On the orbits of the product of two permutations

AbstractWe consider the following problem: given three partitions A,B,C of a finite set Ω, do there exist two permutations α and β such that A,B,C are induced by α, β and αβ respectively? This problem is NP-complete. However it turns out that it can be solved by a polynomial time algorithm when some relations between the number of classes of A,B,C hold

The Adequacy of Retirement Saving

macroeconomics, saving, retirement

Number of right ideals and a qq-analogue of indecomposable permutations

We prove that the number of right ideals of codimension $n$ in the algebra of noncommutative Laurent polynomials in two variables over the finite field $\mathbb F\_q$ is equal to $(q-1)^{n+1} q^{\frac{(n+1)(n-2)}{2}}\sum\_\theta q^{inv(\theta)}$, where the sum is over all indecomposable permutations in $S\_{n+1}$ and where $inv(\theta)$stands for the number of inversions of $\theta$.Comment: submitte

Use of Policy Risk Assessment Results in Political Decision Making

The RAPID project established, during the first period, a thematic network of risk assessment experts, including relevant partners in the ten countries involved, the "Risk assessor database". The project devoted a specific activity, a single work package, to the dissemination and discussion of the methodology developed during" first two years of the project. National workshops were planned in each country to facilitate integrated knowledge translation activity, using a participatory approach to increase potential knowledge-users awareness on the RAPID project, and to engage them in using the RAPID guidance. Workshops were conceived to present case studies and the RAPID guidance to a targeted audience, to discuss and collect further insights, and integrate different perspectives in the final version of the policy evaluation methodology. However, national workshops also actively contributed to develop evidence based methodological guidance and increase its quality and relevance for potential users by bridging know-do gap between researchers and stakeholders; by involving decision makers and potential users in the knowledge creation process; by facilitating diverse stakeholder participation from governmental, academic and private sectors, carefully identified by national RAPID surveys as having direct expertise in the field of risk assessment. The cultural and administrative differences existing in the countries involved in RAPID guarantee the inclusion of a wide range of perspectives. Results of the national workshops helped to identify barriers and solutions for using the guidance, for adapting necessary changes to it and for communicating results to other potential users.

Distance statistics in large toroidal maps

We compute a number of distance-dependent universal scaling functions characterizing the distance statistics of large maps of genus one. In particular, we obtain explicitly the probability distribution for the length of the shortest non-contractible loop passing via a random point in the map, and that for the distance between two random points. Our results are derived in the context of bipartite toroidal quadrangulations, using their coding by well-labeled 1-trees, which are maps of genus one with a single face and appropriate integer vertex labels. Within this framework, the distributions above are simply obtained as scaling limits of appropriate generating functions for well-labeled 1-trees, all expressible in terms of a small number of basic scaling functions for well-labeled plane trees.Comment: 24 pages, 9 figures, minor corrections, new added reference

Random Planar Lattices and Integrated SuperBrownian Excursion

In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous' Integrated SuperBrownian Excursion (ISE). As a consequence, the radius r_n of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r=R-L of the support of the one-dimensional ISE. More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive half-line, reminiscent of Cori and Vauquelin's well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat's construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks to the Brownian snake description of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in two-dimensional pure quantum gravity.Comment: 44 pages, 22 figures. Slides and extended abstract version are available at http://www.loria.fr/~schaeffe/Pub/Diameter/ and http://www.iecn.u-nancy.fr/~chassain

The topological structure of scaling limits of large planar maps

We discuss scaling limits of large bipartite planar maps. If p is a fixed integer strictly greater than 1, we consider a random planar map M(n) which is uniformly distributed over the set of all 2p-angulations with n faces. Then, at least along a suitable subsequence, the metric space M(n) equipped with the graph distance rescaled by the factor n to the power -1/4 converges in distribution as n tends to infinity towards a limiting random compact metric space, in the sense of the Gromov-Hausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4.Comment: 45 pages Second version with minor modification

Integrability of graph combinatorics via random walks and heaps of dimers

We investigate the integrability of the discrete non-linear equation governing the dependence on geodesic distance of planar graphs with inner vertices of even valences. This equation follows from a bijection between graphs and blossom trees and is expressed in terms of generating functions for random walks. We construct explicitly an infinite set of conserved quantities for this equation, also involving suitable combinations of random walk generating functions. The proof of their conservation, i.e. their eventual independence on the geodesic distance, relies on the connection between random walks and heaps of dimers. The values of the conserved quantities are identified with generating functions for graphs with fixed numbers of external legs. Alternative equivalent choices for the set of conserved quantities are also discussed and some applications are presented.Comment: 38 pages, 15 figures, uses epsf, lanlmac and hyperbasic

Combinatorics of bicubic maps with hard particles

We present a purely combinatorial solution of the problem of enumerating planar bicubic maps with hard particles. This is done by use of a bijection with a particular class of blossom trees with particles, obtained by an appropriate cutting of the maps. Although these trees have no simple local characterization, we prove that their enumeration may be performed upon introducing a larger class of "admissible" trees with possibly doubly-occupied edges and summing them with appropriate signed weights. The proof relies on an extension of the cutting procedure allowing for the presence on the maps of special non-sectile edges. The admissible trees are characterized by simple local rules, allowing eventually for an exact enumeration of planar bicubic maps with hard particles. We also discuss generalizations for maps with particles subject to more general exclusion rules and show how to re-derive the enumeration of quartic maps with Ising spins in the present framework of admissible trees. We finally comment on a possible interpretation in terms of branching processes.Comment: 41 pages, 19 figures, tex, lanlmac, hyperbasics, epsf. Introduction and discussion/conclusion extended, minor corrections, references adde

Launch Pad Flame Trench Refractory Materials

The launch complexes at NASA's John F. Kennedy Space Center (KSC) are critical support facilities for the successful launch of space-based vehicles. These facilities include a flame trench that bisects the pad at ground level. This trench includes a flame deflector system that consists of an inverted, V-shaped steel structure covered with a high temperature concrete material five inches thick that extends across the center of the flame trench. One side of the "V11 receives and deflects the flames from the orbiter main engines; the opposite side deflects the flames from the solid rocket boosters. There are also two movable deflectors at the top of the trench to provide additional protection to shuttle hardware from the solid rocket booster flames. These facilities are over 40 years old and are experiencing constant deterioration from launch heat/blast effects and environmental exposure. The refractory material currently used in launch pad flame deflectors has become susceptible to failure, resulting in large sections of the material breaking away from the steel base structure and creating high-speed projectiles during launch. These projectiles jeopardize the safety of the launch complex, crew, and vehicle. Post launch inspections have revealed that the number and frequency of repairs, as well as the area and size of the damage, is increasing with the number of launches. The Space Shuttle Program has accepted the extensive ground processing costs for post launch repair of damaged areas and investigations of future launch related failures for the remainder of the program. There currently are no long term solutions available for Constellation Program ground operations to address the poor performance and subsequent failures of the refractory materials. Over the last three years, significant liberation of refractory material in the flame trench and fire bricks along the adjacent trench walls following Space Shuttle launches have resulted in extensive investigations of failure mechanisms, load response, ejected material impact evaluation, and repair design analysis (environmental and structural assessment, induced environment from solid rocket booster plume, loads summary, and repair integrity), assessment of risk posture for flame trench debris, and justification of flight readiness rationale. Although the configuration of the launch pad, water and exhaust direction, and location of the Mobile Launcher Platform between the flame trench and the flight hardware should protect the Space Vehicle from debris exposure, loss of material could cause damage to a major element of the ground facility (resulting in temporary usage loss); and damage to other facility elements is possible. These are all significant risks that will impact ground operations for Constellation and development of new refractory material systems is necessary to reduce the likelihood of the foreign object debris hazard during launch. KSC is developing an alternate refractory material for the launch pad flame trench protection system, including flame deflector and flame trench walls, that will withstand launch conditions without the need for repair after every launch, as is currently the case. This paper will present a summary of the results from industry surveys, trade studies, life cycle cost analysis, and preliminary testing that have been performed to support and validate the development, testing, and qualification of new refractory materials