RANDOM GEOMETRIC GRAPHS AND ISOMETRIES OF NORMED SPACES

Given a countable dense subset S of a finite-dimensional normed space X, and 0 \u3c p \u3c 1, we form a random graph on S by joining, independently and with probability p, each pair of points at distance less than 1. We say that S is Rado if any two such random graphs are (almost surely) isomorphic. Bonato and Janssen showed that in ℓd∞ almost all S are Rado. Our main aim in this paper is to show that ℓd∞ is the unique normed space with this property: indeed, in every other space almost all sets S are non-Rado. We also determine which spaces admit some Rado set: this turns out to be the spaces that have an ℓ∞ direct summand. These results answer questions of Bonato and Janssen. A key role is played by the determination of which finite-dimensional normed spaces have the property that every bijective step-isometry (meaning that the integer part of distances is preserved) is in fact an isometry. This result may be of independent interest

Strict inequalities of critical values in continuum percolation

We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\rm site} > p_c^{\rm bond}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) {\em strictly} reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.Comment: 38 pages, 8 figure

Braided racks, Hurwitz actions and Nichols algebras with many cubic relations

We classify Nichols algebras of irreducible Yetter-Drinfeld modules over groups such that the underlying rack is braided and the homogeneous component of degree three of the Nichols algebra satisfies a given inequality. This assumption turns out to be equivalent to a factorization assumption on the Hilbert series. Besides the known Nichols algebras we obtain a new example. Our method is based on a combinatorial invariant of the Hurwitz orbits with respect to the action of the braid group on three strands.Comment: v2: 35 pages, 6 tables, 14 figure

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Numerical simulations of stellar SiO maser variability. Investigation of the effect of shocks

A stellar hydrodynamic pulsation model has been combined with a SiO maser model in an attempt to calculate the temporal variability of SiO maser emission in the circumstellar envelope (CE) of a model AGB star. This study investigates whether the variations in local physical conditions brought about by shocks are the predominant contributing factor to SiO maser variability because, in this work, the radiative part of the pump is constant. We find that some aspects of the variability are not consistent with a pump provided by shock-enhanced collisions alone. In these simulations, gas parcels of relatively enhanced SiO abundance are distributed in a model CE by a Monte Carlo method, at a single epoch of the stellar cycle. From this epoch on, Lagrangian motions of individual parcels are calculated according to the velocity fields encountered in the model CE during the stellar pulsation cycle. The potentially masing gas parcels therefore experience different densities and temperatures, and have varying line-of-sight velocity gradients throughout the stellar cycle, which may or may not be suitable to produce maser emission. At each epoch (separated by 16.6 days), emission lines from the parcels are combined to produce synthetic spectra and VLBI-type images. We report here the results for v=1, J=1-0 (43-GHz) and J=2-1 (86-GHz) masers.Comment: 16 pages, 8 figures, accepted by A&

Sampling-based Algorithms for Optimal Motion Planning

During the last decade, sampling-based path planning algorithms, such as Probabilistic RoadMaps (PRM) and Rapidly-exploring Random Trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g., as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g., showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e., such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.Comment: 76 pages, 26 figures, to appear in International Journal of Robotics Researc

The Very Slow Wind From the Pulsating Semiregular Red Giant L2 Pup

We have obtained 11.7 and 17.9 micron images at the Keck I telescope of the circumstellar dust emission from L2 Pup, one of the nearest (D = 61 pc) mass-losing, pulsating, red giants that has a substantial infrared excess. We propose that the wind may be driven by the stellar pulsations with radiation pressure on dust being relatively unimportant, as described in some recent calculations. L2 Pup may serve as the prototype of this phase of stellar evolution where it could lose about 15% of its initial main sequence mass.Comment: ApJ, in pres

Subtended angles

The first is partially supported by NSF grant DMS 1301614. The second author is partially supported by NSF grant DMS 1301614 and MULTIPLEX no. 317532. The third author’s research supported in part by the Hungarian National Science Foundation OTKA 104343, by the Simons Foundation Collaboration Grant #317487, and by the European Research Council Advanced Investigators Grant 267195

Noise Sensitivity in Continuum Percolation

We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first for which the critical probability p_c \ne 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p_c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.Comment: 42 page