Evolution of high-order connected components in random hypergraphs

In this paper we considerj-tuple-connected components in randomk-uniform hypergraphs (thej-tuple-connectedness relation can be defined by letting twoj-sets be connected if they lie in a common edge and considering the transitive closure; the casej= 1 corresponds to the common notion of vertex-connectedness). We show that the existence of aj-tuple-connected component containing Θ(nj)j-sets undergoes a phase transition and show that the threshold occurs at edge probability$$\frac{(k-j)!}{\binom{k}{j}-1}n^{j-k}.$$Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph.Our main original contribution is abounded degree lemma, which controls the structure of the component grown in the search process.

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The Size of the Giant Component in Random Hypergraphs: a Short Proof

The Electronic Journal of Combinatorics ◽

10.37236/7712 ◽

2019 ◽

Vol 26 (3) ◽

Author(s):

Oliver Cooley ◽

Mihyun Kang ◽

Christoph Koch

Keyword(s):

High Probability ◽

Short Proof ◽

The Other ◽

Connected Components ◽

Search Process ◽

Uniform Hypergraph ◽

Breadth First Search ◽

Asymptotic Size ◽

Uniform Hypergraphs ◽

Random Hypergraphs

We consider connected components in $k$-uniform hypergraphs for the following notion of connectedness: given integers $k\ge 2$ and $1\le j \le k-1$, two $j$-sets (of vertices) lie in the same $j$-component if there is a sequence of edges from one to the other such that consecutive edges intersect in at least $j$ vertices.We prove that certain collections of $j$-sets constructed during a breadth-first search process on $j$-components in a random $k$-uniform hypergraph are reasonably regularly distributed with high probability. We use this property to provide a short proof of the asymptotic size of the giant $j$-component shortly after it appears.

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Subcritical random hypergraphs, high-order components, and hypertrees

2019 Proceedings of the Sixteenth Workshop on Analytic Algorithmics and Combinatorics (ANALCO) ◽

10.1137/1.9781611975505.12 ◽

2019 ◽

pp. 111-118 ◽

Cited By ~ 1

Author(s):

Oliver Cooley ◽

Wenjie Fang ◽

Nicola Del Giudice ◽

Mihyun Kang

Keyword(s):

High Order ◽

Order Components ◽

Random Hypergraphs

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Decision letter for "A crabs’ high‐order brain center resolved as a mushroom body‐like structure"

10.1002/cne.24960/v2/decision1 ◽

2020 ◽

Keyword(s):

Mushroom Body ◽

High Order ◽

Brain Center

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Review for "A crabs’ high‐order brain center resolved as a mushroom body‐like structure"

10.1002/cne.24960/v1/review1 ◽

2020 ◽

Keyword(s):

Mushroom Body ◽

High Order ◽

Brain Center

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Determination of the Burgers vector of a dislocation in a Fe-Mn alloy by weak-beam image in HVEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100108799 ◽

1978 ◽

Vol 36 (1) ◽

pp. 330-331

Author(s):

Y. Ishida ◽

H. Ishida ◽

K. Kohra ◽

H. Ichinose

Keyword(s):

High Order ◽

Simulation Technique ◽

The Other ◽

Image Simulation ◽

Diffraction Vector ◽

Burgers Vector ◽

Weak Beam ◽

Beam Image ◽

Edge Component

IntroductionA simple and accurate technique to determine the Burgers vector of a dislocation has become feasible with the advent of HVEM. The conventional image vanishing technique(1) using Bragg conditions with the diffraction vector perpendicular to the Burgers vector suffers from various drawbacks; The dislocation image appears even when the g.b = 0 criterion is satisfied, if the edge component of the dislocation is large. On the other hand, the image disappears for certain high order diffractions even when g.b ≠ 0. Furthermore, the determination of the magnitude of the Burgers vector is not easy with the criterion. Recent image simulation technique is free from the ambiguities but require too many parameters for the computation. The weak-beam “fringe counting” technique investigated in the present study is immune from the problems. Even the magnitude of the Burgers vector is determined from the number of the terminating thickness fringes at the exit of the dislocation in wedge shaped foil surfaces.

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The usefulness of high index low symmetry zone axes for holz line analysis

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100126718 ◽

1987 ◽

Vol 45 ◽

pp. 384-385

Author(s):

C. M. Sung ◽

D. B. Williams

Keyword(s):

Lattice Parameter ◽

High Order ◽

Zone Axis ◽

High Index ◽

High Symmetry ◽

Second Phases ◽

Line Patterns ◽

Low Symmetry ◽

Line Analysis

Researchers have tended to use high symmetry zone axes (e.g. <111> <114>) for High Order Laue Zone (HOLZ) line analysis since Jones et al reported the origin of HOLZ lines and described some of their applications. But it is not always easy to find HOLZ lines from a specific high symmetry zone axis during microscope operation, especially from second phases on a scale of tens of nanometers. Therefore it would be very convenient if we can use HOLZ lines from low symmetry zone axes and simulate these patterns in order to measure lattice parameter changes through HOLZ line shifts. HOLZ patterns of high index low symmetry zone axes are shown in Fig. 1, which were obtained from pure Al at -186°C using a double tilt cooling holder. Their corresponding simulated HOLZ line patterns are shown along with ten other low symmetry orientations in Fig. 2. The simulations were based upon kinematical diffraction conditions.

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Are HOLZ lines kinematic in off-zone-axis orientation?

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100149295 ◽

1993 ◽

Vol 51 ◽

pp. 692-693

Author(s):

J. M. Zuo ◽

A. L. Weickenmeier ◽

R. Holmestad ◽

J. C. H. Spence

Keyword(s):

Structure Determination ◽

Standard Procedure ◽

High Order ◽

Zone Axis ◽

Great Promise ◽

Line Position ◽

Measured Intensity ◽

Constant Intensity ◽

Axis Orientation ◽

Diffraction Condition

The application of high order reflections in a weak diffraction condition off the zone axis center, including those in high order laue zones (HOLZ), holds great promise for structure determination using convergent beam electron diffraction (CBED). It is believed that in this case the intensities of high order reflections are kinematic or two-beam like. Hence, the measured intensity can be related to the structure factor amplitude. Then the standard procedure of structure determination in crystallography may be used for solving unknown structures. The dynamic effect on HOLZ line position and intensity in a strongly diffracting zone axis is well known. In a weak diffraction condition, the HOLZ line position may be approximated by the kinematic position, however, it is not clear whether this is also true for HOLZ intensities. The HOLZ lines, as they appear in CBED patterns, do show strong intensity variations along the line especially near the crossing of two lines, rather than constant intensity along the Bragg condition as predicted by kinematic or two beam theory.

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