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Hermitian and quadratic forms over local classical crossed product orders

100%

Hatzaras Y., Theohari-Apostolidi T.

Colloquium Mathematicum

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2000

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tom 83

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nr 1

43-53

EN

Let R be a complete discrete valuation ring with quotient field K, L/K be a Galois extension with Galois group G and S be the integral closure of R in L. If a is a factor set of G with values in the group of units of S, then (L/K,a) (resp. Λ =(S/R,a)) denotes the crossed product K-algebra (resp. crossed product R -order in A). In this paper hermitian and quadratic forms on Λ -lattices are studied and the existence of at most two irreducible non-singular quadratic Λ -lattices is proved (Theorem 3.5). Further the orthogonal decomposition of an arbitrary non-singular quadratic Λ -lattice is given.

2

On an asymmetric extension of multivariate Archimedean copulas based on quadratic form

80%

Di Bernardino E., Rullière D.

Dependence Modeling

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2016

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tom 4

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nr 1

EN

An important topic in Quantitative Risk Management concerns the modeling of dependence among risk sources and in this regard Archimedean copulas appear to be very useful. However, they exhibit symmetry, which is not always consistent with patterns observed in real world data. We investigate extensions of the Archimedean copula family that make it possible to deal with asymmetry. Our extension is based on the observation that when applied to the copula the inverse function of the generator of an Archimedean copula can be expressed as a linear form of generator inverses. We propose to add a distortion term to this linear part, which leads to asymmetric copulas. Parameters of this new class of copulas are grouped within a matrix, thus facilitating some usual applications as level curve determination or estimation. Some choices such as sub-model stability help associating each parameter to one bivariate projection of the copula. We also give some admissibility conditions for the considered copulas. We propose different examples as some natural multivariate extensions of Farlie-Gumbel-Morgenstern or Gumbel-Barnett.

3

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Structure fractals and para-quaternionic geometry

61%

Ławrynowicz J., Vaccaro M.

Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

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2011

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tom 65

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nr 2

EN

It is well known that starting with real structure, the Cayley-Dickson process gives complex, quaternionic, and octonionic (Cayley) structures related to the Adolf Hurwitz composition formula for dimensions \(p = 2, 4\) and \(8\), respectively, but the procedure fails for \(p = 16\) in the sense that the composition formula involves no more a triple of quadratic forms of the same dimension; the other two dimensions are \(n = 2^7\). Instead, Ławrynowicz and Suzuki (2001) have considered graded fractal bundles of the flower type related to complex and Pauli structures and, in relation to the iteration process \(p \to p + 2 \to p + 4 \to ...\), they have constructed \(2^4\)-dimensional “bipetals” for \(p = 9\) and \(2^7\)-dimensional “bisepals” for \(p = 13\). The objects constructed appear to have an interesting property of periodicity related to the gradating function on the fractal diagonal interpreted as the “pistil” and a family of pairs of segments parallel to the diagonal and equidistant from it, interpreted as the “stamens”. The first named author, M. Nowak-Kepczyk, and S. Marchiafava (2006, 2009a, b) gave an effective, explicit determination of the periods and expressed them in terms of complex and quaternionic structures, thus showing the quaternionic background of that periodicity. In contrast to earlier results, the fractal bundle flower structure, in particular petals, sepals, pistils, and stamens are not introduced ab initio; they are quoted a posteriori, when they are fully motivated. Physical concepts of dual and conjugate objects as well as of antiparticles led us to extend the periodicity theorem to structure fractals in para-quaternionic formulation, applying some results in this direction by the second named author. The paper is concluded by outlining some applications.

Ograniczanie wyników

1 Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

1 Colloquium Mathematicum

1 Dependence Modeling

1 Di Bernardino E.

1 Hatzaras Y.

1 Rullière D.

1 Theohari-Apostolidi T.

1 Vaccaro M.

1 Ławrynowicz J.

1 2016

1 2011

1 2000

Hermitian and quadratic forms over local classical crossed product orders

On an asymmetric extension of multivariate Archimedean copulas based on quadratic form

Structure fractals and para-quaternionic geometry