Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field

We consider the discrete two‐dimensional Gaussian free field on a box of side length $N$, with Dirichlet boundary data, and prove the convergence of the law of the centered maximum of the field.© 2015 Wiley Periodicals, Inc.     

Isoperimetry in Two‐Dimensional Percolation

We study isoperimetric sets, i.e., sets with minimal boundary for a prescribed volume, on the unique infinite connected component of supercritical bond percolation on the square lattice. In the limit of the volume tending to infinity, properly scaled isoperimetric sets are shown to converge (in the Hausdorff metric) to the solution of an isoperimetric problem in ?² with respect to a particular norm. As part of the proof we also show that the anchored isoperimetric profile as well as the Cheeger constant of the giant component in finite boxes scale to deterministic quantities. This settles a conjecture of Itai Benjamini for the square lattice. © 2015 Wiley Periodicals, Inc.     

Nematic Equilibria on a Two‐Dimensional Annulus

We study planar nematic equilibria on a two‐dimensional annulus with strong and weak tangent anchoring, in the Oseen–Frank theoretical framework. We analyze a radially invariant defect‐free state and compute analytic stability criteria for this state in terms of the elastic anisotropy, annular aspect ratio, and anchoring strength. In the strong anchoring case, we define and characterize a new spiral‐like equilibrium which emerges as the defect‐free state loses stability. In the weak anchoring case, we compute stability diagrams that quantify the response of the defect‐free state to radial and azimuthal perturbations. We study sector equilibria on sectors of an annulus, including the effects of weak anchoring and elastic anisotropy, giving novel insights into the correlation between preferred numbers of boundary defects and the geometry. We numerically demonstrate that these sector configurations can approximate experimentally observed equilibria with boundary defects.     

Mixing Times of Critical Two‐Dimensional Potts Models

We study dynamical aspects of the q‐state Potts model on an n × n box at its critical β_c(q). Heat‐bath Glauber dynamics and cluster dynamics such as Swendsen–Wang (that circumvent low‐temperature bottlenecks) are all expected to undergo “critical slowdowns” in the presence of periodic boundary conditions: the inverse spectral gap, which in the subcritical regime is O(1), should at criticality be polynomial in n for 1 < q ≤ 4, and exponential in n for q > 4 in accordance with the predicted discontinuous phase transition. This was confirmed for q = 2 (the Ising model) by the second author and Sly, and for sufficiently large q by Borgs et al. Here we show that the following holds for the critical Potts model on the torus: for q=3, the inverse gap of Glauber dynamics is n^O(1); for q = 4, it is at most n^{O(log n)}; and for every q > 4 in the phase‐coexistence regime, the inverse gaps of both Glauber dynamics and Swendsen‐Wang dynamics are exponential in n. For free or monochromatic boundary conditions and large q, we show that the dynamics at criticality is faster than on the torus (unlike the Ising model where free/periodic boundary conditions induce similar dynamical behavior at all temperatures): the inverse gap of Swendsen‐Wang dynamics is exp(n^o(1)). © 2017 Wiley Periodicals, Inc.     

Spectral Stability of Deep Two‐Dimensional Gravity‐Capillary Water Waves

In this contribution we study the spectral stability problem for periodic traveling gravity‐capillary waves on a two‐dimensional fluid of infinite depth. We use a perturbative approach that computes the spectrum of the linearized water wave operator as an analytic function of the wave amplitude/slope. We extend the highly accurate method of Transformed Field Expansions to address surface tension in the presence of both simple and repeated eigenvalues, then numerically simulate the evolution of the spectrum as the wave amplitude is increased. We also calculate explicitly the first nonzero correction to the flat‐water spectrum, which we observe to accurately predict the stability (or instability) for all amplitudes within the disk of analyticity of the spectrum. With this observation in mind, the disk of analyticity of the flat state spectrum is numerically estimated as a function of the Bond number and the Bloch parameter, and compared to the value of the wave slope at the first finite amplitude eigenvalue collision.     

Uniqueness of Tangent Cones for Two‐Dimensional Almost‐Minimizing Currents

We consider two‐dimensional integer rectifiable currents that are almost area minimizing and show that their tangent cones are everywhere unique. Our argument unifies a few uniqueness theorems of the same flavor, which are all obtained by a suitable modification of White's original theorem for area‐minimizing currents in the euclidean space. This note is also the first step in a regularity program for semicalibrated two‐dimensional currents and spherical cross sections of three‐dimensional area‐minimizing cones.© 2017 Wiley Periodicals, Inc.     

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

The study of optical orthogonal codes has been motivated by an application in an optical code‐division multiple access system. From a practical point of view, compared to one‐dimensional optical orthogonal codes, two‐dimensional optical orthogonal codes tend to require smaller code length. On the other hand, in some circumstances only with good cross‐correlation one can deal with both synchronization and user identification. These motivate the study of two‐dimensional optical orthogonal codes with better cross‐correlation than auto‐correlation. This paper focuses on optimal two‐dimensional optical orthogonal codes with the auto‐correlation and the best cross‐correlation 1. By examining the structures of w‐cyclic group divisible designs and semi‐cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two‐dimensional ‐optical orthogonal codes. When and , the exact number of codewords of an optimal two‐dimensional ‐optical orthogonal code is determined for any positive integers n and .     

On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data

We consider the evolution of two incompressible, immiscible fluids with different densities in porous media, known as the Muskat problem [21], which in two dimensions is analogous to the Hele‐Shaw cell [24]. We establish, for a class of large and monotone initial data, the global existence of weak solutions. The proof is based on a local well‐posedness result for the initial data with certain specific asymptotics at spatial infinity and a new maximum principle for the first derivative of the graph function.© 2016 Wiley Periodicals, Inc.     

Bounds and Constructions for Two‐Dimensional Variable‐Weight Optical Orthogonal Codes

Optical orthogonal codes (1D constant‐weight OOCs or 1D CWOOCs) were first introduced by Salehi as signature sequences to facilitate multiple access in optical fibre networks. In fiber optic communications, a principal drawback of 1D CWOOCs is that large bandwidth expansion is required if a big number of codewords is needed. To overcome this problem, a two‐dimensional (2D) (constant‐weight) coding was introduced. Many optimal 2D CWOOCs were obtained recently. A 2D CWOOC can only support a single QoS (quality of service) class. A 2D variable‐weight OOC (2D VWOOC) was introduced to meet multiple QoS requirements. A 2D VWOOC is a set of 0, 1 matrices with variable weight, good auto, and cross‐correlations. Little is known on the construction of optimal 2D VWOOCs. In this paper, new upper bound on the size of a 2D VWOOC is obtained, and several new infinite classes of optimal 2D VWOOCs are obtained.     

Exponential Asymptotics for Line Solitons in Two‐Dimensional Periodic Potentials

As a first step toward a fully two‐dimensional asymptotic theory for the bifurcation of solitons from infinitesimal continuous waves, an analytical theory is presented for line solitons, whose envelope varies only along one direction, in general two‐dimensional periodic potentials. For this two‐dimensional problem, it is no longer viable to rely on a certain recurrence relation for going beyond all orders of the usual multiscale perturbation expansion, a key step of the exponential asymptotics procedure previously used for solitons in one‐dimensional problems. Instead, we propose a more direct treatment which not only overcomes the recurrence‐relation limitation, but also simplifies the exponential asymptotics process. Using this modified technique, we show that line solitons with any rational line slopes bifurcate out from every Bloch‐band edge; and for each rational slope, two line‐soliton families exist. Furthermore, line solitons can bifurcate from interior points of Bloch bands as well, but such line solitons exist only for a couple of special line angles due to resonance with the Bloch bands. In addition, we show that a countable set of multiline‐soliton bound states can be constructed analytically. The analytical predictions are compared with numerical results for both symmetric and asymmetric potentials, and good agreement is obtained.     

On Results' Accuracy at Two‐Dimensional Transonic Wind Tunnel Testing

The principal factors which influence the accuracy of two‐dimensional wind tunnel test results are analyzed. The influences of Reynolds number, Mach number and wall interference with reference to solid and flow blockage (blockage of wake) as well as the influence of side‐wall boundary layer control are analyzed.     

Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

We study here the initial value problem for a two‐dimensional Korteweg–de Vries (KdV) equation, first derived by Calogero and Bogoyavlenskii, by means of the inverse scattering transform. The dynamics of the discrete spectrum of an associated Schrödinger operator is far richer than that of KdV equation. Even for optimal eigenvalues, generic smooth solutions may develop shocks with multiple branches and/or cusp singularities in finite time. However, evolution may move poles of the transmission coefficient off the imaginary axis, destroy or even create them. We characterize conditions to prevent these pathologies before explosion time and describe ample classes of solutions, corresponding to both continuous and discrete spectrum. We also find that in certain conditions new eigenvalues might be created; in these cases a minimal set of initial spectral data must incorporate additionally the transmission coefficient on the entire plane. The previous results are applied to describe the Cauchy problem corresponding to initial data combinations of delta terms and derivatives and show that for long time the delta singularity may persist or be smoothed to a cusp‐discontinuity. Finally, we give conditions under which the evolution is reduced to the classical KdV.     

Unified Orbital Description of the Envelope Dynamics in Two‐Dimensional Simple Periodic Lattices

The propagation of wave envelopes in two‐dimensional (2‐D) simple periodic lattices is studied. A discrete approximation, known as the tight‐binding (TB) approximation, is employed to find the equations governing a class of nonlinear discrete envelopes in simple 2‐D periodic lattices. Instead of using Wannier function analysis, the orbital approximation of Bloch modes that has been widely used in the physical literature, is employed. With this approximation the Bloch envelope dynamics associated with both simple and degenerate bands are readily studied. The governing equations are found to be discrete nonlinear Schrödinger (NLS)‐type equations or coupled NLS‐type systems. The coefficients of the linear part of the equations are related to the linear dispersion relation. When the envelopes vary slowly, the continuous limit of the general discrete NLS equations are effective NLS equations in moving frames. These continuous NLS equations (from discrete to continuous) also agree with those derived via a direct multiscale expansion. Rectangular and triangular lattices are examples.     

On the Finite‐Time Blowup of a One‐Dimensional Model for the Three‐Dimensional Axisymmetric Euler Equations

In connection with the recent proposal for possible singularity formation at the boundary for solutions of three‐dimensional axisymmetric incompressible Euler's equations (Luo and Hou, Proc. Natl. Acad. Sci. USA (2014)), we study models for the dynamics at the boundary and show that they exhibit a finite‐time blowup from smooth data. © 2017 Wiley Periodicals, Inc.     

Global Solutions of Two‐Dimensional Incompressible Viscoelastic Flows with Discontinuous Initial Data

The global existence of weak solutions of the incompressible viscoelastic flows in two spatial dimensions has been a longstanding open problem, and it is studied in this paper. We show global existence if the initial deformation gradient is close to the identity matrix in L² ∩ L^∞ and the initial velocity is small in L² and bounded in L^p for some p > 2. While the assumption on the initial deformation gradient is automatically satisfied for the classical Oldroyd‐B model, the additional assumption on the initial velocity being bounded in L^p for some p > 2 may due to techniques we employed. The smallness assumption on the L² norm of the initial velocity is, however, natural for global well‐posedness. One of the key observations in the paper is that the velocity and the “ effective viscous flux” are sufficiently regular for positive time. The regularity of leads to a new approach for the pointwise estimate for the deformation gradient without using L^∞ bounds on the velocity gradients in spatial variables. © 2015 Wiley Periodicals, Inc.     

A Synthetical Two‐Component Model with Peakon Solutions

A generalized two‐component model with peakon solutions is proposed in this paper. It allows an arbitrary function to be involved in as well as including some existing integrable peakon equations as special reductions. The generalized two‐component system is shown to possess Lax pair and infinitely many conservation laws. Bi‐Hamiltonian structures and peakon interactions are discussed in detail for typical representative equations of the generalized system. In particular, a new type of N‐peakon solution, which is not in the traveling wave type, is obtained from the generalized system.     

Two‐soliton solutions to the three‐dimensional gravitational Hartree equation

We construct nondispersive two‐soliton solutions to the three‐dimensional gravitational Hartree equation whose trajectories asymptotically reproduce the nontrapped dynamics of the gravitational two‐body problem. © 2009 Wiley Periodicals, Inc.     

Multi‐to One‐Dimensional Optimal Transport

We consider the Monge‐Kantorovich problem of transporting a probability density on to another on the line, so as to optimize a given cost function. We introduce a nestedness criterion relating the cost to the densities, under which it becomes possible to solve this problem uniquely by constructing an optimal map one level set at a time. This map is continuous if the target density has connected support. We use level‐set dynamics to develop and quantify a local regularity theory for this map and the Kantorovich potentials solving the dual linear program. We identify obstructions to global regularity through examples. More specifically, fix probability densities f and g on open sets and with . Consider transporting f onto g so as to minimize the cost . We give a nondegeneracy condition on that ensures the set of x paired with [g‐a.e.] y ∈ Y lie in a codimension‐n submanifold of X. Specializing to the case m > n = 1, we discover a nestedness criterion relating s to (f,g) that allows us to construct a unique optimal solution in the form of a map . When and g and f are bounded, the Kantorovich dual potentials (u,υ) satisfy , and the normal velocity V of with respect to changes in y is given by . Positivity of V locally implies a Lipschitz bound on f; moreover, if intersects transversally. On subsets where this nondegeneracy, positivity, and transversality can be quantified, for each integer the norms of and are controlled by these bounds, , and the smallness of . We give examples showing regularity extends from $X to part of , but not from Y to . We also show that when s remains nested for all (f,g), the problem in reduces to a supermodular problem in . © 2017 Wiley Periodicals, Inc.     

Two‐coloring random hypergraphs

A 2‐coloring of a hypergraph is a mapping from its vertex set to a set of two colors such that no edge is monochromatic. Let H=H(k, n, p) be a random k‐uniform hypergraph on a vertex set V of cardinality n, where each k‐subset of V is an edge of H with probability p, independently of all other k‐subsets. Let $ m = p{{n}\choose{k}}$ denote the expected number of edges in H. Let us say that a sequence of events ?_n holds with high probability (w.h.p.) if lim_n→∞Pr[?_n]=1. It is easy to show that if m=c2^kn then w.h.p H is not 2‐colorable for c>ln 2/2. We prove that there exists a constant c>0 such that if m=(c2^k/k)n, then w.h.p H is 2‐colorable. © 2002 Wiley Periodicals, Inc. Random Struct. Alg. 20: 249–259, 2002     

Two‐cardinal diamond star

Our main results are: (A) It is consistent relative to a large cardinal that holds but fails. (B) If holds and are two infinite cardinals such that and λ carries a good scale, then holds. (C) If are two cardinals such that κ is λ‐Shelah and , then there is no good scale for λ.