Simultaneous Arithmetic Progressions on Algebraic Curves

A simultaneous arithmetic progression (s.a.p.) of length k consists of k points (x_i, y_\sigma(i)), where x_i and y_i are arithmetic progressions and \sigma is a permutation. Garcia-Selfa and Tornero asked whether there is a bound on the length of an s.a.p. on an elliptic curve in Weierstrass form over Q. We show that 4319 is such a bound for curves over R. This is done by considering translates of the curve in a grid as a graph. A simple upper bound is found for the number of crossings and the 'crossing inequality' gives a lower bound. Together these bound the length of an s.a.p. on the curve. We then use a similar method to extend the result to arbitrary real algebraic curves. Instead of considering s.a.p.'s we consider k^2/3 points in a grid. The number of crossings is bounded by Bezout's Theorem. We then give another proof using a result of Jarnik bounding the number of grid points on a convex curve. This result applies as any real algebraic curve can be broken up into convex and concave parts, the number of which depend on the degree. Lastly, these results are extended to complex algebraic curves.Comment: 11 pages, 6 figures, order of email addresses corrected 12 pages, closing remarks, a reference and an acknowledgment adde

Resonant Orbits in Triaxial Galaxies

Box orbits in triaxial potentials are generically thin, that is, they lie close in phase space to a resonant orbit satisfying a relation of the form l\omega_1 +m\omega_2+n\omega_3=0 between the three fundamental frequencies. Resonant orbits are confined to a membrane; they play roughly the same role, in three dimensions, that closed orbits play in two. Stable resonant orbits avoid the center of the potential; orbits that are thick enough to pass near the center are typically stochastic. Very near the center, where the gravitational potential is dominated by the black hole, resonant orbits continue to exist, including at least one family whose elongation is parallel to the long axes of the triaxial figure.Comment: 20 Latex pages, 11 Postscript figures. Submitted to The Astronomical Journa

The effects of social media on political party perception and voting behavior

This study sought to determine to what extent social media influences political party perception (PPP) and political voting behavior. Based on literature a conceptual model was developed which measures political interest, political trust, religion and the use of social media and their effects on PPP and voting behavior. Using an online questionnaire the conceptual model was tested towards and during the Dutch national elections of 2010. Although data analysis indicates several significant effects on PPP, voting behavior is solely determined by political interest. Certain effects of social media seem evident, though further research is necessary in funding and legitimizing its future role in political marketing

Models of Cuspy Triaxial Galaxies

We construct numerical models of mildly triaxial elliptical galaxies with central density cusps. Using a technique we call ``adiabatic squeezing'', we begin with a spherical gamma=1 Hernquist model and apply a drag to the velocities of the particles along each principle axis. The final models are stable in isolation, preserving their density structure and figure shape over many dynamical timescales. The density profile and axial ratios compare well to the observed properties of elliptical galaxies. The orbital structure of these models show a mixture of tubes, boxes, and boxlets, as expected for triaxial systems, with very few chaotic orbits. These N-body realizations of cuspy triaxial galaxies provide a basis for the study of the dynamical evolution of elliptical galaxies.Comment: 14 pages, 7 figures. Accepted by Ap

Vakschool 2.0:sustainable and quality vocational training

Lower vocational-oriented education is under pressure in the Netherlands. Small educational institutions are looking for sustainable quality solutions for offering high quality curricula. Cloud computing and social media may be one part of the solution. This research contributes to develop a sustainable quality vocational education 2.0 concept that can be used in small vocational oriented education. Different experts contributed to a concept, called Vakschool1 2.0, in three different sessions. The concept can be applied within small vocational oriented education in the Netherlands

Simple Three-Integral Scale-Free Galaxy Models

The Jeans equations give the second moments or stresses required to support a stellar population against the gravity field. A general solution of the Jeans equations for arbitrary axisymmetric scale-free densities in flattened scale-free potentials is given. A two-parameter subset of the solution for the second moments for the self-consistent density of the power-law models, which have exactly spheroidal equipotentials, is examined in detail. In the spherical limit, the potential of these models reduces to that of the singular power-law spheres. We build the physical three-integral distribution functions that correspond to the flattened stellar components. Next, we attack the problem of finding distribution functions associated with the Jeans solutions in flattened scale-free potentials. The third or partial integral introduced by de Zeeuw, Evans and Schwarzschild for Binney's model is generalised to thin and near-thin orbits moving in arbitrary axisymmetric scale-free potentials. The partial integral is a modification of the total angular momentum. For the self-consistent power-law models, we show how this enables the construction of simple three-integral distribution functions. The connexion between these approximate distribution functions and the Jeans solutions is discussed in some detail.Comment: 14 pages, 7 postscript figures, to appear in Monthly Notice

The Kinematics of the Outer Halo of M87

Radial velocities are presented for a new sample of globular clusters in the outer halo of M87 at a distance of 300 to 540 arcsec (24 to 43 kpc) from the center of this galaxy. These are used to augment our previously published data and an analysis of the rotation and velocity dispersion of the M87 globular cluster system is carried out. The rotation is $\sim300$ \kms at R = 32 kpc, at which point the velocity dispersion is also still quite high, $\sim450$ \kms. The high rotation is interesting. The outer halo of M87 is, as was found in our previous kinematic analysis, very massive.Comment: Accepted for publication in the AJ. 13 pages with 3 figure