DSpace Community:https://dspace.lboro.ac.uk/2134/892017-01-13T10:28:40Z2017-01-13T10:28:40ZComplete commutative subalgebras in polynomial poisson algebras: a proof of the Mischenko-Fomenko conjectureBolsinov, Alexey V.https://dspace.lboro.ac.uk/2134/236532017-01-10T11:40:32Z2016-01-01T00:00:00ZTitle: Complete commutative subalgebras in polynomial poisson algebras: a proof of the Mischenko-Fomenko conjecture
Authors: Bolsinov, Alexey V.
Abstract: The Mishchenko–Fomenko conjecture says that for each real or
complex finite-dimensional Lie algebra g there exists a complete set of commuting
polynomials on its dual space g*. In terms of the theory of integrable
Hamiltonian systems this means that the dual space g* endowed with the standard
Lie–Poisson bracket admits polynomial integrable Hamiltonian systems.
This conjecture was proved by S. T. Sadetov in 2003. Following his idea, we
give an explicit geometric construction for commuting polynomials on g* and
consider some examples.
Description: This paper is a revised version of: BOLSINOV, A.V., 2006. Complete commutative families of polynomials in Poisson-Lie algebras: a proof of the Mischenko-Fomenko conjecture. Trudy seminara po vektornomu i tenzornomu analizu (ISSN: 0373-4870), 26, pp.87-109.2016-01-01T00:00:00ZA special Cayley octadPrendergast-Smith, Artiehttps://dspace.lboro.ac.uk/2134/236402017-01-09T14:12:11Z2017-01-01T00:00:00ZTitle: A special Cayley octad
Authors: Prendergast-Smith, Artie
Abstract: A Cayley octad is a set of 8 points in P3 which are the base locus of a net of quadrics. Blowing up the points of the octad gives a morphism to P2 defined by the net; the fibres of this morphism are intersections of two quadrics in the net, hence curves of genus 1. The generic fibre therefore has a group structure, and the action of this group on itself extends to a birational action on the whole variety. In particular, if the generic fibre has a
large group of rational points, the birational automorphism group, and hence the birational geometry, of the variety must be complicated. It is natural to ask whether the converse is true: if the generic fibre has only a small group of rational points, is the birational geometry of the variety correspondingly simple? In this paper we study a special Cayley octad with the property that the generic fibre has finitely many rational points. In Section 1 we find that such an octad only exists in characteristic 2, and is unique up to projective transformations. Our main results then show that the simplicity of the generic fibre is indeed reected in the simplicity of the birational geometry of our blowup. In Section 2 we show that the cones of nef and movable divisors are rational polyhedral, as predicted by the Morrison{Kawamata conjecture. Finally, in Section 3 we prove that our blowup has the \best possible" birational geometric properties: it is a Mori dream space.
Description: This paper is in closed access until it is published.2017-01-01T00:00:00ZGeometric aspects of robust testing for normality and sphericityRichter, Wolf-DieterStrelec, LubosAhmadinezhad, HamidStehlik, Milanhttps://dspace.lboro.ac.uk/2134/236342017-01-09T11:03:50Z2017-01-01T00:00:00ZTitle: Geometric aspects of robust testing for normality and sphericity
Authors: Richter, Wolf-Dieter; Strelec, Lubos; Ahmadinezhad, Hamid; Stehlik, Milan
Abstract: Stochastic Robustness of Control Systems under random excitation motivates challenging
developments in geometric approach to robustness. The assumption of normality
is rarely met when analyzing real data and thus the use of classic parametric
methods with violated assumptions can result in the inaccurate computation of pvalues,
e↵ect sizes, and confidence intervals. Therefore, quite naturally, research on
robust testing for normality has become a new trend. Robust testing for normality
can have counter-intuitive behavior, some of the problems have been introduced in
[46]. Here we concentrate on explanation of small-sample e↵ects of normality testing
and its robust properties, and embedding these questions into the more general question
of testing for sphericity. We give geometric explanations for the critical tests. It
turns out that the tests are robust against changes of the density generating function
within the class of all continuous spherical sample distributions.
Description: This paper is in closed access until 12 months after publication.2017-01-01T00:00:00ZTwo-component generalizations of the Camassa-Holm equationHone, Andrew N.W.Novikov, V.S.Wang, Jing Pinghttps://dspace.lboro.ac.uk/2134/236332017-01-09T10:40:17Z2017-01-01T00:00:00ZTitle: Two-component generalizations of the Camassa-Holm equation
Authors: Hone, Andrew N.W.; Novikov, V.S.; Wang, Jing Ping
Abstract: A classification of integrable two-component systems of non-evolutionary partial dif-
ferential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators of specific forms is carried out, in order to obtain bi-Hamiltonian structures for the same systems of equations. Using reciprocal transformations, some exact solutions and Lax pairs are also constructed for the systems considered.
Description: This paper is in closed access until 12 months after publication.2017-01-01T00:00:00Z