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We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on $n$ vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least $s$. We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants $0B\sqrt{n}\log{n}$. This is a factor of $\Theta(\sqrt{\log{n}})$ larger than the lower bound coming from the off-diagonal Ramsey number $r(3,s)$.

The preeminent target of present study is to reveal the speed characteristic of ongoing outbreak COVID-19 due to novel coronavirus. On January 2020, the novel coronavirus infection (COVID-19) detected in India, and the total statistic of cases continuously increased to 7 128 268 cases including 109 285 deceases to October 2020, where 860 601 cases are active in India. In this study, we use the Hermite wavelets basis in order to solve the COVID-19 model with time- arbitrary Caputo derivative. The discussed framework is based upon Hermite wavelets. The operational matrix incorporated with the collocation scheme is used in order to transform arbitrary-order problem into algebraic equations. The corrector scheme is also used for solving the COVID-19 model for distinct value of arbitrary order. Also, authors have investigated the various behaviors of the arbitrary-order COVID-19 system and procured developments are matched with exiting developments by various techniques. The various illustrations of susceptible, exposed, infected, and recovered individuals are given for its behaviors at the various value of fractional order. In addition, the proposed model has been also supported by some numerical simulations and wavelet-based results.

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Flax fiber and oilseed use is increasing in textiles, composites, paper/pulp and industrial/nutritional oil sectors in the U.S. As a winter crop, flax fits well into double-cropping systems in the southeastern U.S. Flax can be harvested as an early crop for fiber, with attached immature seeds, or harvested as a late, mature crop, for both seed and fiber production. Flax can be grown and harvested for fiber using equipment that is readily available and well understood by U.S. farmers.

Let $\Omega\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $\Omega$ and continuous in $\overline \Omega$ which vanishes in a relatively open subset $\Sigma\subset\partial\Omega$ and moreover the normal derivative $\partial_\nu u$ vanishes in a subset of $\Sigma$ with positive surface measure, then $u$ is identically $0$., Comment: More detailed explanation in some argument involving integration by parts and in Remark 3.3. An additional appendix with a self-contained proof of Lemma 4.3, whose proof was not included in the paper previously

We fill in a gap in the proof of the main theorem in our earlier paper [Ol]. At the same time, we prove a slightly stronger version of the theorem needed for another paper.

We present a novel method of analysis and prove finite time asymptotically self-similar blowup of the De Gregorio model \cite{DG90,DG96} for some smooth initial data on the real line with compact support. We also prove self-similar blowup results for the generalized De Gregorio model \cite{OSW08} for the entire range of parameter on $\mathbb{R}$ or $S^1$ for H\"older continuous initial data with compact support. Our strategy is to reformulate the problem of proving finite time asymptotically self-similar singularity into the problem of establishing the nonlinear stability of an approximate self-similar profile with a small residual error using the dynamic rescaling equation. We use the energy method with appropriate singular weight functions to extract the damping effect from the linearized operator around the approximate self-similar profile and take into account cancellation among various nonlocal terms to establish stability analysis. We remark that our analysis does not rule out the possibility that the original De Gregorio model is well posed for smooth initial data on a circle. The method of analysis presented in this paper provides a promising new framework to analyze finite time singularity of nonlinear nonlocal systems of partial differential equations., Comment: Added discussion in Section 2.3 and made some minor edits. Main paper 57 pages, Supplementary material 29 pages. In previous arXiv versions, the hyperlinks of the equation number in the main paper are linked to the supplementary material, which is fixed in this version

Full-waveform inversion (FWI) is today a standard process for the inverse problem of seismic imaging. PDE-constrained optimization is used to determine unknown parameters in a wave equation that represent geophysical properties. The objective function measures the misfit between the observed data and the calculated synthetic data, and it has traditionally been the least-squares norm. In a sequence of papers, we introduced the Wasserstein metric from optimal transport as an alternative misfit function for mitigating the so-called cycle skipping, which is the trapping of the optimization process in local minima. In this paper, we first give a sharper theorem regarding the convexity of the Wasserstein metric as the objective function. We then focus on two new issues. One is the necessary normalization of turning seismic signals into probability measures such that the theory of optimal transport applies. The other, which is beyond cycle skipping, is the inversion for parameters below reflecting interfaces. For the first, we propose a class of normalizations and prove several favorable properties for this class. For the latter, we demonstrate that FWI using optimal transport can recover geophysical properties from domains where no seismic waves travel through. We finally illustrate these properties by the realistic application of imaging salt inclusions, which has been a significant challenge in exploration geophysics.

This paper is concerned with voting processes on graphs where each vertex holds one of two different opinions. In particular, we study the \emph{Best-of-two} and the \emph{Best-of-three}. Here at each synchronous and discrete time step, each vertex updates its opinion to match the majority among the opinions of two random neighbors and itself (the Best-of-two) or the opinions of three random neighbors (the Best-of-three). Previous studies have explored these processes on complete graphs and expander graphs, but we understand significantly less about their properties on graphs with more complicated structures. In this paper, we study the Best-of-two and the Best-of-three on the stochastic block model $G(2n,p,q)$, which is a random graph consisting of two distinct Erdős-Renyi graphs $G(n,p)$ joined by random edges with density $q\leq p$. We obtain two main results. First, if $p=\omega(\log n/n)$ and $r=q/p$ is a constant, we show that there is a phase transition in $r$ with threshold $r^*$ (specifically, $r^*=\sqrt{5}-2$ for the Best-of-two, and $r^*=1/7$ for the Best-of-three). If $r>r^*$, the process reaches consensus within $O(\log \log n+\log n/\log (np))$ steps for any initial opinion configuration with a bias of $\Omega(n)$. By contrast, if $r r^*$, we show that, for any initial opinion configuration, the process reaches consensus within $O(\log n)$ steps. To the best of our knowledge, this is the first result concerning multiple-choice voting for arbitrary initial opinion configurations on non-complete graphs.

This paper concerns the existence of transonic shock solutions to the 2-D steady compressible Euler system in an almost flat finite nozzle ( in the sense that it is a generic small perturbation of a flat one ), under physical boundary conditions proposed by Courant-Friedrichs in \cite{CourantFriedrichs1948}, in which the receiver pressure is prescribed at the exit of the nozzle. In the resulting free boundary problem, the location of the shock-front is one of the most desirable information one would like to determine. However, the location of the normal shock-front in a flat nozzle can be anywhere in the nozzle so that it provides little information on the possible location of the shock-front when the nozzle's boundary is perturbed. So one of the key difficulties in looking for transonic shock solutions is to determine the shock-front. To this end, a free boundary problem for the linearized Euler system will be proposed, whose solution will be taken as an initial approximation for the transonic shock solution. In this paper, a sufficient condition in terms of the geometry of the nozzle and the given exit pressure is derived which yields the existence of the solutions to the proposed free boundary problem. Once an initial approximation is obtained, a further nonlinear iteration could be constructed and proved to lead to a transonic shock solution., 53 pages

We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math

It is well known that when the geometry and/or coefficients allow stable trapped rays, the solution operator of the Helmholtz equation (a.k.a. the resolvent of the Laplacian) grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest-possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial-growth assumption on the solution operator (e.g. $hp$-finite elements, $hp$-boundary elements, certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest-possible trapping, for most frequencies.

In this paper, using Riemann-Liouville integral and Caputo derivative, we study a nonlinear singular integro-differential equation of Lane-Emden type with nonlocal multi-point integral conditions. We prove the existence and uniqueness of solutions by application of Banach contraction principle. Also, we prove an existence result using Schaefer fixed point theorem. Then, we present some examples to show the applicability of the main results. DGRSDT, Direction Generale de la Recherche Scientifique et du Developpement Technologique, Algeria The authors express a special thanks to the associate editor and referees for their motivated comments that made the original manuscript significant and improved. This paper is supported by DGRSDT, Direction Generale de la Recherche Scientifique et du Developpement Technologique, Algeria. WOS:000526582000001 2-s2.0-85084038419

A uniform attachment graph (with parameter $k$), denoted $G_{n,k}$ in the paper, is a random graph on the vertex set $[n]$, where each vertex $v$ makes $k$ selections from $[v-1]$ uniformly and independently, and these selections determine the edge set. We study several aspects of this graph. Our motivation comes from two similarly constructed, well-studied random graphs: $k$-out graphs and preferential attachment graphs. In this paper, we find the asymptotic distribution of its minimum degree and connectivity, and study the expansion properties of $G_{n,k}$ to show that the conductance of $G_{n,k}$ is of order $(\log n)^{-1}$. We also study the bootstrap percolation on $G_{n,k}$, where, each vertex is either initially infected with probability $p$, independently of others, or gets infected later as a result of having $r$ infected neighbors at some point. We show that, for $2\le r\le k-1$, if $p\ll (\log n)^{-r/(r-1)}$, then, with probability approaching 1, the process ends before all vertices get infected. On the other hand, if $p\ge \omega(\log n)^{-r/(r-1)}$, where $\omega$ is a certain very slowly growing function, then all the vertices get infected with probability approaching 1.

The first goal of this paper is to prove multiple asymptotic results for a time-discrete and space-continuous polymer model of a random walk in a random potential. These results include: existence of deterministic free energy density in the infinite volume limit for every fixed asymptotic slope; concentration inequalities for free energy implying a bound on its fluctuation exponent; straightness estimates implying a bound on the transversal fluctuation exponent. The culmination of this program is almost sure existence and uniqueness of polymer measures on one-sided infinite paths with given endpoint and slope, and interpretation of these infinite-volume Gibbs measures as thermodynamic limits. Moreover, we prove that marginals of polymer measures with the same slope and different endpoints are asymptotic to each other. The second goal of the paper is to develop ergodic theory of the Burgers equation with positive viscosity and random kick forcing on the real line without any compactness assumptions. Namely, we prove a One Force -- One Solution principle, using the infinite volume polymer measures to construct a family of stationary global solutions for this system, and proving that each of those solutions is a one-point pullback attractor on the initial conditions with the same spatial average. This provides a natural extension of the same program realized for the inviscid Burgers equation with the help of action minimizers that can be viewed as zero temperature limits of polymer measures., Comment: 67 pages. This is an extension of the ergodic program for the Burgers equation in arXiv:1205.6721 and arXiv:1406.5660 to the positive viscosity case. Minor clarifications and additions to bibliography in this version

The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least $r\geq 2$ infected neighbours becomes infected and remains so forever. Assume that initially $a(t)$ vertices are randomly infected, where $t$ is the total number of vertices of the graph. Suppose also that $r < m$, where $2m$ is the average degree. We determine a critical function $a_c(t)$ such that when $a(t) \gg a_c(t)$, complete infection occurs with high probability as $t \rightarrow \infty$, but when $a(t) \ll a_c (t)$, then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to $a(t)$., This paper is significantly different to the previous version

In a paper that initiated the modern study of the stochastic block model (SBM), Decelle, Krzakala, Moore, and Zdeborova, backed by Mossel, Neeman, and Sly, conjectured that detecting clusters in the symmetric SBM in polynomial time is always possible above the Kesten-Stigum (KS) threshold, while it is possible to detect clusters information theoretically (i.e., not necessarily in polynomial time) below the KS threshold when the number of clusters k is at least 4. Massoulie, Mossel et al., and Bordenave, Lelarge, and Massoulie proved that the KS threshold is in fact efficiently achievable for k = 2, while Mossel et al. proved that it cannot be crossed at k = 2. The above conjecture remained open for k ≥ 3. This paper proves the two parts of the conjecture, further extending the results to general SBMs. For the efficient part, an approximate acyclic belief propagation (ABP) algorithm is developed and proved to detect communities for any k down to the KS threshold in quasi-linear time. Achieving this requires showing optimality of ABP in the presence of cycles, a challenge for message-passing algorithms. The paper further connects ABP to a power iteration method on a nonbacktracking operator of generalized order, formalizing the interplay between message passing and spectral methods. For the information-theoretic part, a nonefficient algorithm sampling a typical clustering is shown to break down the KS threshold at k = 4. The emerging gap is shown to be large in some cases, making the SBM a good case study for information-computation gaps. © 2017 Wiley Periodicals, Inc.

The Bogdanov-Takens bifurcations of a Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting were studied. In the paper “Diff. Equ. Dyn. Syst. 20(2012), 339-366,” Gupta et al proved that the Leslie-Gower predator-prey model with Michaelis-Menten–type prey harvesting has rich dynamics. Some equilibria of codimension 1 and their bifurcations were discussed. In this paper, we find that the model has an equilibrium of codimensions 2 and 3. We also prove analytically that the model undergoes Bogdanov-Takens bifurcations (cusp cases) of codimensions 2 and 3. Hence, the model can have 2 limit cycles, coexistence of a stable homoclinic loop and an unstable limit cycle, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation of codimension 1 as the values of parameters vary. Moreover, several numerical simulations are conducted to illustrate the validity of our results.

This paper studies the dynamics of the generalized Lengyel-Epstein reaction-diffusion model proposed in a recent study by Abdelmalek and Bendoukha. Two main results are shown in this paper. The first of which is sufficient conditions that guarantee the nonexistence of Turing patterns, ie, nonconstant solutions. Second, more relaxed conditions are derived for the stability of the system's unique steady-state solution.

In this paper, a novel simulation methodology based on the reproducing kernels is proposed for solving the fractional order integro-differential transport model for a nuclear reactor. The analysis carried out in this paper thus forms a crucial step in the process of development of fractional calculus as well as nuclear science models. The fractional derivative is described in the Captuo Riemann–Liouville sense. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present scheme is very simple, effective, and appropriate for obtaining numerical simulation of nuclear science models. Copyright © 2017 John Wiley & Sons, Ltd.

In this paper, an iteration process is considered to solve linear ill-posed problems. Based on the randomness of the involved variables, this kind of problems is regarded as simulation problems of the posterior distribution of the unknown variable given the noise data. We construct a new ensemble Kalman filter-based method to seek the posterior target distribution. Despite the ensemble Kalman filter method having widespread applications, there has been little analysis of its theoretical properties, especially in the field of inverse problems. This paper analyzes the propagation of the error with the iteration step for the proposed algorithm. The theoretical analysis shows that the proposed algorithm is convergence. We compare the numerical effect with the Bayesian inversion approach by two numerical examples: backward heat conduction problem and the first kind of integral equation. The numerical tests show that the proposed algorithm is effective and competitive with the Bayesian method. Copyright © 2017 John Wiley & Sons, Ltd.

Ren and Zeng (2013) introduced a new kind of q-Bernstein-Schurer operators and studied some approximation properties. Acu etal. (2016) defined the Durrmeyer modification of these operators and studied the rate of convergence and statistical approximation. The purpose of this paper is to introduce a Kantorovich modification of these operators by using q-Riemann integral and investigate the rate of convergence by means of the Lipschitz class and the Peetre's K-functional. Next, we introduce the bivariate case of q-Bernstein-Schurer-Kantorovich operators and study the degree of approximation with the aid of the partial modulus continuity, Lipschitz space, and the Peetre's K-functional. Finally, we define the generalized Boolean sum operators of the q-Bernstein-Schurer-Kantorovich type and investigate the approximation of the Bogel continuous and Bogel differentiable functions by using the mixed modulus of smoothness. Furthermore, we illustrate the convergence of the operators considered in the paper for the univariate case and the associated generalized Boolean sum operators to certain functions by means of graphics using Maple algorithms. Copyright (c) 2017 John Wiley & Sons, Ltd.

This paper studies a higher dimensional generalization of Frieze's $\zeta(3)$-limit theorem in the Erd\"os-R\'enyi graph process. Frieze's theorem states that the expected weight of the minimum spanning tree converges to $\zeta(3)$ as the number of vertices goes to infinity. In this paper, we study the $d$-Linial-Meshulam process as a model for random simplicial complexes, where $d=1$ corresponds to the Erd\"os-R\'enyi graph process. First, we define spanning acycles as a higher dimensional analogue of spanning trees, and connect its minimum weight to persistent homology. Then, our main result shows that the expected weight of the minimum spanning acycle behaves in $O(n^{d-1})$., Comment: 24 pages

The k-core, defined as the maximal subgraph of minimum degree at least k, of the random graph G(n,p) has been studied extensively. In a landmark paper Pittel, Wormald and Spencer [J Combin Theory Ser B 67 (1996), 111–151] determined the threshold dk for the appearance of an extensive k-core. The aim of the present paper is to describe how the k-core is “embedded” into the random graph in the following sense. Let k≥3 and fix d=np>dk. Colour each vertex that belongs to the k-core of G(n,p) in black and all remaining vertices in white. Here we derive a multi-type branching process that describes the local structure of this coloured random object as n tends to infinity. This generalises prior results on, e.g., the internal structure of the k-core. In the physics literature it was suggested to characterize the core by means of a message passing algorithm called Warning Propagation. Ibrahimi, Kanoria, Kraning and Montanari [Ann Appl Probab 25 (2015), 2743–2808] used this characterization to describe the 2-core of random hypergraphs. To derive our main result we use a similar approach. A key observation is that a bounded number of iterations of this algorithm is enough to give a good approximation of the k-core. Based on this the study of the k-core reduces to the analysis of Warning Propagation on a suitable Galton-Watson tree. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 00, 000–000, 2017

We consider consecutive random subdivision of polygons described as follows. Given an initial convex polygon with $d\ge 3$ edges, we choose a point at random on each edge, such that the proportions in which these points divide edges are i.i.d. copies of some random variable $\xi$. These new points form a new (smaller) polygon. By repeatedly implementing this procedure we obtain a sequence of random polygons. The aim of this paper is to show that under very mild non-degenerateness conditions on $\xi$, the shapes of these polygons eventually become "flat" The convergence rate to flatness is also investigated; in particular, in the case of triangles ($d=3$), we show how to calculate the exact value of the rate of convergence, connected to Lyapunov exponents. Using the theory of products of random matrices our paper greatly generalizes the results of Volkov (2013) which are achieved mostly by using ad hoc methods., Comment: 35 pages, 2 figures

This paper discusses uncertainty principles of images defined on the square, or, equivalently, uncertainty principles of signals on the 2-torus. Means and variances of time and frequency for signals on the 2-torus are defined. A set of phase and amplitude derivatives are introduced. Based on the derivatives, we obtain three comparable lower bounds of the product of variances of time and frequency, of which the largest lower bound corresponds to the strongest uncertainty principles known for periodic signals. Examples, including simulations, are provided to illustrate the obtained results. To the authors' knowledge, it is in the present paper, and for the first time, that uncertainty principles on the torus are studied. Copyright © 2016 John Wiley & Sons, Ltd.

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016

This paper is concerned with interval general bidirectional associative memory (BAM) neural networks with proportional delays. Using appropriate nonlinear variable transformations, the interval general BAM neural networks with proportional delays can be equivalently transformed into the interval general BAM neural networks with constant delays. The sufficient condition for the existence and uniqueness of equilibrium point of the model is established by applying Brouwer's fixed point theorem. By constructing suitable delay differential inequalities, some sufficient conditions for the global exponential stability of the model are obtained. Two examples are given to illustrate the effectiveness of the obtained results. This paper ends with a brief conclusion. Copyright © 2016 John Wiley & Sons, Ltd.

In this paper, a class of neutral-type neural networks with delays in the leakage term on time scales are considered. By using the Banach fixed point theorem and the theory of calculus on time scales, some sufficient conditions are obtained for the existence and exponential stability of almost periodic solutions for this class of neural networks. The results of this paper are new and complementary to the previously known results. Finally, an example is presented to illustrate the effectiveness of our results. Copyright © 2016 John Wiley & Sons, Ltd.

This is the first of a series of papers devoted to the initial value problem for the one-dimensional Euler system of compressible fluids and augmented versions containing higher-order terms. In the present paper, we are interested in dispersive shock waves and analyze the zero viscosity-capillarity limit associated with the Navier-Stokes-Korteweg system. Specifically, we establish the existence of finite energy solutions as well as their convergence toward entropy solutions to the Euler system. Our method of proof combines energy and effective energy estimates, a nonlinear Sobolev inequality, high-integrability properties for the mass density and for the velocity, and compactness properties based on entropies.© 2015 Wiley Periodicals, Inc.

The regression-based Monte Carlo methods for backward stochastic differential equations (BSDEs) have been the object of considerable research, particularly for solving nonlinear partial differential equations (PDEs). Unfortunately, such methods often become unstable when implemented with small time steps because the variance of gradient estimates is inversely proportional to the time step (σ2∼ 1/Δ t). Recently new variance reduction techniques were introduced to address this problem in~a paper by the author and Avellaneda. The purpose of this paper is to provide a rigorous justification for these techniques in the context of the discrete-time BSDE scheme of Bouchard and Touzi. We also suggest a new higher-order scheme that makes the variance proportional to the time step (σ2∼Δ t). These techniques are easy to implement. Numerical examples strongly indicate that they render the regression-based Monte Carlo methods stable for small time steps and thus viable for numerical solution of nonlinear PDEs.© 2016 Wiley Periodicals, Inc.

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non-linear differential-difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential-difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich-type theorem and Malmquist-type theorem about differential equations to the case of differential-difference equations. Copyright © 2015 John Wiley & Sons, Ltd.

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most C|VG|. In this paper, we show that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph Gn,p, which improves upon existing results showing that asymptotically almost surely the cop number of Gn,p is Onlogn provided that pni¾?2+elogn for some e>0. We do this by first showing that the conjecture holds for a general class of graphs with some specific expansion-type properties. This will also be used in a separate paper on random d-regular graphs, where we show that the conjecture holds asymptotically almost surely when d=dni¾?3. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 396-421, 2016