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Jablan S.,The Mathematical Institute | Radovic L.,University of Niš | Sazdanovic R.,University of Pennsylvania
Journal of Mathematical Chemistry | Year: 2011

Virtual knot theory offers the possibility to consider knots and links embedded on different surfaces. This paper analyzes nonplanarity of graphs obtained from Gauss codes of virtual knots and links and their potential applications in chemistry. © 2011 Springer Science+Business Media, LLC.

Mason L.J.,The Mathematical Institute | Reid-Edwards R.A.,The Mathematical Institute | Taghavi-Chabert A.,Masaryk University
Journal of Geometry and Physics | Year: 2012

This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space-time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the six-dimensional case in which twistor space is the 6-quadric Q in CP7 with a view to applications to the self-dual (0, 2)-theory. We show how spinor-helicity momentum eigenstates have canonically defined distributional representatives on twistor space (a story that we extend to arbitrary dimension). These yield an elementary proof of the surjectivity of the Penrose transform. We give a direct construction of the twistor transform between the two different representations of massless fields on twistor space (H2 and H3) in which the H3s arise as obstructions to extending the H2s off Q into CP7.We also develop the theory of Sparling's 'Ξ-transform', the analogous totally real split signature story based now on real integral geometry where cohomology no longer plays a role. We extend Sparling's Ξ-transform to all helicities and homogeneities on twistor space and show that it maps kernels and cokernels of conformally invariant powers of the ultrahyperbolic wave operator on twistor space to conformally invariant massless fields on space-time. This is proved by developing the six-dimensional analogue of the half-Fourier transform between functions on twistor space and momentum space. We give a treatment of the elementary conformally invariant Φ3 amplitude on twistor space and finish with a discussion of conformal field theories in twistor space. © 2012 Elsevier B.V.

Mason L.,The Mathematical Institute | Skinner D.,The Mathematical Institute
Communications in Mathematical Physics | Year: 2010

We give a self-contained proof of the formula for the MHV amplitudes for gravity conjectured by Berends, Giele & Kuijf and use the associated twistor generating function to define a twistor action for the MHV diagram approach to gravity. Starting from a background field calculation on a spacetime with anti-self-dual curvature, we obtain a simple spacetime formula for the scattering of a single, positive helicity linearized graviton into one of negative helicity. Re-expressing our integral in terms of twistor data allows us to consider a spacetime that is asymptotic to a superposition of plane waves. Expanding these out perturbatively yields the gravitational MHV amplitudes of Berends, Giele & Kuijf. We go on to take the twistor generating function off-shell at the perturbative level. Combining this with a twistor action for the anti-self-dual background, the generating function provides the MHV vertices for the MHV diagram approach to perturbative gravity. We finish by extending these results to supergravity, in particular N = 4 and N = 8. © Springer-Verlag 2009.

Mason L.,The Mathematical Institute | Taghavi-Chabert A.,The Mathematical Institute
Journal of Geometry and Physics | Year: 2010

We show that the Euclidean Kerr-NUT-(A)dS metric in 2m dimensions locally admits 2m Hermitian complex structures. These are derived from the existence of a non-degenerate closed conformal Killing-Yano tensor with distinct eigenvalues. More generally, a conformal Killing-Yano tensor, provided its exterior derivative satisfies a certain condition, algebraically determines 2m almost complex structures that turn out to be integrable as a consequence of the conformal Killing-Yano equations. In the complexification, these lead to 2m maximal isotropic foliations of the manifold and, in Lorentz signature, these lead to two congruences of null geodesics. These are not shear-free, but satisfy a weaker condition that also generalises the shear-free condition from four dimensions to higher dimensions. In odd dimensions, a conformal Killing-Yano tensor leads to similar integrable distributions in the complexification. We show that the recently discovered five-dimensional solution of Lü, Mei and Pope also admits such integrable distributions, although this does not quite fit into the story as the obvious associated two-form is not conformal Killing-Yano. We give conditions on the Weyl curvature tensor imposed by the existence of a non-degenerate conformal Killing-Yano tensor; these give an appropriate generalisation of the type D condition on a Weyl tensor from four dimensions. © 2010 Elsevier B.V.

Singh K.,the Mathematical Institute | Michelin S.,Ecole Polytechnique - Palaiseau | De Langre E.,Ecole Polytechnique - Palaiseau
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2012

The problem of energy harvesting from flutter instabilities in flexible slender structures in axial flows is considered. In a recent study, we used a reduced-order theoretical model of such a system to demonstrate the feasibility for harvesting energy from these structures. Following this preliminary study, we now consider a continuous fluid-structure system. Energy harvesting is modelled as strain-based damping, and the slender structure under investigation lies in a moderate fluid loading range, for which the flexible structure may be destabilized by damping. The key goal of this work is to analyse the effect of damping distribution and intensity on the amount of energy harvested by the system. The numerical results indeed suggest that non-uniform damping distributions may significantly improve the power-harvesting capacity of the system. For low-damping levels, clustered dampers at the position of peak curvature are shown to be optimal. Conversely for higher damping, harvesters distributed over the whole structure are more effective. © 2012 The Royal Society.

Mason L.J.,The Mathematical Institute | Nicolas J.-P.,University of Western Brittany
Journal of Geometry and Physics | Year: 2012

We study the peeling of Dirac and Maxwell fields on a Schwarzschild background following the approach developed by the authors in Mason and Nicolas (2009). [12] for the wave equation. The method combines a conformal compactification with vector field techniques in order to work out the optimal space of initial data for a given transverse regularity of the rescaled field across null infinity. The results show that analogous decay and regularity assumptions in Minkowski and Schwarzschild produce the same regularity across null infinity. The results are valid also for the classes of asymptotically simple spacetimes constructed by Corvino-Schoen/Chruściel-Delay. © 2012 .

Jablan S.,The Mathematical Institute | Radovic L.,University of Niš | Sazdanovic R.,The Mathematical Institute
Journal of Mathematical Chemistry | Year: 2011

In order to facilitate recognition of polymer graphs and patterns obtained by graphical recombination, we analyze polynomial invariants, graphs and knots associated to polyominoes, polyiamonds, and polyhexes. © 2010 Springer Science+Business Media, LLC.

Mason L.,The Mathematical Institute | Skinner D.,Perimeter Institute for Theoretical Physics
Journal of High Energy Physics | Year: 2010

We show that the complete planar S-matrix of N = 4 super Yang-Mills - including all N kMHV partial amplitudes to all loops - is equivalent to the correlation function of a supersymmetric Wilson loop in twistor space. Remarkably, the entire classical S-matrix arises from evaluating the correlation function in the self-dual sector, while the expansion of the correlation function in powers of the Yang-Mills coupling constant provides the loop expansion of the amplitudes. We support our proposal with explicit computations of the n particle NMHV and N 2MHV trees, the integrands of the 1-loop MHV and NMHV amplitudes, and the n particle 2-loop MHV amplitude. These calculations are performed using the twistor action in axial gauge. In this gauge, the Feynman diagrams of the correlation function are the planar duals of the usual MHV diagrams for the scattering amplitude. The results are presented in the form of a sum of products of dual superconformal invariants in (momentum) twistor space, and agree with the expressions derived in the companion paper [1] directly from the MHV rules. The twistor space Wilson loop is a natural supersymmetric generalization of the standard Wilson loop used to compute MHV amplitudes. We show how the Penrose-Ward transform can be used to determine a corresponding supersymmetrization on space-time and give the corresponding superconnection in the abelian case. © 2010 SISSA.

Jablan S.,The Mathematical Institute | Radovic L.,University of Niš
Kybernetes | Year: 2011

Purpose: The purpose of this paper is to consider the history of certain modular elements: Truchet tiles, Op-tiles, Kufic tiles, and key-patterns, which occur as ornamental archetypes from Paleolithic times until the present. The appearance of the same ornamental archetypes at the same level of the development in different cultures, distant in space and time can be described from the cybernetics point of view as a specific kind of self-referential systems or cellular automata present in the intellectual and cultural development of mankind. The aim of this research is to show a continuity of the development of ornamental structures based on modular elements used as ornamental archetypes. Design/methodology/approach: Research of the material from archaeological findings, history of art, painting, architecture, and applied arts. Findings: Existence of universal geometrical construction principles based on modularity. Practical implications: Creation of new patterns or designs (e.g. TeX-fonts, tiles, etc.) based on modularity. Originality/value: The paper presents a new explanation of constructions of labyrinths and different Islamic patterns. © Emerald Group Publishing Limited.

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