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Fibrations of groupoids
 J. Algebra
, 1970
"... theory, and change of base for groupoids and multiple ..."
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theory, and change of base for groupoids and multiple
The Lyapunov Characteristic Exponents and their
 Computation, Lect. Notes Phys
, 2010
"... For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. For Want of a Nail (proverbial rhyme) Summary. We present ..."
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For want of a nail the shoe was lost. For want of a shoe the horse was lost. For want of a horse the rider was lost. For want of a rider the battle was lost. For want of a battle the kingdom was lost. And all for the want of a horseshoe nail. For Want of a Nail (proverbial rhyme) Summary. We present a survey of the theory of the Lyapunov Characteristic Exponents (LCEs) for dynamical systems, as well as of the numerical techniques developed for the computation of the maximal, of few and of all of them. After some historical notes on the first attempts for the numerical evaluation of LCEs, we discuss in detail the multiplicative ergodic theorem of Oseledec [99], which provides the theoretical basis for the computation of the LCEs. Then, we analyze the algorithm for the computation of the maximal LCE, whose value has been extensively used as an indicator of chaos, and the algorithm of the so–called ‘standard method’, developed by Benettin et al. [14], for the computation of many LCEs. We also consider different discrete and continuous methods for computing the LCEs based on the QR or the singular value decomposition techniques. Although, we are mainly interested in finite–dimensional conservative systems, i. e. autonomous Hamiltonian systems and symplectic maps, we also briefly refer to the evaluation of LCEs of dissipative systems and time series. The relation of two chaos detection techniques, namely the fast Lyapunov indicator (FLI) and the generalized alignment index (GALI), to the computation of the LCEs is also discussed. 1
The BrauerSeveri variety associated with a central simple algebra: a survery, http://www.mathematik.unibielefeld.de/lag/man/052.pdf
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Homotopy Theory, and Change of Base for Groupoids and Multiple Groupoids
, 1996
"... This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids. ..."
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Cited by 7 (6 self)
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This survey article shows how the notion of "change of base", used in some applications to homotopy theory of the fundamental groupoid, has surprising higher dimensional analogues, through the use of certain higher homotopy groupoids with values in forms of multiple groupoids.
Cubic surfaces with a Galois invariant pair of Steiner trihedra
"... We present a method to construct nonsingular cubic surfaces over with a Galois invariant pair of Steiner trihedra. We start with cubic surfaces in a form generalizing that of A. Cayley and G. Salmon. For these, we develop an explicit version of Galois descent. 1 ..."
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Cited by 6 (4 self)
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We present a method to construct nonsingular cubic surfaces over with a Galois invariant pair of Steiner trihedra. We start with cubic surfaces in a form generalizing that of A. Cayley and G. Salmon. For these, we develop an explicit version of Galois descent. 1
Varieties of Mathematical Prose
 Primus
, 1997
"... This article begins the development of a taxonomy of mathematical prose, describing the precise function and meaning of specific types mathematical exposition. This article further discusses the merits and demerits of a style of mathematical writing that labels each passage according to its function ..."
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Cited by 5 (3 self)
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This article begins the development of a taxonomy of mathematical prose, describing the precise function and meaning of specific types mathematical exposition. This article further discusses the merits and demerits of a style of mathematical writing that labels each passage according to its function as described in the taxonomy. Key words Mathematical exposition, writing style, mathematical argument, formal reasoning, symbolic logic, definitions, proofs, terminology, hypertext. 1 Introduction 1.1 Rationale Many students of mathematics are not experienced in reading mathematics texts. They may not understand the nature and use of definitions. Even if they do, they may not easily distinguish between a definition and an informal discussion of a topic. They may not pick up on the use of a word such as "group" that has a meaning in ordinary discourse but that has been given a special technical meaning in their text. They may not distinguish a plausibility argument from a careful proof, an...
COMPARISON OF CONGRUENCES AND STRICT EQUIVALENCES FOR REAL, COMPLEX, AND QUATERNIONIC MATRIX PENCILS WITH SYMMETRIES ∗
"... Abstract. The equivalence relations of strict equivalence and congruence of real and complex matrix pencils with symmetries are compared, depending on whether the congruence matrices are real, complex, or quaternionic. The obtained results are applied to comparison of congruences of matrices, over t ..."
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Abstract. The equivalence relations of strict equivalence and congruence of real and complex matrix pencils with symmetries are compared, depending on whether the congruence matrices are real, complex, or quaternionic. The obtained results are applied to comparison of congruences of matrices, over the reals, the complexes, and the quaternions. Key words. Quaternionic matrices, Congruence, Strict equivalence. AMS subject classifications. 15A33. 1. Introduction. Let F be the real field R, the complex field C, or the skew field of real quaternions H. Fix an involutory antiautomorphism φ of F, inotherwords,a bijective map φ: F − → F having the properties that and φ(xy) =φ(y)φ(x) and φ(x + y) =φ(x)+φ(y) ∀ x, y ∈ F
PI DEGREE PARITY IN qSKEW POLYNOMIAL RINGS
, 2007
"... For k a field of arbitrary characteristic, and R a kalgebra, we show that the PI degree of an iterated skew polynomial ring R[x1; τ1, δ1] · · · [xn; τn, δn] agrees with the PI degree of R[x1; τ1] · · ·[xn; τn] when each (τi, δi) satisfies a qiskew relation for qi ∈ k × and extends to a highe ..."
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For k a field of arbitrary characteristic, and R a kalgebra, we show that the PI degree of an iterated skew polynomial ring R[x1; τ1, δ1] · · · [xn; τn, δn] agrees with the PI degree of R[x1; τ1] · · ·[xn; τn] when each (τi, δi) satisfies a qiskew relation for qi ∈ k × and extends to a higher qiskew τiderivation. We confirm the quantum Gel’fandKirillov conjecture for various quantized coordinate rings, and calculate their PI degrees. We extend these results to completely prime factor algebras.
The algebra of the parallel endomorphisms of a pseudoRiemannian manifold, in preparation
"... Abstract. On a (pseudo)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) may be parallel for g. They form an associative algebra e, which is also the commutant of the holonomy group of g. We study it. As any associative algebra, e is the sum of its radical and of a se ..."
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Abstract. On a (pseudo)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) may be parallel for g. They form an associative algebra e, which is also the commutant of the holonomy group of g. We study it. As any associative algebra, e is the sum of its radical and of a semisimple algebra s. We show the following: s may be of eight different types, including the generic type s = RId, and the KÃďhler and hyperkÃďhler types s ≃ C and s ≃ H. For N any self adjoint nilpotent element of the commutant of such an s in End(TM), the set of germs of metrics the holonomy group of which is included in the commutant of s∪{N} in O 0 (g) is non empty. We parametrise it. Generically, the holonomy group of those metrics is the full commutant O 0 (g) s∪{N}. Apart from some “degenerate ” cases, the algebra of the parallel endomorphisms of those metrics is s × 〈N〉. To prove it, we introduce an analogy with complex Differential Calculus, the ring R[X]/(X n) replacing the field C. This describes totally the local situation when the radical of e is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal, and give the constraints imposed to the Ricci curvature when e is non trivial.