ABpercolation on plane triangulations is unimodal

Martin J. B. Appel

doi:10.2307/3215246

AB percolation on plane triangulations is unimodal

Journal of Applied Probability ◽

10.1017/s0021900200107442 ◽

1994 ◽

Vol 31 (01) ◽

pp. 193-204

Author(s):

Martin J. B. Appel

Keyword(s):

Probability Function ◽

Product Measure ◽

Bounded Region ◽

Infinite Path ◽

Percolation Probability ◽

Vertex Set ◽

Alternate States

Let ℱ be a countable plane triangulation embedded in ℝ2in such a way that no bounded region contains more than finitely many vertices, and letPpbe Bernoulli (p) product measure on the vertex set of ℱ. LetEbe the event that a fixed vertex belongs to an infinite path whose vertices alternate states sequentially. It is shown that theAB percolation probability function θΑΒ(p) =Pp(E) is non-decreasing inpfor 0 ≦p≦ ½. By symmetry,θAΒ(p) is therefore unimodal on [0, 1]. This result partially verifies a conjecture due to Halley and stands in contrast to the examples of Łuczak and Wierman.

On the continuity of the percolation probability function

Conference on Modern Analysis and Probability - Contemporary Mathematics ◽

10.1090/conm/026/737388 ◽

1984 ◽

pp. 61-65 ◽

Cited By ~ 19

Author(s):

J. van den Berg ◽

M. Keane

Keyword(s):

Probability Function ◽

Percolation Probability

Random transceiver networks

Advances in Applied Probability ◽

10.1017/s0001867800003311 ◽

2009 ◽

Vol 41 (02) ◽

pp. 323-343 ◽

Cited By ~ 1

Author(s):

Paul Balister ◽

Béla Bollobás ◽

Mark Walters

Keyword(s):

Large Distance ◽

Two Dimensions ◽

Critical Area ◽

Bounded Region ◽

Directed Path ◽

Expected Number ◽

Fixed Angle ◽

Starting Point ◽

Vertex Set ◽

Fixed Distribution

Consider randomly scattered radio transceivers in ℝ d , each of which can transmit signals to all transceivers in a given randomly chosen region about itself. If a signal is retransmitted by every transceiver that receives it, under what circumstances will a signal propagate to a large distance from its starting point? Put more formally, place points {x i } in ℝ d according to a Poisson process with intensity 1. Then, independently for each x i , choose a bounded region A x i from some fixed distribution and let be the random directed graph with vertex set whenever x j ∈ x i + A x i . We show that, for any will almost surely have an infinite directed path, provided the expected number of transceivers that can receive a signal directly from x i is at least 1 + η, and the regions x i + A x i do not overlap too much (in a sense that we shall make precise). One example where these conditions hold, and so gives rise to percolation, is in ℝ d , with each A x i a ball of volume 1 + η centred at x i , where η → 0 as d → ∞. Another example is in two dimensions, where the A x i are sectors of angle ε γ and area 1 + η, uniformly randomly oriented within a fixed angle (1 + ε)θ. In this case we can let η → 0 as ε → 0 and still obtain percolation. The result is already known for the annulus, i.e. that the critical area tends to 1 as the ratio of the radii tends to 1, while it is known to be false for the square (l∞) annulus. Our results show that it does however hold for the randomly oriented square annulus.

Random transceiver networks

Advances in Applied Probability ◽

10.1239/aap/1246886613 ◽

2009 ◽

Vol 41 (2) ◽

pp. 323-343 ◽

Cited By ~ 1

Author(s):

Paul Balister ◽

Béla Bollobás ◽

Mark Walters

Keyword(s):

Large Distance ◽

Two Dimensions ◽

Critical Area ◽

Bounded Region ◽

Directed Path ◽

Expected Number ◽

Fixed Angle ◽

Starting Point ◽

Vertex Set ◽

Fixed Distribution

Consider randomly scattered radio transceivers in ℝd, each of which can transmit signals to all transceivers in a given randomly chosen region about itself. If a signal is retransmitted by every transceiver that receives it, under what circumstances will a signal propagate to a large distance from its starting point? Put more formally, place points {xi} in ℝd according to a Poisson process with intensity 1. Then, independently for each xi, choose a bounded region Axi from some fixed distribution and let be the random directed graph with vertex set whenever xj ∈ xi + Axi. We show that, for any will almost surely have an infinite directed path, provided the expected number of transceivers that can receive a signal directly from xi is at least 1 + η, and the regions xi + Axi do not overlap too much (in a sense that we shall make precise). One example where these conditions hold, and so gives rise to percolation, is in ℝd, with each Axi a ball of volume 1 + η centred at xi, where η → 0 as d → ∞. Another example is in two dimensions, where the Axi are sectors of angle ε γ and area 1 + η, uniformly randomly oriented within a fixed angle (1 + ε)θ. In this case we can let η → 0 as ε → 0 and still obtain percolation. The result is already known for the annulus, i.e. that the critical area tends to 1 as the ratio of the radii tends to 1, while it is known to be false for the square (l∞) annulus. Our results show that it does however hold for the randomly oriented square annulus.

Density of Monochromatic Infinite Paths

The Electronic Journal of Combinatorics ◽

10.37236/7758 ◽

2018 ◽

Vol 25 (4) ◽

Author(s):

Allan Lo ◽

Nicolás Sanhueza-Matamala ◽

Guanghui Wang

Keyword(s):

Complete Graph ◽

Infinite Path ◽

Vertex Set ◽

Upper Density ◽

Edge Colouring

For any subset $A \subseteq \mathbb{N}$, we define its upper density to be $\limsup_{ n \rightarrow \infty } |A \cap \{ 1, \dotsc, n \}| / n$. We prove that every $2$-edge-colouring of the complete graph on $\mathbb{N}$ contains a monochromatic infinite path, whose vertex set has upper density at least $(9 + \sqrt{17})/16 \approx 0.82019$. This improves on results of Erdős and Galvin, and of DeBiasio and McKenney.

Almost empty polygons

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.40.2003.3.1 ◽

2003 ◽

Vol 40 (3) ◽

pp. 269-286 ◽

Cited By ~ 4

Author(s):

H. Nyklová

Keyword(s):

Exact Results ◽

Point Sets ◽

Planar Point ◽

Convex Position ◽

Vertex Set ◽

Point Set ◽

Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

Parallel Residues

10.23943/princeton/9780691166902.003.0021 ◽

2017 ◽

Author(s):

Bernhard M¨uhlherr ◽

Holger P. Petersson ◽

Richard M. Weiss

Keyword(s):

Distance Function ◽

Length Function ◽

Coxeter System ◽

Vertex Set ◽

Ordered Pairs

This chapter considers the notion of parallel residues in a building. It begins with the assumption that Δ‎ is a building of type Π‎, which is arbitrary except in a few places where it is explicitly assumed to be spherical. Δ‎ is not assumed to be thick. The chapter then elaborates on a hypothesis which states that S is the vertex set of Π‎, (W, S) is the corresponding Coxeter system, d is the W-distance function on the set of ordered pairs of chambers of Δ‎, and ℓ is the length function on (W, S). It also presents a notation in which the type of a residue R is denoted by Typ(R) and concludes with the condition that residues R and T of a building will be called parallel if R = projR(T) and T = projT(R).

Random Forest Refinement of Pairwise Potentials for Protein-ligand Decoy Detection

10.26434/chemrxiv.8047820.v1 ◽

2019 ◽

Cited By ~ 1

Author(s):

Jun Pei ◽

Zheng Zheng ◽

Hyunji Kim ◽

Lin Song ◽

Sarah Walworth ◽

...

Keyword(s):

Machine Learning ◽

Random Forest ◽

Probability Function ◽

Pair Potential ◽

Scoring Function ◽

Stable Structure ◽

Scoring Functions ◽

Atom Pair ◽

Data Set ◽

Atom Pairs

An accurate scoring function is expected to correctly select the most stable structure from a set of pose candidates. One can hypothesize that a scoring function’s ability to identify the most stable structure might be improved by emphasizing the most relevant atom pairwise interactions. However, it is hard to evaluate the relevant importance for each atom pair using traditional means. With the introduction of machine learning methods, it has become possible to determine the relative importance for each atom pair present in a scoring function. In this work, we use the Random Forest (RF) method to refine a pair potential developed by our laboratory (GARF6) by identifying relevant atom pairs that optimize the performance of the potential on our given task. Our goal is to construct a machine learning (ML) model that can accurately differentiate the native ligand binding pose from candidate poses using a potential refined by RF optimization. We successfully constructed RF models on an unbalanced data set with the ‘comparison’ concept and, the resultant RF models were tested on CASF-2013.5 In a comparison of the performance of our RF models against 29 scoring functions, we found our models outperformed the other scoring functions in predicting the native pose. In addition, we used two artificial designed potential models to address the importance of the GARF potential in the RF models: (1) a scrambled probability function set, which was obtained by mixing up atom pairs and probability functions in GARF, and (2) a uniform probability function set, which share the same peak positions with GARF but have fixed peak heights. The results of accuracy comparison from RF models based on the scrambled, uniform, and original GARF potential clearly showed that the peak positions in the GARF potential are important while the well depths are not. <br>

Valued Field Graph and Some Related Parameters

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.1.30 ◽

2020 ◽

pp. 389-395

Author(s):

G. Suresh Singh ◽

P. K. Prasobha

Keyword(s):

Finite Field ◽

Independence Number ◽

Valued Field ◽

Vertex Set ◽

Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

Conservation and Restoration of Grasslands

10.1093/oso/9780198744511.003.0008 ◽

2018 ◽

Author(s):

Brian J. Wilsey

Keyword(s):

Community Assembly ◽

Woody Plants ◽

Woody Plant ◽

Priority Effects ◽

Woody Plant Encroachment ◽

Humpty Dumpty ◽

Alternate States ◽

Density And Biomass ◽

Acid Test ◽

Stocking Rates

Conservation programs alter herbivore stocking rates and find and protect the remaining areas that have not been plowed or converted to crops. Restoration is an ‘Acid Test’ for ecology. If we fully understand how grassland systems function and assemble after disturbance, then it should be easy to restore them after they have been degraded or destroyed. Alternatively, the idea that restorations will not be equivalent to remnants has been termed the ‘Humpty Dumpty’ hypothesis—once lost, it cannot be put back together again. Community assembly may follow rules, and if these rules are uncovered, then we may be able to accurately predict final species composition after assembly. Priority effects are sometimes found depending on species arrival orders, and they can result in alternate states. Woody plant encroachment is the increase in density and biomass of woody plants, and it is strongly affecting grassland C and water cycles.