On distinguishing trees by their chromatic symmetric functions

Let $T$ be an unrooted tree. The \emph{chromatic symmetric function} $X_T$, introduced by Stanley, is a sum of monomial symmetric functions corresponding to proper colorings of $T$. The \emph{subtree polynomial} $S_T$, first considered under a different name by Chaudhary and Gordon, is the bivariate generating function for subtrees of $T$ by their numbers of edges and leaves. We prove that $S_T =$, where $$ is the Hall inner product on symmetric functions and $\Phi$ is a certain symmetric function that does not depend on $T$. Thus the chromatic symmetric function is a stronger isomorphism invariant than the subtree polynomial. As a corollary, the path and degree sequences of a tree can be obtained from its chromatic symmetric function. As another application, we exhibit two infinite families of trees (\emph{spiders} and some \emph{caterpillars}), and one family of unicyclic graphs (\emph{squids}) whose members are determined completely by their chromatic symmetric functions.Comment: 16 pages, 3 figures. Added references [2], [13], and [15

Measurement of the Higgs mass via the channel : e+e- -> ZH -> e+e- + X

In this communication, the mass declined for the decay channel, e+e- -> ZH -> e+e- + X, as measured by the ILD detector was studied. The Higgs mass is assumed to be 120 GeV and the center of mass energy is 250 GeV. For an integrated luminosity of 250 fb-1, the accuracy of the reconstruction and the good knowledge of the initial state allow for the measurement of the Higgs boson mass with a precision of about 100 MeV.Comment: 7 pages, 14 figures, LCWS/ILC 2010 (International Linear Collider Workshop 2010 LCWS10 and ILC10

A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra

We study finite loop models on a lattice wrapped around a cylinder. A section of the cylinder has N sites. We use a family of link modules over the periodic Temperley-Lieb algebra EPTL_N(\beta, \alpha) introduced by Martin and Saleur, and Graham and Lehrer. These are labeled by the numbers of sites N and of defects d, and extend the standard modules of the original Temperley-Lieb algebra. Beside the defining parameters \beta=u^2+u^{-2} with u=e^{i\lambda/2} (weight of contractible loops) and \alpha (weight of non-contractible loops), this family also depends on a twist parameter v that keeps track of how the defects wind around the cylinder. The transfer matrix T_N(\lambda, \nu) depends on the anisotropy \nu and the spectral parameter \lambda that fixes the model. (The thermodynamic limit of T_N is believed to describe a conformal field theory of central charge c=1-6\lambda^2/(\pi(\lambda-\pi)).) The family of periodic XXZ Hamiltonians is extended to depend on this new parameter v and the relationship between this family and the loop models is established. The Gram determinant for the natural bilinear form on these link modules is shown to factorize in terms of an intertwiner i_N^d between these link representations and the eigenspaces of S^z of the XXZ models. This map is shown to be an isomorphism for generic values of u and v and the critical curves in the plane of these parameters for which i_N^d fails to be an isomorphism are given.Comment: Replacement of "The Gram matrix as a connection between periodic loop models and XXZ Hamiltonians", 31 page

What controls the large-scale magnetic fields of M dwarfs?

Observations of active M dwarfs show a broad variety of large-scale magnetic fields encompassing dipole-dominated and multipolar geometries. We detail the analogy between some anelastic dynamo simulations and spectropolarimetric observations of 23 M stars. In numerical models, the relative contribution of inertia and Coriolis force in the global force balance -estimated by the so-called local Rossby number- is known to have a strong impact on the magnetic field geometry. We discuss the relevance of this parameter in setting the large-scale magnetic field of M dwarfs.Comment: 4 pages, 3 figures, conference proceeding, IAUS 302 'Magnetic Fields Throughout the Stellar Evolution', (26-30 Aug 2013, Biarritz, France

What controls the magnetic geometry of M dwarfs?

Context: observations of rapidly rotating M dwarfs show a broad variety of large-scale magnetic fields encompassing dipole-dominated and multipolar geometries. In dynamo models, the relative importance of inertia in the force balance -- quantified by the local Rossby number -- is known to have a strong impact on the magnetic field geometry. Aims: we aim to assess the relevance of the local Rossby number in controlling the large-scale magnetic field geometry of M dwarfs. Methods: we explore the similarities between anelastic dynamo models in spherical shells and observations of active M-dwarfs, focusing on field geometries derived from spectropolarimetric studies. To do so, we construct observation-based quantities aimed to reflect the diagnostic parameters employed in numerical models. Results: the transition between dipole-dominated and multipolar large-scale fields in early to mid M dwarfs is tentatively attributed to a Rossby number threshold. We interpret late M dwarfs magnetism to result from a dynamo bistability occurring at low Rossby number. By analogy with numerical models, we expect different amplitudes of differential rotation on the two dynamo branches.Comment: 4 pages, 4 figures, accepted for publication in A&