It is shown that the hard-core model on ${{\mathbb Z}}^d$ exhibits a phase transition at activities above some function $\lambda(d)$ which tends to zero as $d\rightarrow \infty$. More precisely, consider the usual nearest neighbour graph on ${{\mathbb Z}}^d$, and write ${\cal E}$ and ${\cal O}$ for the sets of even and odd vertices (defined in the obvious way). Set $${\cal G}L_M={\cal G}L_M^d =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}\leq M\},\quad \partial^{\star} {\cal G}L_M =\{z\in{{\mathbb Z}}^d:\|z\|_{\infty}= M\},$$ and write ${\cal I}({\cal G}L_M)$ for the collection of independent sets (sets of vertices spanning no edges) in ${\cal G}L_M$. For $\lambda>0$ let ${\bf I}$ be chosen from ${\cal I}({\cal G}L_M)$ with $\Pr({\bf I}=I) \propto \lambda^{|I|}$.

TheoremThere is a constant$C$such that if$\lambda > Cd^{-1/4}\log^{3/4}d$, then$$\lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}|{\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal E})~> \lim_{M\rightarrow\infty}\Pr(\underline{0}\in{\bf I}| {\bf I}\supseteq \partial^{\star} {\cal G}L_M\cap {\cal O}).$$ Thus, roughly speaking, the influence of the boundary on behaviour at the origin persists as the boundary recedes.

We consider two interrelated tasks in a synchronous $n$-node ring: distributed constant colouring and local communication. We investigate the impact of the amount of knowledge available to nodes on the time of completing these tasks. Every node knows the labels of nodes up to a distance $r$ from it, called the knowledge radius. In distributed constant colouring every node has to assign itself one out of a constant number of colours, so that adjacent nodes get different colours. In local communication every node has to communicate a message to both of its neighbours. We study these problems in two popular communication models: the one-way model, in which each node can only either transmit to one neighbour or receive from one neighbour, in any round, and the radio model, in which simultaneous receiving from two neighbours results in interference noise. Hence the main problem in fast execution of the above tasks is breaking symmetry with restricted knowledge of the ring.

We show that distributed constant colouring and local communication are tightly related and one can be used to accomplish the other. Also, in most situations the optimal time is the same for both of them, and it strongly depends on knowledge radius. For knowledge radius $r=0$, i.e., when each node knows only its own label, our bounds on time for both tasks are tight in both models: the optimal time in the one-way model is $\Theta(n)$, while in the radio model it is $\Theta(\log n)$. For knowledge radius $r=1$ both tasks can be accomplished in time $O(\log \log n)$ in the one-way model, if the ring is oriented. For $2 \leq r \leq c \log ^* n$, where $c < 1/2$, the upper bounds on time are $O(\log^{(2r)} n)$ in the one-way model and $O(\log ^{(2\lfloor r/2 \rfloor)} n)$ in the radio model; the lower bound is $\Omega (\log^* n)$, in both models. For $r \geq (\log^*n)/2$ both tasks can be completed in constant time, in the one-way model, and distributed constant colouring also in the radio model. Finally, if $r \geq \log^*n$ then constant time is also enough for local communication in the radio model.

We consider random planar graphs on $n$ labelled nodes, and show in particular that if the graph is picked uniformly at random then the expected number of edges is at least $\frac{13}{7}n +o(n)$. To prove this result we give a lower bound on the size of the set of edges that can be added to a planar graph on $n$ nodes and $m$ edges while keeping it planar, and in particular we see that if $m$ is at most $\frac{13}{7}n - c$ (for a suitable constant~$c$) then at least this number of edges can be added.

We describe how a modification of a common technique for developing asymptotic expansions of solutions of linear differential equations can be used to derive Hadamard expansions of solutions of differential equations. Hadamard expansions are convergent series that share some of the features of hyperasymptotic expansions, particularly that of having exponentially small remainders when truncated, and, as a consequence, provide a useful computational tool for evaluating special functions. The methods we discuss can be applied to linear differential equations of hypergeometric type and may have wider applicability.

We consider the Dirac equation given bywith initial condition y1 (0) cos α + y2(0) sin α = 0, α ε [0; π ) and suppose the equation is in the limit-point case at infinity. Using to denote the derivative of the corresponding spectral function, a formula for is given when is known and positive for three distinct values of α. In general, if is known and positive for only two distinct values of α, then is shown to be one of two possibilities. However, in special cases of the Dirac equation, can be uniquely determined given for only two values of α.

We prove the uniqueness of the very singular solution towhen 1 < p < (N + 2)/(N + 1), thus completing the previous result by Qi and Wang, restricted to self-similar solutions.

For any n, m ∈ N, we prove the existence of 2mπ-periodic solutions, with n bouncings in each period, for a second-order forced equation with attractive singularity by using the approach of successor map and Poincaré-Birkhoff twist theorem.

We consider an elliptic boundary problem defined in a bounded region Ω ⊂ Rn and where the spectral parameter is multiplied by a weight function ω(x). We suppose that ω(x) ≠ 0 for x ∈ Ω, but vanishes in a specified manner on the boundary of Ω. Under limited smoothness assumptions, we derive results pertaining to existence and uniqueness of and a priori estimates for solutions of the boundary problem. If S(λ) denotes the operator pencil induced in L2(Ω) by the boundary problem with zero boundary conditions, then results are also derived pertaining to the spectral properties of S(λ).

In this paper we obtain, for a semilinear elliptic problem in RN, families of solutions bifurcating from the bottom of the spectrum of −Δ. The problem is variational in nature and we apply a nonlinear reduction method that allows us to search for solutions as critical points of suitable functionals defined on finite-dimensional manifolds.

We provide an estimate of the energy of the solutions to the Poisson equation with constant data and Dirichlet boundary conditions in a convex domain Ω ⊂ Rn. This estimate is obtained by restricting the variational formulation of the problem to the space of functions depending only on the distance from the boundary of Ω. The main tool in the proof is an isoperimetric inequality for convex domains, which is a consequence of the Brunn-Minkowski theorem.

Consider the systemwhere λ is a positive parameter and Ω is a bounded domain in RN. We prove the existence of a large positive solution for λ large when limx → ∞ (f(Mg(x))/x) = 0 for every M > 0. In particular, we do not need any monotonicity assumptions on f, g, nor any sign conditions on f(0), g(0).

Let V be an infinite-dimensional locally convex complex space, X a closed subset of P(V) defined by finitely many continuous homogeneous equations and E a holomorphic vector bundle on X with finite rank. Here we show that E is holomorphically trivial if it is topologically trivial and spanned by its global sections and in a few other cases.

In this paper we investigate three-dimensional complete minimal hypersurfaces with constant Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0). We prove that if the scalar curvature of a such hypersurface is bounded from below, then its Gauss-Kronecker curvature vanishes identically. Examples of complete minimal hypersurfaces which are not totally geodesic in the Euclidean space E4 and the hyperbolic space H4(c) with vanishing Gauss-Kronecker curvature are also presented. It is also proved that totally umbilical hypersurfaces are the only complete hypersurfaces with non-zero constant mean curvature and with zero quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature is bounded from below. In particular, we classify complete hypersurfaces with constant mean curvature and with constant quasi-Gauss-Kronecker curvature in a space form M4(c) (c ≤ 0) if the scalar curvature r satisfies r≥ ⅔c.

Using techniques of bifurcation theory, we give exact multiplicity and uniqueness results for the fourth-order Dirichlet problem, which describes deflection of an elastic beam, subjected to a nonlinear force, and clamped at the end points. The crucial part of this approach was to show positivity of non-trivial solutions of the corresponding linearized problem.

We consider the equation−Δu = |u|4/(N−2)u + εf(x) under zero Dirichlet boundary conditions in a bounded domain Ω in RN exhibiting certain symmetries, with f ≥ 0, f ≠ 0. In particular, we find that the number of sign-changing solutions goes to infinity for radially symmetric f, as ε → 0 if Ω is a ball. The same is true for the number of negative solutions if Ω is an annulus and the support of f is compact in Ω.

We prove the existence of a positive solution for the Schrödinger-type equation on RN,−Δu + V(x)u = f(u) for u ∈ H1(RN), provided thatThe result is obtained in two cases. (a) lim|x|→ ∞V(x) = V (∞) ∈ (0,∞), f is subcritical and supt>0 2F(t)t−2 < V (∞). (b) There exist V > 0 and R* > 0 such that f(t)t−1 < V ∞ ≤ V (x) for all t > 0 and |x| ≥ R*.

We study the existence of closed characteristics on three-dimensional energy manifolds of second-order Lagrangian systems. These manifolds are always non-compact, connected and not necessarily of contact type. Using the specific geometry of these manifolds, we prove that the number of closed characteristics on a prescribed energy manifold is bounded below by its second Betti number, which is easily computable from the Lagrangian.

We study a class of initial-boundary-value problems for which an auxiliary condition of the formis prescribed. We determine bounds on an energy expression by means of differential inequalities and derive pointwise bounds for the solution and its gradient by use of a parabolic maximum principle.

Two nonlinear diffusion equations for thin film epitaxy, with or without slope selection, are studied in this work. The nonlinearity models the Ehrlich–Schwoebel effect – the kinetic asymmetry in attachment and detachment of adatoms to and from terrace boundaries. Both perturbation analysis and numerical simulation are presented to show that such an atomistic effect is the origin of a nonlinear morphological instability, in a rough-smooth-rough pattern, that has been experimentally observed as transient in an early stage of epitaxial growth on rough surfaces. Initial-boundary-value problems for both equations are proven to be well-posed, and the solution regularity is also obtained. Galerkin spectral approximations are studied to provide both a priori bounds for proving the well-posedness and numerical schemes for simulation. Numerical results are presented to confirm part of the analysis and to explore the difference between the two models on coarsening dynamics.

We describe various blow-up patterns for the fourth-order one-dimensional semilinear parabolic equation \[ u_t = - u_{xxxx} + \b [(u_x)^3]_x + e^{u} \] with a parameter $\b \ge 0$, which is a model equation from explosion-convection theory. Unlike the classical Frank-Kamenetskii equation $u_t=u_{xx} +e^u$ (a solid fuel model), by using analytical and numerical evidence, we show that the generic blow-up in this fourth-order problem is described by a similarity solution $u_*(x,t) = -\ln(T\,{-}\,t) \,{+}\, f_1(x/(T\,{-}\,t)^{1/4})(T>0$ is the blow-up time), with a non-trivial profile $f_1 \not \equiv 0$. Numerical solution of the PDE shows convergence to the self-similar solution with the profile $f_1$ from a wide variety of initial data. We also construct a countable subset of other, not self-similar, blow-up patterns by using a spectral analysis of an associated linearized operator and matching with similarity solutions of a first-order Hamilton–Jacobi equation.