Mathematical Institute

Chalfont Saint Giles, United Kingdom

Mathematical Institute

Chalfont Saint Giles, United Kingdom

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Penrose R.,Mathematical Institute
Foundations of Physics | Year: 2014

The 2nd Law of thermodynamics was driven by the Big Bang being extraordinary special, with hugely suppressed gravitational degrees of freedom. This cannot have been simply the result of a conventional quantum gravity. Conformal cyclic cosmology proposes a different picture, of a classical evolution from an aeon preceding our own. The ultimate Hawking evaporation of black holes is key to the 2nd Law and requires information loss, violating unitarity in a strongly gravitational context. © 2013 The Author(s).


Penrose R.,Mathematical Institute
Foundations of Physics | Year: 2014

This paper argues that the case for "gravitizing" quantum theory is at least as strong as that for quantizing gravity. Accordingly, the principles of general relativity must influence, and actually change, the very formalism of quantum mechanics. Most particularly, an "Einsteinian", rather than a "Newtonian" treatment of the gravitational field should be adopted, in a quantum system, in order that the principle of equivalence be fully respected. This leads to an expectation that quantum superpositions of states involving a significant mass displacement should have a finite lifetime, in accordance with a proposal previously put forward by Diósi and the author. © 2014 The Author(s).


Dellar P.J.,Mathematical Institute
Computers and Mathematics with Applications | Year: 2013

The lattice Boltzmann space/time discretisation, as usually derived from integration along characteristics, is shown to correspond to a Strang splitting between decoupled streaming and collision steps. Strang splitting offers a second-order accurate approximation to evolution under the combination of two non-commuting operators, here identified with the streaming and collision terms in the discrete Boltzmann partial differential equation. Strang splitting achieves second-order accuracy through a symmetric decomposition in which one operator is applied twice for half timesteps, and the other operator is applied once for a full timestep. We show that a natural definition of a half timestep of collisions leads to the same change of variables that was previously introduced using different reasoning to obtain a second-order accurate and explicit scheme from an integration of the discrete Boltzmann equation along characteristics. This approach extends easily to include general matrix collision operators, and also body forces. Finally, we show that the validity of the lattice Boltzmann discretisation for grid-scale Reynolds numbers larger than unity depends crucially on the use of a Crank-Nicolson approximation to discretise the collision operator. Replacing this approximation with the readily available exact solution for collisions uncoupled from streaming leads to a scheme that becomes much too diffusive, due to the splitting error, unless the grid-scale Reynolds number remains well below unity. © 2013 Published by Elsevier Ltd.


Olver S.,Mathematical Institute
BIT Numerical Mathematics | Year: 2010

We present a numerically stable way to compute oscillatory integrals. For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the amount of work needed decreases as the frequency increases. Moreover, we can modify the method for computing oscillatory integrals with stationary points. This is the first stable algorithm for oscillatory integrals with stationary points which does not lose accuracy as the frequency increases and does not require deformation into the complex plane. © Springer Science + Business Media B.V. 2010.


Tod P.,Mathematical Institute
General Relativity and Gravitation | Year: 2012

We present a calculation of the angular size of the circles in the CMB predicted by Penrose on the basis of his conformal cyclic cosmology. If these circles are detected, the existence of an upper limit on their angular radius would provide a challenge for inflation. © 2012 Springer Science+Business Media, LLC.


Dellar P.J.,Mathematical Institute
Journal of Computational Physics | Year: 2014

The vast majority of lattice Boltzmann algorithms produce a non-Galilean invariant viscous stress. This defect arises from the absence of a term in the third moment, the equilibrium heat flow tensor, proportional to the cube of the fluid velocity. This moment cannot be specified independently of the lower moments on the standard lattices such as D2Q9, D3Q15, D3Q19 or D3Q27. A partial correction has recently been demonstrated that restores some of these missing cubic terms on the D2Q9 and D3Q27 tensor product lattices. This correction restores Galilean invariance for shear flows aligned with the coordinate axes, but flows inclined at arbitrary angles may show larger errors than before. These remaining errors are due to the diagonal terms of the equilibrium heat flow tensor, which cannot be corrected on standard lattices. However, the remaining errors may be largely absorbed by introducing a matrix collision operator with velocity-dependent collision rates for the diagonal components of the momentum flux tensor. This completely restores Galilean invariance for flows with uniform density, and in general reduces the magnitude of the defect in Galilean invariance from Mach number cubed to Mach number to the fifth power. The effectiveness of the resulting algorithm is demonstrated by comparisons with the standard and partially corrected lattice Boltzmann algorithms for two- and three-dimensional flows. © 2013.


Vincent J.-P.,UK National Institute for Medical Research | Fletcher A.G.,Mathematical Institute | Baena-Lopez L.A.,UK National Institute for Medical Research
Nature Reviews Molecular Cell Biology | Year: 2013

When fast-growing cells are confronted with slow-growing cells in a mosaic tissue, the slow-growing cells are often progressively eliminated by apoptosis through a process known as cell competition. The underlying signalling pathways remain unknown, but recent findings have shown that cell crowding within an epithelium leads to the eviction of cells from the epithelial sheet. This suggests that mechanical forces could contribute to cell elimination during cell competition. © 2013 Macmillan Publishers Limited. All rights reserved.


Hodges A.,Mathematical Institute
Journal of High Energy Physics | Year: 2013

This note addresses the problem of spurious poles in gauge-theoretic scattering amplitudes. New twistor coordinates for the momenta are introduced, based on the concept of dual conformal invariance. The cancellation of spurious poles for a class of NMHV amplitudes is greatly simplified in these coordinates. The poles are eliminated altogether by defining a new type of twistor integral, dual to twistor diagrams as previously studied, and considerably simpler. The geometric features indicate a supersymmetric extension of the formalism at least to all NMHV amplitudes, allowing the dihedral symmetry of the super-amplitude to be made manifest. More generally, the definition of 'momentum-twistor' coordinates suggests a powerful new approach to the study of scattering amplitudes. © 2013 SISSA, Trieste, Italy.


Hitchin N.,Mathematical Institute
Communications in Mathematical Physics | Year: 2013

A hyperkähler manifold with a circle action fixing just one complex structure admits a natural hyperholomorphic line bundle. This observation forms the basis for the construction of a corresponding quaternionic Kähler manifold in the work of A.Haydys. In this paper the corresponding holomorphic line bundle on twistor space is described and many examples computed, including monopole and Higgs bundle moduli spaces. Finally a twistor version of the hyperkähler/quaternion Kähler correspondence is established. © 2013 Springer-Verlag Berlin Heidelberg.


Hodges A.,Mathematical Institute
Journal of High Energy Physics | Year: 2013

New methods are introduced for evaluating tree-level gravitational scattering amplitudes. A new N=7 super-symmetric recursion yields amplitudes free from the spurious double poles of the N=8 theory. This is illustrated by a new nine-term expression for the six-graviton NMHV amplitude. The general scheme also implies a simplified recurrence relation for MHV amplitudes. We show how this relation is satisfied by a new expression for MHV amplitudes, far simpler than those hitherto identified, and exhibiting manifest Sn symmetry. This reformulation is related to a new momentum-twistor representation of the MHV amplitudes. © 2013 SISSA, Trieste, Italy.

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