Cycles in the burnt pancake graphs

The pancake graph $P_n$ is the Cayley graph of the symmetric group $S_n$ on $n$ elements generated by prefix reversals. $P_n$ has been shown to have properties that makes it a useful network scheme for parallel processors. For example, it is $(n-1)$-regular, vertex-transitive, and one can embed cycles in it of length $\ell$ with $6\leq\ell\leq n!$. The burnt pancake graph $BP_n$, which is the Cayley graph of the group of signed permutations $B_n$ using prefix reversals as generators, has similar properties. Indeed, $BP_n$ is $n$-regular and vertex-transitive. In this paper, we show that $BP_n$ has every cycle of length $\ell$ with $8\leq\ell\leq 2^n n!$. The proof given is a constructive one that utilizes the recursive structure of $BP_n$. We also present a complete characterization of all the $8$-cycles in $BP_n$ for $n \geq 2$, which are the smallest cycles embeddable in $BP_n$, by presenting their canonical forms as products of the prefix reversal generators.Comment: Added a reference, clarified some definitions, fixed some typos. 42 pages, 9 figures, 20 pages of appendice

A detailed view of filaments and sheets in the warm-hot intergalactic medium. I. Pancake formation

Numerical simulations predict a considerable fraction of the missing baryons at redshift z ~ 0 resting in the so called warm-hot intergalactic medium (WHIM). The filaments and sheets of the WHIM have high temperatures 10^5 - 10^7 K) and a high degree of ionization while having only low to intermediate densities. The particular physical conditions of the WHIM structures, e.g. density and temperature profiles, velocity fields, are expected to leave their special imprint on spectroscopic observations. In order to get further insight into these conditions, we perform hydrodynamical simulations of the WHIM. Instead of analyzing large simulations of cosmological structure formation, we simulate particular well-defined structures and study the impact of different physical processes as well as of the scale dependencies. We start with the comprehensive study of the one-dimensional collapse (pancake) and examine the influence of radiative cooling, heating due to an UV background, and thermal conduction. We investigate the effect of small scale perturbations given according to the initial cosmological power spectrum. If the initial perturbation length scale L exceeds ~ 2 Mpc the collapse leads to shock confined structures. As a result of radiative cooling and of heating due to an UV background a relatively cold and dense core forms in the one-dimensional case. The properties of the core (extension, density, and temperature) are correlated with L. For larger L the core sizes are more concentrated. Thermal conduction enhances this trend and may even result in an evaporation of the core. Our estimates predict that a core may start to evaporate for perturbation lengths larger than L ~ 30 Mpc. The obtained detailed profiles for density and temperature for prototype WHIM structures allow for the determination of possible spectral signatures by the WHIM.Comment: 14 pages, 9 figures, accepted for publication in A&

VideoGraph: Recognizing Minutes-Long Human Activities in Videos

Many human activities take minutes to unfold. To represent them, related works opt for statistical pooling, which neglects the temporal structure. Others opt for convolutional methods, as CNN and Non-Local. While successful in learning temporal concepts, they are short of modeling minutes-long temporal dependencies. We propose VideoGraph, a method to achieve the best of two worlds: represent minutes-long human activities and learn their underlying temporal structure. VideoGraph learns a graph-based representation for human activities. The graph, its nodes and edges are learned entirely from video datasets, making VideoGraph applicable to problems without node-level annotation. The result is improvements over related works on benchmarks: Epic-Kitchen and Breakfast. Besides, we demonstrate that VideoGraph is able to learn the temporal structure of human activities in minutes-long videos

First-Order Melting and Dynamics of Flux Lines in a Model for YBa2_2Cu3_3O7−δ_{7-\delta}

We have studied the statics and dynamics of flux lines in a model for YBCO, using both Monte Carlo simulations and Langevin dynamics. For a clean system, both approaches yield the same melting curve, which is found to be weakly first order with a heat of fusion of about $0.02 k_BT_m$ per vortex pancake at a field of $50 {\rm kG}.$ The time averaged magnetic field distribution experienced by a fixed spin is found to undergo a qualitative change at freezing, in agreement with NMR and $\mu {\rm SR}$ experiments. Melting in the clean system is accompanied by a proliferation of free disclinations which show a clear B-dependent 3D-2D crossover from long disclination lines parallel to the c-axis at low fields, to 2D ``pancake'' disclinations at higher fields. Strong point pins produce a logarithmical $\ln t$ relaxation which results from slow annealing out of disclinations in disordered samples.Comment: 31 pages, latex, revtex, 12 figures available upon request, No major changes to the original text, but some errors in the axes scale for Figures 6 and 7 were corrected(new figures available upon request), to be published in Physical Review B, July 199

Continuum mechanics at nanoscale. A tool to study trees' watering and recovery

The cohesion-tension theory expounds the crude sap ascent thanks to the negative pressure generated by evaporation of water from leaves. Nevertheless, trees pose multiple challenges and seem to live in unphysical conditions: the negative pressure increases cavitation; it is possible to obtain a water equilibrium between connected parts where one is at a positive pressure and the other one is at negative pressure; no theory is able to satisfactorily account for the refilling of vessels after embolism events. A theoretical form of our paper in the Journal of Theoretical Biology is proposed together with new results: a continuum mechanics model of the disjoining pressure concept refers to the Derjaguin School of physical chemistry. A comparison between liquid behaviour both in tight-filled microtubes and in liquid thin-films is offered when the pressure is negative in liquid bulks and is positive in liquid thin-films and vapour bulks. In embolized xylem microtubes, when the air-vapour pocket pressure is greater than the air-vapour bulk pressure, a refilling flow occurs between the air-vapour domains to empty the air-vapour pockets although the liquid-bulk pressure remains negative. The model has a limit of validity taking the maximal size of trees into account. These results drop inkling that the disjoining pressure is an efficient tool to study biological liquids in contact with substrates at a nanoscale range.Comment: The paper is a review and overlap of my different papers about the watering of trees as a mathematical development of my paper in The Journal of Theoretical Biology. These results are presented together with new researches: transfer of liquid water and vapour between xylem microtubes, an explanation of ultrasounds generated in the watering network considered as sound pipes, numerical calculations of flows in thin liquid films and of Poiseuille flows in xylem microtubes, an estimation of the velocity for the ascent of crude sap and of the recovery time of trees during the spring perio

Cosmic bubble and domain wall instabilities II: Fracturing of colliding walls

We study collisions between nearly planar domain walls including the effects of small initial nonplanar fluctuations. These perturbations represent the small fluctuations that must exist in a quantum treatment of the problem. In a previous paper, we demonstrated that at the linear level a subset of these fluctuations experience parametric amplification as a result of their coupling to the planar symmetric background. Here we study the full three-dimensional nonlinear dynamics using lattice simulations, including both the early time regime when the fluctuations are well described by linear perturbation theory as well as the subsequent stage of fully nonlinear evolution. We find that the nonplanar fluctuations have a dramatic effect on the overall evolution of the system. Specifically, once these fluctuations begin to interact nonlinearly the split into a planar symmetric part of the field and the nonplanar fluctuations loses its utility. At this point the colliding domain walls dissolve, with the endpoint of this being the creation of a population of oscillons in the collision region. The original (nearly) planar symmetry has been completely destroyed at this point and an accurate study of the system requires the full three-dimensional simulation.Comment: 23 pages + references, 13 figures. Submitted to JCAP. v2: Acknowledgements updated, no other change