arXiv subjects

Alan D. Rendall

Publications and source records attributed to Alan D. Rendall.

At least 37 records · Page 2Linked to original sources

arXiv2014

A proof of bistability for the dual futile cycle

The multiple futile cycle is an important building block in networks of chemical reactions arising in molecular biology. A typical process which it describes is the addition of n phosphate groups to a protein. It can be modelled by a system of ordinary differential equations depending on parameters. The special case n = 2 is called the dual futile cycle. The main result of this paper is a proof that there are parameter values for which the system of ODE describing the dual futile cycle has two distinct stable stationary solutions. The proof is based on bifurcation theory and geometric singular perturbation theory. An important entity built of three coupled multiple futile cycles is the MAPK cascade. It is explained how the ideas used to prove bistability for the dual futile cycle might help to prove the existence of periodic solutions for the MAPK cascade.

q-bio.MN↗

arXiv2013

Dynamical properties of models for the Calvin cycle

Modelling the Calvin cycle of photosynthesis leads to various systems of ordinary differential equations and reaction-diffusion equations. They differ by the choice of chemical substances included in the model, the choices of stoichiometric coefficients and chemical kinetics and whether or not diffusion is taken into account. This paper studies the long-time behaviour of solutions of several of these systems, concentrating on the ODE case. In some examples it is shown that there exist two positive stationary solutions. In several cases it is shown that there exist solutions where the concentrations of all substrates tend to zero at late times and others (runaway solutions) where the concentrations of all substrates increase without limit. In another case, where the concentration of ATP is explicitly included, runaway solutions are ruled out.

q-bio.MN↗

arXiv2013

The spatially homogeneous relativistic Boltzmann equation with a hard potential

In this paper, we study spatially homogeneous solutions of the Boltzmann equation in special relativity and in Robertson-Walker spacetimes. We obtain an analogue of the Povzner inequality in the relativistic case and use it to prove global existence theorems. We show that global solutions exist for a certain class of collision cross sections of the hard potential type in Minkowski space and in spatially flat Robertson-Walker spacetimes.

gr-qc↗

arXiv2012

Construction of oscillatory singularities

One way to understand more about spacetime singularities is to construct solutions of the Einstein equations containing singularities with prescribed properties. The heuristic ideas of the BKL picture suggest that oscillatory singularities should be very common and give a detailed picture of how these could look. The more straightforward case of singularities without oscillations is reviewed and existing results on that subject are surveyed. Then recent theorems proving the existence of spatially homogeneous solutions with oscillatory singularities of a specific type are presented. The proofs of these involve applications of some ideas concerning heteroclinic chains and their stability. Some necessary background from the theory of dynamical systems is explained. Finally some directions in which this research might be generalized in the future are pointed out.

gr-qc↗

arXiv2012

Oscillatory singularities in Bianchi models with magnetic fields

An idea which has been around in general relativity for more than forty years is that in the approach to a big bang singularity solutions of the Einstein equations can be approximated by the Kasner map, which describes a succession of Kasner epochs. This is already a highly non-trivial statement in the spatially homogeneous case. There the Einstein equations reduce to ordinary differential equations and it becomes a statement that the solutions of the Einstein equations can be approximated by heteroclinic chains of the corresponding dynamical system. For a long time progress on proving a statement of this kind rigorously was very slow but recently there has been new progress in this area, particularly in the case of the vacuum Einstein equations. In this paper we generalize some of these results to the Einstein-Maxwell equations. It turns out that this requires new techniques since certain eigenvalues are in a less favourable configuration in the case with a magnetic field. The difficulties which arise in that case are overcome by using the fact that the dynamical system of interest is of geometrical origin and thus has useful invariant manifolds.

gr-qc↗

arXiv2012

The Einstein-Boltzmann system and positivity

The Einstein-Boltzmann system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and by Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the Einstein-Boltzmann system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the Einstein-Boltzmann system so as to include scattering kernels which are physically well-motivated.

gr-qc↗

arXiv2012

Mathematics of the NFAT signalling pathway

This paper is a mathematical study of some aspects of the signalling pathway leading to the activation of the transcription factor NFAT (nuclear factor of activated T cells). Activation takes place by dephosphorylation at multiple sites. This has been modelled by Salazar and Höfer using a large system of ordinary differential equations depending on many parameters. With the help of chemical reaction network theory we show that for any choice of the parameters this system has a unique stationary solution for each value of the conserved quantity given by the total amount of NFAT and that all solutions converge to this stationary solution at late times. The dephosphorylation is carried out by calcineurin, which in turn is activated by a rise in calcium concentration. We study the way in which the dynamics of the calcium concentration influences NFAT activation, an issue also considered by Salazar and Höfer with the help of a model arising from work of Somogyi and Stucki. Criteria are obtained for convergence to equilibrium of solutions of the model for the calcium concentration.

q-bio.MN↗

arXiv2011

Multiple steady states in a mathematical model for interactions between T cells and macrophages

The aim of this paper is to prove results about the existence and stability of multiple steady states in a system of ordinary differential equations introduced by R. Lev Bar-Or to model the interactions between T cells and macrophages. Previous results showed that for certain values of the parameters these equations have three stationary solutions, two of which are stable. Here it is shown that there are values of the parameters for which the number of stationary solutions is at least seven and the number of stable stationary solutions at least four. This requires approaches different to those used in existing work on this subject. In addition, a rather explicit characterization is obtained of regions of parameter space for which the system has a given number of stationary solutions.

q-bio.CB↗

arXiv2011

A global foliation of Einstein-Euler spacetimes with Gowdy-symmetry on T3

We investigate the initial value problem for the Einstein-Euler equations of general relativity under the assumption of Gowdy symmetry on T3, and we construct matter spacetimes with low regularity. These spacetimes admit, both, impulsive gravitational waves in the metric (for instance, Dirac mass curvature singularities propagating at light speed) and shock waves in the fluid (i.e., discontinuities propagating at about the sound speed). Given an initial data set, we establish the existence of a future development and we provide a global foliation in terms of a globally and geometrically defined time-function, closely related to the area of the orbits of the symmetry group. The main difficulty lies in the low regularity assumed on the initial data set which requires a distributional formulation of the Einstein-Euler equations.

gr-qc↗

arXiv2010

Dynamics of solutions of the Einstein equations with twisted Gowdy symmetry

Some of the most interesting results on the global dynamics of solutions of the vacuum Einstein equations concern the Gowdy spacetimes whose spatial topology is that of a three-dimensional torus. In this paper certain of these ideas are extended to a wider class of vacuum spacetimes where the spatial topology is that of a non-trivial torus bundle over a circle. Compared to the case of the torus these are topologically twisted. They include inhomogeneous generalizations of the spatially homogeneous vacuum spacetimes of Bianchi types II and VI${}_0$. Using similar procedures it is shown that the vacuum solutions of Bianchi type VII${}_0$ are isometric to a class of Gowdy spacetimes, the circular loop spacetimes, thus establishing links between results in the literature which were not previously known to be related to each other.

gr-qc↗

arXiv2010

A class of dust-like self-similar solutions of the massless Einstein-Vlasov system

In this paper the existence of a class of self-similar solutions of the Einstein-Vlasov system is proved. The initial data for these solutions are not smooth, with their particle density being supported in a submanifold of codimension one. They can be thought of as intermediate between smooth solutions of the Einstein-Vlasov system and dust. The motivation for studying them is to obtain insights into possible violation of weak cosmic censorship by solutions of the Einstein-Vlasov system. By assuming a suitable form of the unknowns it is shown that the existence question can be reduced to that of the existence of a certain type of solution of a four-dimensional system of ordinary differential equations depending on two parameters. This solution starts at a particular point $P_0$ and converges to a stationary solution $P_1$ as the independent variable tends to infinity. The existence proof is based on a shooting argument and involves relating the dynamics of solutions of the four-dimensional system to that of solutions of certain two- and three-dimensional systems obtained from it by limiting processes.

gr-qc↗

arXiv2010

Analysis of a mathematical model for interactions between T cells and macrophages

The aim of this paper is to carry out a mathematical analysis of a system of ordinary differential equations introduced by R. Lev Bar-Or to model the interactions between T cells and macrophages. Under certain restrictions on the parameters of the model, theorems are proved about the number of stationary solutions and their stability. In some cases the existence of periodic solutions or heteroclinic cycles is ruled out. Evidence is presented that the same biological phenomena could be equally well described by a simpler model.

q-bio.CB↗

arXiv2010

Future geodesic completeness of some spatially homogeneous solutions of the vacuum Einstein equations in higher dimensions

It is known that all spatially homogeneous solutions of the vacuum Einstein equations in four dimensions which exist for an infinite proper time towards the future are future geodesically complete. This paper investigates whether the analogous statement holds in higher dimensions. A positive answer to this question is obtained for a large class of models which can be studied with the help of Kaluza-Klein reduction to solutions of the Einstein-scalar field equations in four dimensions. The proof of this result makes use of a criterion for geodesic completeness which is applicable to more general spatially homogeneous models.

gr-qc↗

arXiv2009

Asymptotics of linearized cosmological perturbations

In cosmology an important role is played by homogeneous and isotropic solutions of the Einstein-Euler equations and linearized perturbations of these. This paper proves results on the asymptotic behaviour of scalar perturbations both in the approach to the initial singularity of the background model and at late times. The main equation of interest is a linear hyperbolic equation whose coefficients depend only on time. Expansions for the solutions are obtained in both asymptotic regimes. In both cases it is shown how general solutions with a linear equation of state can be parametrized by certain functions which are coefficients in the asymptotic expansion. For some nonlinear equations of state it is found that the late-time asymptotic behaviour is qualitatively different from that in the linear case.

math.AP↗

arXiv2009

Strong cosmic censorship for solutions of the Einstein-Maxwell field equations with polarized Gowdy symmetry

A proof of strong cosmic censorship is presented for a class of solutions of the Einstein-Maxwell equations, those with polarized Gowdy symmetry. A key element of the argument is the observation that by means of a suitable choice of variables the central equations in this problem can be written in a form where they are identical to the central equations for general (i.e. non-polarized) vacuum Gowdy spacetimes. Using this it is seen that the deep results of Ringström on strong cosmic censorship in the vacuum case have implications for the Einstein-Maxwell case. Working out the geometrical meaning of these analytical results leads to the main conclusion.

gr-qc↗

arXiv2008

Shock Waves in Plane Symmetric Spacetimes

We consider Einstein's equations coupled to the Euler equations in plane symmetry, with compact spatial slices and constant mean curvature time. We show that for a wide variety of equations of state and a large class of initial data, classical solutions break down in finite time. The key mathematical result is a new theorem on the breakdown of solutions of systems of balance laws. We also show that an extension of the solution is possible if the spatial derivatives of the energy density and the velocity are bounded, indicating that the breakdown is really due to the formation of shock waves.

gr-qc↗

arXiv2008

Isotropization in the approach to big rip singularities for Cardassian models

Cardassian models are an alternative to general relativity which have been proposed as an approach to explaining accelerated cosmic expansion while avoiding directly introducing dark energy. They are generally formulated only in the homogeneous and isotropic case. In this paper an extension of the usual formulation to general spatially homogeneous geometries is given. A characteristic feature of many classes of Cardassian models is the occurrence of big rip singularities where the scale factor tends to infinity after a finite time. It is shown that big rip singularities are also widespread in more general homogeneous cases. It is also shown that there is isotropization in the approach to a big rip singularity which bears a strong resemblance to the late-time isotropization observed in cosmological spacetimes which accelerate forever in the future.

gr-qc↗

arXiv2007

Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter

This paper addresses strong cosmic censorship for spacetimes with self-gravitating collisionless matter, evolving from surface-symmetric compact initial data. The global dynamics exhibit qualitatively different features according to the sign of the curvature $k$ of the symmetric surfaces and the cosmological constant $Λ$. With a suitable formulation, the question of strong cosmic censorship is settled in the affirmative if $Λ=0$ or $k\le0$, $Λ>0$. In the case $Λ>0$, $k=1$, we give a detailed geometric characterization of possible "boundary" components of spacetime; the remaining obstruction to showing strong cosmic censorship in this case has to do with the possible formation of extremal Schwarzschild-de Sitter-type black holes. In the special case that the initial symmetric surfaces are all expanding, strong cosmic censorship is shown in the past for all $k,Λ$. Finally, our results also lead to a geometric characterization of the future boundary of black hole interiors for the collapse of asymptotically flat data: in particular, in the case of small perturbations of Schwarzschild data, it is shown that these solutions do not exhibit Cauchy horizons emanating from $i^+$ with strictly positive limiting area radius.

gr-qc↗