Abstract:
Measuring radio source counts is critical for characterizing new extragalactic populations, brings a wealth of science within reach and will inform forecasts for SKA and its pathfinders. Yet there is currently great debate (and few measurements) about the behaviour of the 1.4-GHz counts in the microJy regime. One way to push the counts to these levels is via 'stacking', the covariance of a map with a catalogue at higher resolution and (often) a different wavelength. For the first time, we cast stacking in a fully bayesian framework, applying it to (i) the SKADS simulation and (ii) VLA data stacked at the positions of sources from the VIDEO survey. In the former case, the algorithm recovers the counts correctly when applied to the catalogue, but is biased high when confusion comes into play. This needs to be accounted for in the analysis of data from any relatively-low-resolution SKA pathfinders. For the latter case, the observed radio source counts remain flat below the 5-sigma level of 85 microJy as far as 40 microJy, then fall off earlier than the flux hinted at by the SKADS simulations and a recent P(D) analysis (which is the only other measurement from the literature at these flux-density levels, itself extrapolated in frequency). Division into galaxy type via spectral-energy distribution reveals that normal spiral galaxies dominate the counts at these fluxes.

Abstract:
We developed a Keplerian-based Hamiltonian splitting for solving the gravitational $N$-body problem. This splitting allows us to approximate the solution of a general $N$-body problem by a composition of multiple, independently evolved $2$-body problems. While the Hamiltonian splitting is exact, we show that the composition of independent $2$-body problems results in a non-symplectic non-time-symmetric first-order map. A time-symmetric second-order map is then constructed by composing this basic first-order map with its self-adjoint. The resulting method is precise for each individual $2$-body solution and produces quick and accurate results for near-Keplerian $N$-body systems, like planetary systems or a cluster of stars that orbit a supermassive black hole. The method is also suitable for integration of $N$-body systems with intrinsic hierarchies, like a star cluster with primordial binaries. The superposition of Kepler solutions for each pair of particles makes the method excellently suited for parallel computing; we achieve $\gtrsim 64\%$ efficiency for only $8$ particles per core, but close to perfect scaling for $16384$ particles on a $128$ core distributed-memory computer. We present several implementations in \texttt{Sakura}, one of which is publicly available via the AMUSE framework.

Abstract:
We develop a method to analyze the effect of an asymmetric supernova on hierarchical multiple star systems and we present analytical formulas to calculate orbital parameters for surviving binaries or hierarchical triples and runaway velocities for their dissociating equivalents. The effect of an asymmetric supernova on the orbital parameters of a binary system has been studied to great extent (e.g. Hills 1983; Kalogera 1996; Tauris & Takens 1998), but this effect on higher multiplicity hierarchical systems has not been explored before. With our method, the supernova effect can be computed by reducing the hierarchical multiple to an effective binary by means of recursively replacing the inner binary by an effective star at the center of mass of that binary. We apply our method to a hierarchical triple system similar to the progenitor of PSR J1903+0327 suggested by Portegies Zwart et al. (2011). We confirm their earlier finding that PSR J1903+0327 could have evolved from a hierarchical triple that became unstable and ejected the secondary star of the inner binary. Furthermore, if such as system did evolve via this mechanism the most probable configuration would be a small supernova kick velocity, an inner binary with a large semi-major axis, and the fraction of mass accreted onto the neutron star to the mass lost by the secondary would most likely be between 0.35 and 0.5

Abstract:
We calculate the tadpole equations and their solutions for a class of four-dimensional orientifolds with orbifold group Z_N X Z_M, and we present the massless bosonic spectra of these models. Surprisingly we find no consistent solutions for the models with Z_2 X Z_4 and Z_4 X Z_4 orbifold groups.

Abstract:
We construct the Matrix theory descriptions of M-theory on the Mobius strip and the Klein bottle. In a limit, these provide the matrix string theories for the CHL string and an orbifold of type IIA string theory.

Abstract:
String theory compactified on a three-torus possesses an SL(5,Z) U-duality group. We investigate the realisation of this symmetry on the Born-Infeld theory on a three-brane, and discuss a U-duality covariant formulation of the BPS sector of the theory where the rank of the gauge group is treated on an equal footing with the fluxes.

Abstract:
The Born-Infeld theory of a toroidal D3-brane has an SL(5,Z) U-duality symmetry. We investigate how this symmetry is reflected in the supersymmetry algebra. We propose an action of the group on the gauge theory fields in the BPS sector by introducing an extra field together with an additional symmetry, and argue for the U-invariance of the degeneracies of the BPS spectrum.

Abstract:
We give sufficient conditions such that the exponential stability of the linearization of a non-linear system implies that the non-linear system is (locally) exponentially stable. One of these conditions is that the non-linear system is Fr\'{e}chet differential at the equilibrium, if it is only Gateaux differentiable, then we show by means of an example that the result does not hold.

Abstract:
Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most $K$ elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the groups form a Riesz family. This implies that if in every range of the spectral projection we construct an orthonormal basis, then the union of these bases is a Riesz basis in the Hilbert space.

Abstract:
Given a Hilbert space and the generator $A$ of a strongly continuous, exponentially stable, semigroup on this Hilbert space. For any $g(-s) \in {\mathcal H}_{\infty}$ we show that there exists an infinite-time admissible output operator $g(A)$. If $g$ is rational, then this operator is bounded, and equals the "normal" definition of $g(A)$. In particular, when $g(s)=1/(s + \alpha)$, $ \alpha \in {\mathbb C}_0^+$, then this admissible output operator equals $(\alpha I - A)^{-1}$. Although in general $g(A)$ may be unbounded, we always have that $g(A)$ multiplied by the semigroup is a bounded operator for every (strictly) positive time instant. Furthermore, when there exists an admissible output operator $C$ such that $(C,A)$ is exactly observable, then $g(A)$ is bounded for all $g$'s with $g(-s) \in {\mathcal H}_{\infty}$, i.e., there exists a bounded ${\mathcal H}_{\infty}$-calculus. Moreover, we rediscover some well-known classes of generators also having a bounded ${\mathcal H}_{\infty}$-calculus.