10,795 research outputs found
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, 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
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