Waltzing peakons and compacton pairs in a cross-coupled Camassa-Holm equation

We consider singular solutions of a system of two cross-coupled Camassa-Holm (CCCH) equations. This CCCH system admits peakon solutions, but it is not in the two-component CH integrable hierarchy. The system is a pair of coupled Hamiltonian partial differential equations for two types of solutions on the real line, each of which separately possesses exp(-|x|) peakon solutions with a discontinuity in the first derivative at the peak. However, there are no self-interactions, so each of the two types of peakon solutions moves only under the induced velocity of the other type. We analyse the `waltzing' solution behaviour of the cases with a single bound peakon pair (a peakon couple), as well as the over-taking collisions of peakon couples and the antisymmetric case of the head-on collision of a peakon couple and a peakon anti-couple. We then present numerical solutions of these collisions, which are inelastic because the waltzing peakon couples each possess an internal degree of freedom corresponding to their `tempo' -- that is, the period at which the two peakons of opposite type in the couple cycle around each other in phase space. Finally, we discuss compacton couple solutions of the cross-coupled Euler-Poincar\'e (CCEP) equations and illustrate the same types of collisions as for peakon couples, with triangular and parabolic compacton couples. We finish with a number of outstanding questions and challenges remaining for understanding couple dynamics of the CCCH and CCEP equations

The Square Root Depth Wave Equations

We introduce a set of coupled equations for multilayer water waves that removes the ill-posedness of the multilayer Green-Naghdi (MGN) equations in the presence of shear. The new well-posed equations are Hamiltonian and in the absence of imposed background shear they retain the same travelling wave solutions as MGN. We call the new model the Square Root Depth equations, from the modified form of their kinetic energy of vertical motion. Our numerical results show how the Square Root Depth equations model the effects of multilayer wave propagation and interaction, with and without shear.Comment: 10 pages, 5 figure

Error bounds on complex floating-point multiplication

Given floating-point arithmetic with t-digit base-β significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values z0 and z1 can be computed with maximum absolute error |z0||z1|1/2β 1-t√5. In particular, this provides relative error bounds of 2-24√5 and 2-53√5. for IEEE 754 single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for IEEE 754 single and double precision arithmetic

An Euler Poincar\'e framework for the multilayer Green Nagdhi equations

The Green Nagdhi equations are frequently used as a model of the wave-like behaviour of the free surface of a fluid, or the interface between two homogeneous fluids of differing densities. Here we show that their multilayer extension arises naturally from a framework based on the Euler Poincare theory under an ansatz of columnar motion. The framework also extends to the travelling wave solutions of the equations. We present numerical solutions of the travelling wave problem in a number of flow regimes. We find that the free surface and multilayer waves can exhibit intriguing differences compared to the results of single layer or rigid lid models

Error Bounds on Complex Floating-Point Multiplication

International audienceGiven floating-point arithmetic with $t$-digit base-$\beta$ significands in which all arithmetic operations are performed as if calculated to infinite precision and rounded to a nearest representable value, we prove that the product of complex values $z_0$ and $z_1$ can be computed with maximum absolute error \abs{z_0} \abs{z_1} \frac{1}{2} \beta^{1 - t} \sqrt{5}. In particular, this provides relative error bounds of $2^{-24} \sqrt{5}$ and $2^{-53} \sqrt{5}$ for {IEEE 754} single and double precision arithmetic respectively, provided that overflow, underflow, and denormals do not occur. We also provide the numerical worst cases for {IEEE 754} single and double precision arithmetic

Hydro-morphodynamics 2D modelling using a discontinuous Galerkin discretisation

The development of morphodynamic models to simulate sediment transport accurately is a challenging process that is becoming ever more important because of our increasing exploitation of the coastal zone, as well as sea-level rise and the potential increase in strength and frequency of storms due to a changing climate. Morphodynamic models are highly complex given the non-linear and coupled nature of the sediment transport problem. Here we implement a new depth-averaged coupled hydrodynamic and sediment transport model within the coastal ocean model Thetis, built using the code generating framework Firedrake which facilitates code flexibility and optimisation benefits. To the best of our knowledge, this represents the first full morphodynamic model including both bedload and suspended sediment transport which uses a discontinuous Galerkin based finite element discretisation. We implement new functionalities within Thetis extending its existing capacity to model scalar transport to modelling suspended sediment transport, incorporating within Thetis options to model bedload transport and bedlevel changes. We apply our model to problems with non-cohesive sediment and account for effects of gravity and helical flow by adding slope gradient terms and parametrising secondary currents. For validation purposes and in demonstrating model capability, we present results from test cases of a migrating trench and a meandering channel comparing against experimental data and the widely-used model Telemac-Mascaret

The velocity field of 2MRS Ks=11.75 galaxies: constraints on beta and bulk flow from the luminosity function

Using the nearly full sky Ks=11.75 2MASS Redshift Survey [2MRS]of ~45,000 galaxies we reconstruct the underlying peculiar velocity field and constrain the cosmological bulk flow within ~100. These results are obtained by maximizing the probability to estimate the absolute magnitude of a galaxy given its observed apparent magnitude and redshift. At a depth of ~60 Mpc/h we find a bulk flow Vb=(90\pm65,-230\pm65,50\pm65) km/s in agreement with the theoretical predictions of the LCDM model. The reconstructed peculiar velocity field that maximizes the likelihood is characterized by the parameter beta=0.323 +/- 0.08. Both results are in agreement with those obtained previously using the ~23,000 galaxies of the shallower Ks=11.25 2MRS survey. In our analysis we find that the luminosity function of 2MRS galaxies is poorly fitted by the Schechter form and that luminosity evolves such that objects become fainter with increasing redshift according to L(z)=L(z=0)(1+z)^(+2.7 +/-0.15).Comment: 10 pages, 6 figure

Non-Standard Structure Formation Scenarios

Observations on galactic scales seem to be in contradiction with recent high resolution N-body simulations. This so-called cold dark matter (CDM) crisis has been addressed in several ways, ranging from a change in fundamental physics by introducing self-interacting cold dark matter particles to a tuning of complex astrophysical processes such as global and/or local feedback. All these efforts attempt to soften density profiles and reduce the abundance of satellites in simulated galaxy halos. In this contribution we are exploring the differences between a Warm Dark Matter model and a CDM model where the power on a certain scale is reduced by introducing a narrow negative feature (''dip''). This dip is placed in a way so as to mimic the loss of power in the WDM model: both models have the same integrated power out to the scale where the power of the Dip model rises to the level of the unperturbed CDM spectrum again. Using N-body simulations we show that that the new Dip model appears to be a viable alternative to WDM while being based on different physics: where WDM requires the introduction of a new particle species the Dip stems from a non-standard inflationary period. If we are looking for an alternative to the currently challenged standard LCDM structure formation scenario, neither the LWDM nor the new Dip model can be ruled out with respect to the analysis presented in this contribution. They both make very similar predictions and the degeneracy between them can only be broken with observations yet to come.Comment: 4 pages, 3 figures; to appear in "The Evolution of Galaxies III. From Simple Approaches to Self-Consistent Models", proceedings of the 3rd EuroConference on the evolution of galaxies, held in Kiel, Germany, July 16-20, 200

Time Protection: the Missing OS Abstraction

Timing channels enable data leakage that threatens the security of computer systems, from cloud platforms to smartphones and browsers executing untrusted third-party code. Preventing unauthorised information flow is a core duty of the operating system, however, present OSes are unable to prevent timing channels. We argue that OSes must provide time protection in addition to the established memory protection. We examine the requirements of time protection, present a design and its implementation in the seL4 microkernel, and evaluate its efficacy as well as performance overhead on Arm and x86 processors

Amplitude distribution of eigenfunctions in mixed systems

We study the amplitude distribution of irregular eigenfunctions in systems with mixed classical phase space. For an appropriately restricted random wave model a theoretical prediction for the amplitude distribution is derived and good agreement with numerical computations for the family of limacon billiards is found. The natural extension of our result to more general systems, e.g. with a potential, is also discussed.Comment: 13 pages, 3 figures. Some of the pictures are included in low resolution only. For a version with pictures in high resolution see http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab