1,675 research outputs found

    An Algorithm for constructing Hjelmslev planes

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    Projective Hjelmslev planes and Affine Hjelmselv planes are generalisations of projective planes and affine planes. We present an algorithm for constructing a projective Hjelmslev planes and affine Hjelsmelv planes using projective planes, affine planes and orthogonal arrays. We show that all 2-uniform projective Hjelmslev planes, and all 2-uniform affine Hjelsmelv planes can be constructed in this way. As a corollary it is shown that all 2-uniform Affine Hjelmselv planes are sub-geometries of 2-uniform projective Hjelmselv planes.Comment: 15 pages. Algebraic Design Theory and Hadamard matrices, 2014, Springer Proceedings in Mathematics & Statistics 13

    Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations

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    We prove the following special case of Mazur's conjecture on the topology of rational points. Let EE be an elliptic curve over Q\mathbb{Q} with jj-invariant 17281728. For a class of elliptic pencils which are quadratic twists of EE by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic polynomials. For the proof, we investigate a highly singular rational curve on the Kummer surface KK associated to a product of two elliptic curves over Q\mathbb{Q}, which previously appeared in publications by Mestre, Kuwata-Wang and Satg\'e. We produce this curve in a simpler manner by finding algebraic equations which give a direct proof of rationality. We find that the same equations give rise to rational curves on a class of more general surfaces extending the Kummer construction. This leads to further applications apart from Mazur's conjecture, for example the existence of rational points on simultaneous twists of superelliptic curves. Finally, we give a proof of Mazur's conjecture for the Kummer surface KK without any restrictions on the jj-invariants of the two elliptic curves.Comment: 14 pages, same content as published version except for added remark acknowledging overlap with prior work by Ula

    Solitons and admissible families of rational curves in twistor spaces

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    It is well known that twistor constructions can be used to analyse and to obtain solutions to a wide class of integrable systems. In this article we express the standard twistor constructions in terms of the concept of an admissible family of rational curves in certain twistor spaces. Examples of of such families can be obtained as subfamilies of a simple family of rational curves using standard operations of algebraic geometry. By examination of several examples, we give evidence that this construction is the basis of the construction of many of the most important solitonic and algebraic solutions to various integrable differential equations of mathematical physics. This is presented as evidence for a principal that, in some sense, all soliton-like solutions should be constructable in this way.Comment: 15 pages, Abstract and introduction rewritten to clarify the objectives of the paper. This is the final version which will appear in Nonlinearit

    String Dualities and Toric Geometry: An Introduction

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    This note is supposed to be an introduction to those concepts of toric geometry that are necessary to understand applications in the context of string and F-theory dualities. The presentation is based on the definition of a toric variety in terms of homogeneous coordinates, stressing the analogy with weighted projective spaces. We try to give both intuitive pictures and precise rules that should enable the reader to work with the concepts presented here.Comment: 17 pages, Latex, 7 figures, invited paper to appear in the special issue of the Journal of Chaos, Solitons and Fractals on "Superstrings, M, F, S, ... Theory" (M.S. El Naschie and C. Castro, editors
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