We report on the discovery of three transiting planets around GJ9827. The planets have radii of 1.75±0.18, 1.36±0.14, and 2.11 0.21 0.22 -+ R ⊕ , and periods of 1.20896, 3.6480, and 6.2014 days, respectively. The detection was made in Campaign 12 observations as part of our K2 survey of nearby stars. GJ9827 is a V=10.39 mag K6V star at a distance of 30.3±1.6 parsecs and the nearest star to be found hosting planets by Kepler and K2. The radial velocity follow-up, high-resolution imaging, and detection of multiple transiting objects near commensurability drastically reduce the false positive probability. The orbital periods of GJ9827b, c, and d planets are very close to the 1:3:5 mean motion resonance. Our preliminary analysis shows that GJ9827 planets are excellent candidates for atmospheric observations. Besides, the planetary radii span both sides of the rocky and gaseous divide, hence the system will be an asset in expanding our understanding of the threshold.

We discuss the detailed structure of the spectrum of the Hamiltonian for the polymerized harmonic oscillator and compare it with the spectrum in the standard quantization. As we will see the non-separability of the Hilbert space implies that the point spectrum consists of bands similar to the ones appearing in the treatment of periodic potentials. This feature of the spectrum of the polymeric harmonic oscillator may be relevant for the discussion of the polymer quantization of the scalar field and may have interesting consequences for the statistical mechanics of these models.

The purpose of this paper is to study in detail the constraint structure of the Hamiltonian and symplectic-Lagrangian descriptions for the scalar and electromagnetic fields in the presence of spatial boundaries. We carefully discuss the implementation of the geometric constraint algorithm of Gotay, Nester and Hinds with special emphasis on the relevant functional analytic aspects of the problem. This is an important step towards the rigorous understanding of general field theories in the presence of boundaries, very especially when these fail to be regular. The geometric approach developed in the paper is also useful with regard to the interpretation of the physical degrees of freedom and the nature of the constraints when both gauge symmetries and boundaries are present.

This article discusses and explains the Hamiltonian formulation for a class of simple gauge invariant mechanical systems consisting of point masses and idealized rods. The study of these models may be helpful to advanced undergraduate or graduate students in theoretical physics to understand, in a familiar context, some concepts relevant to the study of classical and quantum field theories. We use a geometric approach to derive the Hamiltonian formulation for the model considered in the paper: four equal masses connected by six ideal rods. We obtain and discuss the meaning of several important elements, in particular, the constraints and the Hamiltonian vector fields that define the dynamics of the system, the constraint manifold, gauge symmetries, gauge orbits, gauge fixing, and the reduced phase space.