Note on cubature formulae and designs obtained from group orbits

In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t. In this paper, we find some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and moreover gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007) which classifies tight Euclidean designs invariant under the Weyl group of type B to other finite reflection groups.Comment: 18 pages, no figur

On finite T-groups

This paper has been published in Journal of the Australian Mathematical Society 75(2):181-192 (2003). The final publication is available at Cambridge University Press Journals, http://journals.cambridge.org/abstract_S1446788700003712 http://dx.doi.org/10.1017/S1446788700003712[EN] Characterisations of finite groups in which normality is a transitive relation are presented in the paper. We also characterise the finite groups in which every subgroup is either permutable or coincides with its permutiser as the groups in which every subgroup is permutable.Supported by Proyecto PB97-0674-C02-02 and Proyecto PB97-O6O4 from DGICYT, Ministerio de Education y Cienciahttp://journals.cambridge.org/abstract_S1446788700003712Ballester Bolinches, A.; Esteban Romero, R. (2003). On finite T-groups. Journal of the Australian Mathematical Society. 2(75). doi:10.1017/S144678870000371227

Constructing Cubature Formulas of Degree 5 with Few Points

This paper will devote to construct a family of fifth degree cubature formulae for $n$-cube with symmetric measure and $n$-dimensional spherically symmetrical region. The formula for $n$-cube contains at most $n^2+5n+3$ points and for $n$-dimensional spherically symmetrical region contains only $n^2+3n+3$ points. Moreover, the numbers can be reduced to $n^2+3n+1$ and $n^2+n+1$ if $n=7$ respectively, the later of which is minimal.Comment: 13 page

Mastering Existential Concepts in Fiction

Экзистенциализм принято считать философским направлением. Однако в художественной литературе можно обнаружить достаточное количество произведений, где представлены экзистенциальные идеи. Автор находит примеры экзистенциальных концептов в художественной литературе, показывая ее состоятельность в данном вопросе.Existentialism is considered to be a philosophical trend. However, in fiction one can find a sufficient number of works where existential ideas are presented. The author finds examples of existential concepts in fiction, showing its consistency in this matter

Reconstruction from Radon projections and orthogonal expansion on a ball

The relation between Radon transform and orthogonal expansions of a function on the unit ball in \RR^d is exploited. A compact formula for the partial sums of the expansion is given in terms of the Radon transform, which leads to algorithms for image reconstruction from Radon data. The relation between orthogonal expansion and the singular value decomposition of the Radon transform is also exploited.Comment: 15 page

The kinetic energy and and the geometric structure in the B=2B=2 sector of the Skyrme model: A study using the Atiyah-Manton Ansatz

We study the construction of the collective-coordinate manifold in the baryon number two sector of the Skyrme model. To that end we use techniques of adiabatic large amplitude collective motion, which treat potential and kinetic energy on an equal footing. In this paper the starting point is the Ansatz proposed by Atiyah and Manton (Phys.~Lett.~{\bf 438B}, 222 (1989)), which allows a study of the dynamics using a finite and small number of variables. From these variables we choose a subset of collective ones. We then study the behavior of inertial parameters along parts of the collective manifold, and study the dynamical parts of the interaction.Comment: FAU-T3-94/1, 42 pages. 21 postscript figures can be included in the text using epsf.sty. Postscript file of complete manuscript avalailabe as ftp://theorie3.physik.uni-erlangen.de/pub/publications/NRWAMSk.ps.g

On the construction of general cubature formula by flat extensions

International audienceWe describe a new method to compute general cubature formulae. The problem is initially transformed into the computation of truncated Hankel operators with flat extensions. We then analyse the algebraic properties associated to flat extensions and show how to recover the cubature points and weights from the truncated Hankel operator. We next present an algorithm to test the flat extension property and to additionally compute the decomposition. To generate cubature formulae with a minimal number of points, we propose a new relaxation hierarchy of convex optimization problems minimizing the nuclear norm of the Hankel operators. For a suitably high order of convex relaxation, the minimizer of the optimization problem corresponds to a cubature formula. Furthermore cubature formulae with a minimal number of points are associated to faces of the convex sets. We illustrate our method on some examples, and for each we obtain a new minimal cubature formula