the case of a general n-dimensional quaternions, etc. Relying on those characteristics (among others), polytope). a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares perform calculus with hypercomplex quantities, i.e., elements (LMS) adaptive filter variants: real-entries LMS, complex LMS, that generalize complex numbers for higher dimensions [2]– and quaternions LMS. Mean-square analysis and simulations in [10]. a system identification scenario are provided, showing very good agreement for different levels of measurement noise. GA-based AFs were first introduced in [11], [12], where they were successfully employed to estimate the geometric Index Terms—Adaptive filtering, geometric algebra, quater- transformation (rotation and translation) that aligns a pair of nions.
    [Show full text]
  • Sedeonic Theory of Massive Fields
    Sedeonic theory of massive fields Sergey V. Mironov and Victor L. Mironov∗ Institute for physics of microstructures RAS, Nizhniy Novgorod, Russia (Submitted on August 5, 2013) Abstract In the present paper we develop the description of massive fields on the basis of space-time algebra of sixteen-component sedeons. The generalized sedeonic second-order equation for the potential of baryon field is proposed. It is shown that this equation can be reformulated in the form of a system of Maxwell-like equations for the field intensities. We calculate the baryonic fields in the simple model of point baryon charge and obtain the expression for the baryon- baryon interaction energy. We also propose the generalized sedeonic first-order equation for the potential of lepton field and calculate the energies of lepton-lepton and lepton-baryon interactions. Introduction Historically, the first theory of nuclear forces based on hypothesis of one-meson exchange has been formulated by Hideki Yukawa in 1935 [1]. In fact, he postulated a nonhomogeneous equation for the scalar field similar to the Klein-Gordon equation, whose solution is an effective short-range potential (so-called Yukawa potential). This theory clarified the short-ranged character of nuclear forces and was successfully applied to the explaining of low-energy nucleon-nucleon scattering data and properties of deuteron [2]. Afterwards during 1950s - 1990s many-meson exchange theories and phenomenological approach based on the effective nucleon potentials were proposed for the description of few-nucleon systems and intermediate-range nucleon interactions [3-11]. Nowadays the main fundamental concepts of nuclear force theory are developed in the frame of quantum chromodynamics (so-called effective field theory [13-15], see also reviews [16,17]) however, the meson theories and the effective potential approaches still play an important role for experimental data analysis in nuclear physics [18, 19].
    [Show full text]
  • Determinants Math 122 Calculus III D Joyce, Fall 2012
    Determinants Math 122 Calculus III D Joyce, Fall 2012 What they are. A determinant is a value associated to a square array of numbers, that square array being called a square matrix. For example, here are determinants of a general 2 × 2 matrix and a general 3 × 3 matrix. a b = ad − bc: c d a b c d e f = aei + bfg + cdh − ceg − afh − bdi: g h i The determinant of a matrix A is usually denoted jAj or det (A). You can think of the rows of the determinant as being vectors. For the 3×3 matrix above, the vectors are u = (a; b; c), v = (d; e; f), and w = (g; h; i). Then the determinant is a value associated to n vectors in Rn. There's a general definition for n×n determinants. It's a particular signed sum of products of n entries in the matrix where each product is of one entry in each row and column. The two ways you can choose one entry in each row and column of the 2 × 2 matrix give you the two products ad and bc. There are six ways of chosing one entry in each row and column in a 3 × 3 matrix, and generally, there are n! ways in an n × n matrix. Thus, the determinant of a 4 × 4 matrix is the signed sum of 24, which is 4!, terms. In this general definition, half the terms are taken positively and half negatively. In class, we briefly saw how the signs are determined by permutations.
    [Show full text]
  • Finite Projective Geometries 243
    FINITE PROJECTÎVEGEOMETRIES* BY OSWALD VEBLEN and W. H. BUSSEY By means of such a generalized conception of geometry as is inevitably suggested by the recent and wide-spread researches in the foundations of that science, there is given in § 1 a definition of a class of tactical configurations which includes many well known configurations as well as many new ones. In § 2 there is developed a method for the construction of these configurations which is proved to furnish all configurations that satisfy the definition. In §§ 4-8 the configurations are shown to have a geometrical theory identical in most of its general theorems with ordinary projective geometry and thus to afford a treatment of finite linear group theory analogous to the ordinary theory of collineations. In § 9 reference is made to other definitions of some of the configurations included in the class defined in § 1. § 1. Synthetic definition. By a finite projective geometry is meant a set of elements which, for sugges- tiveness, are called points, subject to the following five conditions : I. The set contains a finite number ( > 2 ) of points. It contains subsets called lines, each of which contains at least three points. II. If A and B are distinct points, there is one and only one line that contains A and B. HI. If A, B, C are non-collinear points and if a line I contains a point D of the line AB and a point E of the line BC, but does not contain A, B, or C, then the line I contains a point F of the line CA (Fig.
    [Show full text]
  • New Foundations for Geometric Algebra1
    Text published in the electronic journal Clifford Analysis, Clifford Algebras and their Applications vol. 2, No. 3 (2013) pp. 193-211 New foundations for geometric algebra1 Ramon González Calvet Institut Pere Calders, Campus Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Spain E-mail : [email protected] Abstract. New foundations for geometric algebra are proposed based upon the existing isomorphisms between geometric and matrix algebras. Each geometric algebra always has a faithful real matrix representation with a periodicity of 8. On the other hand, each matrix algebra is always embedded in a geometric algebra of a convenient dimension. The geometric product is also isomorphic to the matrix product, and many vector transformations such as rotations, axial symmetries and Lorentz transformations can be written in a form isomorphic to a similarity transformation of matrices. We collect the idea Dirac applied to develop the relativistic electron equation when he took a basis of matrices for the geometric algebra instead of a basis of geometric vectors. Of course, this way of understanding the geometric algebra requires new definitions: the geometric vector space is defined as the algebraic subspace that generates the rest of the matrix algebra by addition and multiplication; isometries are simply defined as the similarity transformations of matrices as shown above, and finally the norm of any element of the geometric algebra is defined as the nth root of the determinant of its representative matrix of order n. The main idea of this proposal is an arithmetic point of view consisting of reversing the roles of matrix and geometric algebras in the sense that geometric algebra is a way of accessing, working and understanding the most fundamental conception of matrix algebra as the algebra of transformations of multiple quantities.
    [Show full text]