Interlace Polynomials for Multimatroids and Delta-Matroids

We provide a unified framework in which the interlace polynomial and several related graph polynomials are defined more generally for multimatroids and delta-matroids. Using combinatorial properties of multimatroids rather than graph-theoretical arguments, we find that various known results about these polynomials, including their recursive relations, are both more efficiently and more generally obtained. In addition, we obtain several interrelationships and results for polynomials on multimatroids and delta-matroids that correspond to new interrelationships and results for the corresponding graphs polynomials. As a tool we prove the equivalence of tight 3-matroids and delta-matroids closed under the operations of twist and loop complementation, called vf-safe delta-matroids. This result is of independent interest and related to the equivalence between tight 2-matroids and even delta-matroids observed by Bouchet.Comment: 35 pages, 3 figure

Children and Families With Incarcerated Parents: Exploring Development in the Field and Opportunities for Growth

Summarizes discussions with experts, practitioners, advocates, policy makers, and funders about meeting the needs of prisoners' children, including through public policy and system reform. Offers principles to guide efforts to support family connections

Nullity and Loop Complementation for Delta-Matroids

We show that the symmetric difference distance measure for set systems, and more specifically for delta-matroids, corresponds to the notion of nullity for symmetric and skew-symmetric matrices. In particular, as graphs (i.e., symmetric matrices over GF(2)) may be seen as a special class of delta-matroids, this distance measure generalizes the notion of nullity in this case. We characterize delta-matroids in terms of equicardinality of minimal sets with respect to inclusion (in addition we obtain similar characterizations for matroids). In this way, we find that, e.g., the delta-matroids obtained after loop complementation and after pivot on a single element together with the original delta-matroid fulfill the property that two of them have equal "null space" while the third has a larger dimension.Comment: Changes w.r.t. v4: different style, Section 8 is extended, and in addition a few small changes are made in the rest of the paper. 15 pages, no figure

CMB Anisotropies, Cosmological Parameters and Fundamental Physics: Current Status & Perspectives

I describe briefly the Cosmic Microwave Background (hereafter CMB) physics which explains why high accuracy observations of its spatial structure are a unique observational tool both for the determination of the global cosmological parameters and to constrain observationally the physics of the early universe. I also briefly survey the many experiments which have measured the anisotropies of the CMB and led to crucial advances in observational Cosmology. The somewhat frantic series of new results has recently culminated with the outcome of the WMAP satellite which confirmed earlier results, set new standards of accuracy, and suggested that the Universe may have reionised earlier than anticipated. Many more CMB experiments are currently taking data or being planned, with the Planck satellite on the 2007 Horizon poised to extract all the cosmological information in the temperature anisotropies, and foray deeply into polarisation.Comment: To appear in the proceedings of "Where Cosmology and Fundamental Physics Meet", 23-26 June, 2003, Marseille, Franc

The Planck mission

These lecture from the 100th Les Houches summer school on "Post-planck cosmology" of July 2013 discuss some aspects of the Planck mission, whose prime objective was a very accurate measurement of the temperature anisotropies of the Cosmic Microwave Background (CMB). We announced our findings a few months ago, on March 21$^{st}$, 2013. I describe some of the relevant steps we took to obtain these results, sketching the measurement process, how we processed the data to obtain full sky maps at 9 different frequencies, and how we extracted the CMB temperature anisotropies map and angular power spectrum. I conclude by describing some of the main cosmological implications of the statistical characteristics of the CMB we found. Of course, this is a very much shortened and somewhat biased view of the \Planck\ 2013 results, written with the hope that it may lead some of the students to consult the original papers.Comment: 53 p.-34 fig; for spacetime consideration, the file here is not paying justice to the actual thing; a closer approximation of it can be found at https://www.researchgate.net/profile/Francois_Bouchet/publication/262004262_The_Planck_Mission/file/e0b495363b042e81dd.pd

Nullity Invariance for Pivot and the Interlace Polynomial

We show that the effect of principal pivot transform on the nullity values of the principal submatrices of a given (square) matrix is described by the symmetric difference operator (for sets). We consider its consequences for graphs, and in particular generalize the recursive relation of the interlace polynomial and simplify its proof.Comment: small revision of Section 8 w.r.t. v2, 14 pages, 6 figure

The Group Structure of Pivot and Loop Complementation on Graphs and Set Systems

We study the interplay between principal pivot transform (pivot) and loop complementation for graphs. This is done by generalizing loop complementation (in addition to pivot) to set systems. We show that the operations together, when restricted to single vertices, form the permutation group S_3. This leads, e.g., to a normal form for sequences of pivots and loop complementation on graphs. The results have consequences for the operations of local complementation and edge complementation on simple graphs: an alternative proof of a classic result involving local and edge complementation is obtained, and the effect of sequences of local complementations on simple graphs is characterized.Comment: 21 pages, 7 figures, significant additions w.r.t. v3 are Thm 7 and Remark 2