Results 1  10
of
15
A treatise on quantum Clifford Algebras
, 2002
"... ... on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are ve ..."
Abstract

Cited by 15 (12 self)
 Add to MetaCart
... on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five alternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of GraßmannCayley algebras including comeet and cojoin for GraßmannCayley cogebras which are very efficient and may be used in Robotics, left and right contractions, left and right cocontractions, Clifford and coClifford products, etc. The Chevalley deformation, using a Clifford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a biconvolution. Antipode and crossing are consequences of the product and coproduct structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the definition of nonlocal products and interacting Hopf gebras which are generically nonperturbative. A ‘spinorial ’ generalization of the antipode is given. The nonexistence of nontrivial integrals in lowdimensional Clifford cogebras is shown. Generalized cliffordization is discussed which is based on nonexponentially generated bilinear forms in general resulting in non unital, nonassociative products. Reasonable assumptions lead to bilinear forms based on 2cocycles. Cliffordization is used to derive time and normalordered generating functionals for the SchwingerDyson hierarchies of nonlinear spinor field theory and spinor electrodynamics. The relation between the vacuum structure, the operator ordering, and the Hopf gebraic counit is discussed. QCAs are proposed as the natural language for (fermionic) quantum field theory.
The nonEuclidean style of Minkowskian relativity
 The Symbolic Universe: Geometry and Physics, 1890–1930
, 1999
"... walter @ univnancy2.fr ..."
Between laws and models: Some philosophical morals of Lagrangian mechanics
, 2004
"... I extract some philosophical morals from some aspects of Lagrangian mechanics. (A companion paper will present similar morals from Hamiltonian mechanics and HamiltonJacobi theory.) One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
I extract some philosophical morals from some aspects of Lagrangian mechanics. (A companion paper will present similar morals from Hamiltonian mechanics and HamiltonJacobi theory.) One main moral concerns methodology: Lagrangian mechanics provides a level of description of phenomena which has been largely ignored by philosophers, since it falls between their accustomed levels—“laws of nature ” and “models”. Another main moral concerns ontology: the ontology of Lagrangian mechanics is both more subtle and more problematic than philosophers often realize. The treatment of Lagrangian mechanics provides an introduction to the subject for philosophers, and is technically elementary. In particular, it is confined to systems with a finite number of degrees of freedom, and for the most part eschews modern geometry. Newton’s fundamental discovery, the one which he considered necessary to keep secret and published only in the form of an anagram, consists of the following: Data aequatione quotcunque fluentes quantitae involvente fluxiones invenire et vice versa. In contemporary mathematical language, this means: “It is useful to solve differential equations”.
Breaking in the 4vectors: the fourdimensional movement ingravitation,1905–1910.InJürgenRenn, editor, The Genesis of General Relativity
, 2007
"... The law of gravitational attraction is a window on three formal approaches to laws of nature based on Lorentzinvariance: Poincaré’s fourdimensional vector space (1906), Minkowski’s matrix calculus and spacetime geometry (1908), and Sommerfeld’s 4vector algebra (1910). In virtue of a common appeal ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
The law of gravitational attraction is a window on three formal approaches to laws of nature based on Lorentzinvariance: Poincaré’s fourdimensional vector space (1906), Minkowski’s matrix calculus and spacetime geometry (1908), and Sommerfeld’s 4vector algebra (1910). In virtue of a common appeal to 4vectors for the characterization of gravitational attraction, these three contributions track the emergence and early development of fourdimensional physics.
CLASSICAL SYMMETRIC FUNCTIONS IN SUPERSPACE
, 2005
"... Abstract. We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the s ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We present the basic elements of a generalization of symmetric function theory involving functions of commuting and anticommuting (Grassmannian) variables. These new functions, called symmetric functions in superspace, are invariant under the diagonal action of the symmetric group on the sets of commuting and anticommuting variables. In this work, we present the superspace extension of the classical bases, namely, the monomial symmetric functions, the elementary symmetric functions, the completely symmetric functions, and the power sums. Various basic results, such as the generating functions for the multiplicative bases, Cauchy formulas, involution operations as well as the combinatorial scalar product are also generalized. 1.
Historical Projects in Discrete Mathematics and Computer Science
"... A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itse ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
A course in discrete mathematics is a relatively recent addition, within the last 30 or 40 years, to the modern American undergraduate curriculum, born out of a need to instruct computer science majors in algorithmic thought. The roots of discrete mathematics, however, are as old as mathematics itself, with the notion of counting a discrete operation, usually cited as the first mathematical development
The role of models and analogies in the electromagnetic theory: a historical case study. Science and Education 16(4):835–848
, 2007
"... Abstract. Despite its great importance, many students and even their teachers still cannot recognize the relevance of models to build up physical knowledge and are unable to develop qualitative explanations for mathematical expressions that exist within physics. Thus, it is not a surprise that analo ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. Despite its great importance, many students and even their teachers still cannot recognize the relevance of models to build up physical knowledge and are unable to develop qualitative explanations for mathematical expressions that exist within physics. Thus, it is not a surprise that analogies play an important role in science education, since students ’ construction of mental models of abstract phenomena need to be rooted in some existing or previous experience in order to interpret more complex ideas. The present article focuses on some of these issues by analyzing some specific instances of the historical development of the electromagnetic theory. Using the mental models framework, the importance of mechanical analogies to understand some of the electromagnetic concepts are emphasized.
Quaternions and the Heuristic Role of Mathematical Structures in Physics
, 1992
"... One of the important ways development takes place in mathematics is via a process of generalization. On the basis of a recent characterization of this process we propose a principle that generalizations of mathematical structures that are already part of successful physical theories serve as good gu ..."
Abstract
 Add to MetaCart
(Show Context)
One of the important ways development takes place in mathematics is via a process of generalization. On the basis of a recent characterization of this process we propose a principle that generalizations of mathematical structures that are already part of successful physical theories serve as good guides for the development of new physical theories. The principle is a more formal presentation and extension of a position stated earlier this century by Dirac. Quaternions form an excellent example of such a generalization, and we consider a number of the ways in which their use in physical theories illustrates this principle. Key words: Quaternions, heuristics, mathematics and physics, quaternionic In recent decades the necessary role mathematical structures play in the formulation of physical theories has been the subject of ongoing interest. Wigner’s reference in a well known essay of 1960 [1] to the “unreasonable effectiveness” of mathematics in this role has captured what is undoubtedly a widespread feeling
Knot theory from Vandermonde to Jones
"... . Leibniz wrote in 1679: "I consider that we need yet another kind of analysis, : : : which deals directly with position." He called it "geometry of position"(geometria situs) . The first convincing example of geometria situs was Euler's solution to the bridges of Konigsberg ..."
Abstract
 Add to MetaCart
(Show Context)
. Leibniz wrote in 1679: "I consider that we need yet another kind of analysis, : : : which deals directly with position." He called it "geometry of position"(geometria situs) . The first convincing example of geometria situs was Euler's solution to the bridges of Konigsberg problem (1735). The first mathematical paper which mentions knots was written by A. T. Vandermonde in 1771: "Remarques sur les problemes de situation". We will sketch in this essay 1 the history of knot theory from Vandermonde to Jones stressing the combinatorial aspect of the theory that is so visible in Jones type invariants. "When Alexander reached Gordium, he was seized with a longing to ascend to the acropolis, where the palace of Gordius and his son Midas was situated, and to see Gordius' wagon and the knot of the wagon's yoke: : :. Over and above this there was a legend about the wagon, that anyone who untied the knot of the yoke would rule Asia. The knot was of cornel bark, and you could not see whe...