Generating sequences and Poincar\'e series for a finite set of plane divisorial valuations

Let $V$ be a finite set of divisorial valuations centered at a 2-dimensional regular local ring $R$. In this paper we study its structure by means of the semigroup of values, $S_V$, and the multi-index graded algebra defined by $V$, \gr_V R. We prove that $S_V$ is finitely generated and we compute its minimal set of generators following the study of reduced curve singularities. Moreover, we prove a unique decomposition theorem for the elements of the semigroup. The comparison between valuations in $V$, the approximation of a reduced plane curve singularity $C$ by families of sets $V^{(k)}$ of divisorial valuations, and the relationship between the value semigroup of $C$ and the semigroups of the sets $V^{(k)}$, allow us to obtain the (finite) minimal generating sequences for $C$ as well as for $V$. We also analyze the structure of the homogeneous components of \gr_V R. The study of their dimensions allows us to relate the Poincar\'e series for $V$ and for a general curve $C$ of $V$. Since the last series coincides with the Alexander polynomial of the singularity, we can deduce a formula of A'Campo type for the Poincar\'e series of $V$. Moreover, the Poincar\'e series of $C$ could be seen as the limit of the series of $V^{(k)}$, $k\ge 0$

Dirac Triplet Extension of the MSSM

In this paper we explore extensions of the Minimal Supersymmetric Standard Model involving two $SU(2)_L$ triplet chiral superfields that share a superpotential Dirac mass yet only one of which couples to the Higgs fields. This choice is motivated by recent work using two singlet superfields with the same superpotential requirements. We find that, as in the singlet case, the Higgs mass in the triplet extension can easily be raised to $125\,\text{GeV}$ without introducing large fine-tuning. For triplets that carry hypercharge, the regions of least fine tuning are characterized by small contributions to the $\mathcal T$ parameter, and light stop squarks, $m_{\tilde t_1} \sim 300-450\,\text{GeV}$; the latter is a result of the $\tan\beta$ dependence of the triplet contribution to the Higgs mass. Despite such light stop masses, these models are viable provided the stop-electroweakino spectrum is sufficiently compressed.Comment: 26 pages, 4 figure

The stellar content of the Local Group dwarf galaxy Phoenix

We present new deep $VI$ ground-based photometry of the Local Group dwarf galaxy Phoenix. Our results confirm that this galaxy is mainly dominated by red stars, with some blue plume stars indicating recent (100 Myr old) star formation in the central part of the galaxy. We have performed an analysis of the structural parameters of Phoenix based on an ESO/SRC scanned plate, in order to search for differentiated component. The results were then used to obtain the color-magnitude diagrams for three different regions of Phoenix in order to study the variation of the properties of its stellar population. The young population located in the central component of Phoenix shows a clear asymmetry in its distribution, that could indicate a propagation of star formation across the central component. The HI cloud found at 6 arcmin Southwest by Young & Lo (1997) could have been involved in this process. We also find the presence of a substantial intermediate-age population in the central region of Phoenix that would be less abundant or absent in its outer regions. This result is also consistent with the gradient found in the number of horizontal branch stars, whose frequency relative to red giant branch stars increases towards the outer part of the galaxy. These results, together with those of our morphological study, suggest the existence of an old, metal-poor population with a spheroidal distribution surrounding the younger inner component of Phoenix. This two-component structure may resemble the halo-disk structure observed in spirals, although more data, in particular on kinematics, are necessary to confirm this.Comment: 46 pages, 21 figures, 9 Tables, to be published in AJ, August 9

Authoring courses with rich adaptive sequencing for IMS learning design

This paper describes the process of translating an adaptive sequencing strategy designed using Sequencing Graphs to the semantics of IMS Learning Design. The relevance of this contribution is twofold. First, it combines the expressive power and flexibility of Sequencing Graphs, and the interoperability capabilities of IMS. Second, it shows some important limitations of IMS specifications (focusing on Learning Design) for the sequencing of learning activities

Semiquantitative theory of electronic Raman scattering from medium-size quantum dots

A consistent semiquantitative theoretical analysis of electronic Raman scattering from many-electron quantum dots under resonance excitation conditions has been performed. The theory is based on random-phase-approximation-like wave functions, with the Coulomb interactions treated exactly, and hole valence-band mixing accounted for within the Kohn-Luttinger Hamiltonian framework. The widths of intermediate and final states in the scattering process, although treated phenomenologically, play a significant role in the calculations, particularly for well above band gap excitation. The calculated polarized and unpolarized Raman spectra reveal a great complexity of features and details when the incident light energy is swept from below, through, and above the quantum dot band gap. Incoming and outgoing resonances dramatically modify the Raman intensities of the single particle, charge density, and spin density excitations. The theoretical results are presented in detail and discussed with regard to experimental observations.Comment: Submitted to Phys. Rev.

Equivariant Poincar\'e series of filtrations and topology

Earlier, for an action of a finite group $G$ on a germ of an analytic variety, an equivariant $G$-Poincar\'e series of a multi-index filtration in the ring of germs of functions on the variety was defined as an element of the Grothendieck ring of $G$-sets with an additional structure. We discuss to which extend the $G$-Poincar\'e series of a filtration defined by a set of curve or divisorial valuations on the ring of germs of analytic functions in two variables determines the (equivariant) topology of the curve or of the set of divisors