On the density of sets of the Euclidean plane avoiding distance 1

A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number $m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb R^2)$ represents the highest proportion of the plane that can be filled by a set avoiding distance 1. This parameter is related to the fractional chromatic number $\chi_f(\mathbb R^2)$ of the plane. We establish that $m_1(\mathbb R^2) \leq 0.25646$ and $\chi_f(\mathbb R^2) \geq 3.8992$.Comment: 11 pages, 5 figure

On the theta number of powers of cycle graphs

We give a closed formula for Lovasz theta number of the powers of cycle graphs and of their complements, the circular complete graphs. As a consequence, we establish that the circular chromatic number of a circular perfect graph is computable in polynomial time. We also derive an asymptotic estimate for this theta number.Comment: 17 page

On the density of sets avoiding parallelohedron distance 1

The maximal density of a measurable subset of R^n avoiding Euclidean distance1 is unknown except in the trivial case of dimension 1. In this paper, we consider thecase of a distance associated to a polytope that tiles space, where it is likely that the setsavoiding distance 1 are of maximal density 2^-n, as conjectured by Bachoc and Robins. We prove that this is true for n = 2, and for the Vorono\"i regions of the lattices An, n >= 2

Partitionable graphs arising from near-factorizations of finite groups

AbstractIn 1979, two constructions for making partitionable graphs were introduced in (by Chvátal et al. (Ann. Discrete Math. 21 (1984) 197)). The graphs produced by the second construction are called CGPW graphs. A near-factorization (A,B) of a finite group is roughly speaking a non-trivial factorization of G minus one element into two subsets A and B. Every CGPW graph with n vertices turns out to be a Cayley graph of the cyclic group Zn, with connection set (A−A)⧹{0}, for a near-factorization (A,B) of Zn. Since a counter-example to the Strong Perfect Graph Conjecture would be a partitionable graph (Padberg, Math. Programming 6 (1974) 180), any ‘new’ construction for making partitionable graphs is of interest. In this paper, we investigate the near-factorizations of finite groups in general, and their associated Cayley graphs which are all partitionable. In particular, we show that near-factorizations of the dihedral groups produce every CGPW graph of even order. We present some results about near-factorizations of finite groups which imply that a finite abelian group with a near-factorization (A,B) such that |A|⩽4 must be cyclic (already proved by De Caen et al. (Ars Combin. 29 (1990) 53)). One of these results may be used to speed up exhaustive calculations. At last, we prove that there is no counter-example to the Strong Perfect Graph Conjecture arising from near-factorizations of a finite abelian group of even order

How unique is Lovász's theta function?

International audienceThe famous Lovász's ϑ function is computable in polynomial time for every graph, as a semi-definite program (Grötschel, Lovász and Schrijver, 1981). The chromatic number and the clique number of every perfect graph G are computable in polynomial time. Despite numerous efforts since the last three decades, stimulated by the Strong Perfect Graph Theo-rem (Chudnovsky, Robertson, Seymour and Thomas, 2006), no combinatorial proof of this result is known. In this work, we try to understand why the "key properties" of Lovász's ϑ function make it so "unique". We introduce an infinite set of convex functions, which includes the clique number ω and ϑ . This set includes a sequence of linear programs which are monotone increasing and converging to ϑ . We provide some evidences that ϑ is the unique function in this setting allowing to compute the chromatic number of perfect graphs in polynomial time

Exhumation, crustal deformation, and thermal structure of the Nepal Himalaya derived from the inversion of thermochronological and thermobarometric data and modeling of the topography

Two end‐member kinematic models of crustal shortening across the Himalaya are currently debated: one assumes localized thrusting along a single major thrust fault, the Main Himalayan Thrust (MHT) with nonuniform underplating due to duplexing, and the other advocates for out‐of‐sequence (OOS) thrusting in addition to thrusting along the MHT and underplating. We assess these two models based on the modeling of thermochronological, thermometric, and thermobarometric data from the central Nepal Himalaya. We complement a data set compiled from the literature with 114 ^(40)Ar/^(39)Ar, 10 apatite fission track, and 5 zircon (U‐Th)/He thermochronological data. The data are predicted using a thermokinematic model (PECUBE), and the model parameters are constrained using an inverse approach based on the Neighborhood Algorithm. The model parameters include geometric characteristics as well as overthrusting rates, radiogenic heat production in the High Himalayan Crystalline (HHC) sequence, the age of initiation of the duplex or of out-of-sequence thrusting. Both models can provide a satisfactory fit to the inverted data. However, the model with out-of-sequence thrusting implies an unrealistic convergence rate ≥30 mm yr^(−1). The out-of-sequence thrust model can be adjusted to fit the convergence rate and the thermochronological data if the Main Central Thrust zone is assigned a constant geometry and a dip angle of about 30° and a slip rate of <1 mm yr^(−1). In the duplex model, the 20 mm yr^(−1) convergence rate is partitioned between an overthrusting rate of 5.8 ± 1.4 mm yr^(−1) and an underthrusting rate of 14.2 ± 1.8 mm yr^(−1). Modern rock uplift rates are estimated to increase from about 0.9 ± 0.31 mm yr^(−1) in the Lesser Himalaya to 3.0 ± 0.9 mm yr^(−1) at the front of the high range, 86 ± 13 km from the Main Frontal Thrust. The effective friction coefficient is estimated to be 0.07 or smaller, and the radiogenic heat production of HHC units is estimated to be 2.2 ± 0.1 µWm^(−3). The midcrustal duplex initiated at 9.8 ± 1.7 Ma, leading to an increase of uplift rate at front of the High Himalaya from 0.9 ± 0.31 to 3.05 ± 0.9 mm yr^(−1). We also run 3-D models by coupling PECUBE with a landscape evolution model (CASCADE). This modeling shows that the effect of the evolving topography can explain a fraction of the scatter observed in the data but not all of it, suggesting that lateral variations of the kinematics of crustal deformation and exhumation are likely. It has been argued that the steep physiographic transition at the foot of the Greater Himalayan Sequence indicates OOS thrusting, but our results demonstrate that the best fit duplex model derived from the thermochronological and thermobarometric data reproduces the present morphology of the Nepal Himalaya equally well

Consecutive ones matrices for multi-dimensional orthogonal packing problems

International audienceThe multi-dimensional orthogonal packing problem (OPP) is a well studied decisional problem. Given a set of items with rectangular shapes, the problem is to decide whether there is a non-overlapping packing of these items in a rectangular bin. The rotation of items is not allowed. A powerful caracterization of packing configurations by means of interval graphs was recently introduced. In this paper, we propose a new algorithm using consecutive ones matrices as data structure. This new algorithm is then used to solve the two-dimensional orthogonal knapsack problem. Computational results are reported, which show its effectiveness

On the density of sets of the Euclidean plane avoiding distance

International audienc

Presence and geodynamic significance of Cambro-Ordovician series of SE Karakaram (N. Pakistan)

New geological, geochemical and geochronological data from the Southern Karakoram (NE Pakistan) indicate the presence of several unexpectedly old and well preserved units along the Asian margin: (1) a Precambrian basement, displaying a minimum amphibole Ar-Ar age of 651 Ma; (2) a thick Cambro-Ordovician platform-type sedimentary unit overlying the Precam-brian basement. These series are dated by graptolite and crinoid faunas, and are confirmed by concordant 87Sr/86Sr and 13C “ages” of the marbles; (3) a dismembered ophiolitic series formed by slices of metagabbros and metabasalts separated by ultramafic lenses (the Masherbrum Greenstone Complex). The occurrence of such Cambro-Ordovician series overlying a Precambrian basement in south-eastern Karakoram similar to the south-western Karakoram shows that the Karakoram constitutes a continuous tectonic block. The petrology and geochemistry of the Masherbrum Greenstone Complex (mineral chemistry, major and trace element and Sr-Nd isotopic data) are indicative of a supra-subductive environment. The presence of LREE-enriched calc-alkaline rocks [(La/Yb)N = 4.45.6; (Nb/La)N = 0.2-0.3; eNd565 = 5.1-7.1] and LREE-depleted tholeiitic rocks [(La/Yb)N =0.5-1.3; (Nb/La)N = 0.6-0.9; eNd565 = 5.6-7.8] are consistent with arc and back-arc settings, respectively. A high-Mg andesitic dolerite and an OIB-type metabasalt, with lower eNd ratios (eNd565 = 0.5 and 4.5) are in accordance with source heterogeneity beneath the arc. The Masherbrum Greenstone Complex, along with other Cambro-Ordovician central-eastern volcanic series give evidence of a tectonic situation governed by micro-plate convergent-divergent systems with occurrence of arc - back-arc settings during the Lower Palaeozoic, comparable to that of the current SW Pacific area

Thermokinematic evolution of the Annapurna-Dhaulagiri Himalaya, central Nepal: The composite orogenic system

The Himalayan orogen represents a ‘‘Composite Orogenic System’’ in which channel flow, wedge extrusion, and thrust stacking operate in separate ‘‘Orogenic Domains’’ with distinct rheologies and crustal positions. We analyze 104 samples from the metamorphic core (Greater Himalayan Sequence, GHS) and bounding units of the Annapurna-Dhaulagiri Himalaya, central Nepal. Optical microscopy and electron backscatter diffraction (EBSD) analyses provide a record of deformation microstructures and an indication of active crystal slip systems, strain geometries, and deformation temperatures. These data, combined with existing thermobarometry and geochronology data are used to construct detailed deformation temperature profiles for the GHS. The profiles define a three-stage thermokinematic evolution from midcrustal channel flow (Stage 1, >7008C to 550–6508C), to rigid wedge extrusion (Stage 2, 400–6008C) and duplexing (Stage 3, <280–4008C). These tectonic processes are not mutually exclusive, but are confined to separate rheologically distinct Orogenic Domains that form the modular components of a Composite Orogenic System. These Orogenic Domains may be active at the same time at different depths/positions within the orogen. The thermokinematic evolution of the Annapurna-Dhaulagiri Himalaya describes the migration of the GHS through these Orogenic Domains and reflects the spatial and temporal variability in rheological boundary conditions that govern orogenic systems