Image inpainting is an image restoration problem, in which image models play a critical role, as demonstrated by Chan, Kang & Shen's [12] recent inpainting schemes based on the bounded variation and the elastica [11] image models. In this paper, we propose two novel inpainting models based on the Mumford–Shah image model [41], and its high order correction – the Mumford–Shah–Euler image model. We also present their efficient numerical realization based on the Γ-convergence approximations of Ambrosio & Tortorelli [2, 3] and De Giorgi [21].

Let {Av}v∈V be a finite collection of events and G = (V, E) be a chordal graph. Our main result – the chordal graph sieve – is a Bonferroni-type inequality where the selection of intersections in the estimates is determined by a chordal graph G. It interpolates between Boole's inequality (G empty) and the sieve formula (G complete). By varying G, several inequalities both well-known and new are obtained in a concise and unified way.

We show that the limiting distribution of the number of comparisons used by Hoare's quickselect algorithm when given a random permutation of n elements for finding the mth-smallest element, where m = o(n), is the Dickman function. The limiting distribution of the number of exchanges is also derived.

We show that, if G is a graph of order n with maximal degree Δ(G) and minimal degree δ(G) whose complement contains no K2,s, s [ges ] 2, then G contains every tree T of order n−s+1 whose maximal degree is at most Δ(G) and whose vertex of second-largest degree is at most δ(G). We then show that this result implies that special cases of two conjectures are true. We verify that the Erdös–Sós conjecture, which states that a graph whose average degree is larger than k−1 contains every tree of order k+1, is true for graphs whose complement does not contain a K2,4, and the Komlós–Sós conjecture, which states that every graph of median degree at least k contains every tree of order k+1, is true for graphs whose complement does not contain a K2,3.

Consider a finite alphabet Ω and patterns which consist of characters from Ω. For a given pattern w, let cor(w) denote its autocorrelation, which can be seen as a measure of the amount of overlap in w. Letting aw(n) denote the number of strings over Ω of length n which do not contain w as a substring, the main result of this paper is: If cor(w) > cor(w′) then aw(n)−aw′(n) > (|Ω|−1)(aw(n−1)−aw′(n−1)) for n [ges ] N, and the value of N is given. This result confirms a conjecture by Eriksson [2], which was previously proved to be true by Cakir, Chryssaphinou and Månsson [1] when |Ω| [ges ] 3.

In this note it is shown that resistance and resistance dimension are rough isometry invariant.

Consider the class of graphs on n vertices which have maximum degree at most 1/2n−1+τ, where τ [ges ] −n1/2+ε for sufficiently small ε > 0. We find an asymptotic formula for the number of such graphs and show that their number of edges has a normal distribution whose parameters we determine. We also show that expectations of random variables on the degree sequences of such graphs can often be estimated using a model based on truncated binomial distributions.

Let r = r(n) → ∞ with 3 [les ] r [les ] n1−η for an arbitrarily small constant η > 0, and let Gr denote a graph chosen uniformly at random from the set of r-regular graphs with vertex set {1, 2, …, n}. We prove that, with probability tending to 1 as n → ∞, Gr has the following properties: the independence number of Gr is asymptotically 2n log r/r and the chromatic number of Gr is asymptotically r/2nlogr.

We study the complexity of computing the coefficients of three classical polynomials, namely the chromatic, flow and reliability polynomials of a graph. Each of these is a specialization of the Tutte polynomial Σtijxiyj. It is shown that, unless NP = RP, many of the relevant coefficients do not even have good randomized approximation schemes. We consider the quasi-order induced by approximation reducibility and highlight the pivotal position of the coefficient t10 = t01, otherwise known as the beta invariant.

Our nonapproximability results are obtained by showing that various decision problems based on the coefficients are NP-hard. A study of such predicates shows a significant difference between the case of graphs, where, by Robertson–Seymour theory, they are computable in polynomial time, and the case of matrices over finite fields, where they are shown to be NP-hard.

It is proved that elliptic boundary-value problems have a global smoothing property in Lebesgue spaces, provided the underlying space of weak solutions admits a Sobolev-type inequality. The results apply to all standard boundary conditions, and a wide range of non-smooth domains, even if the classical estimates fail. The dependence on the data is explicit. In particular, this provides good control over the domain dependence, which is important for applications involving varying domains.

We consider the eigenvalue problemin an arbitrary OrliczSobolev space. We show that the existence of an eigenvalue can be derived from a generalized version of Lagrange multiplier rule. Our approach also applies to more general problems. We emphasize that no 2 condition is imposed.

We study some asymptotic behaviour of phase interfaces with variable chemical potential under the uniform energy bound. The problem is motivated by the Cahn-Hilliard equation, where one has a control of the total energy and chemical potential. We show that the limit interface is an integral varifold with generalized Lp mean curvature. The convergence of interfaces as 0 is in the Hausdorff distance sense.

We study fine properties of currents in the framework of geometric measure theory on metric spaces developed by Ambrosio and Kirchheim, and we prove a rectifiability criterion for flat currents of finite mass. We apply these tools to study the structure of the distributional Jacobians of functions in the space BnV, defined by Jerrard and Soner. We define the subspace of special functions of bounded higher variation and we prove a closure theorem.

Letbe the open unit disc in C. It is proved that a holomorphic embedding f : C2 can grow arbitrarily fast near b. It is also proved that a holomorphic embedding f : CC2 can grow arbitrarily fast near infinity.

We introduce the concept of flat ridges for submanifolds of codimension 2 from the viewpoint of their contact with hyperplanes. We characterize them geometrically, studying some of their properties. In particular, we see that the highest-order flat ridges coincide with the flattenings of the asymptotic lines and from this we obtain some lower bounds for their numbers under appropriate conditions.

Let M be a compact Riemannian manifold with non-empty boundary M. In this paper we consider an inverse problem for the second-order hyperbolic initial-boundary-value problem utt + but + a(x, D)u = 0 in MR+, u|MR+ = f, u|t=0 = ut|t=0 = 0. Our goal is to determine (M, g), b and a(x, D) from the knowledge of the non-stationary Dirichlet-to-Neumann map (the hyperbolic response operator) RT, with sufficiently large T0. The response operator RT is the map , where is the normal derivative of the solution of the initial-boundary-value problem.

More specifically, we show the following.

1. (i) It is possible to determine Rt for any t0 if we know RT for sufficiently large T and some geometric condition upon the geodesic behaviour on (M, g) is satisfied.

2. (ii) It is then possible to determine (M, g) and b uniquely and the elliptic operator a(x, D) modulo generalized gauge transformations.

Radial deformations of a ball composed of a nonlinear elastic material and corresponding to cavitation have been much studied. In this paper we use rescalings to show that each such deformation can be used to construct infinitely many non-symmetric singular weak solutions of the equations of nonlinear elasticity for the same displacement boundary-value problem. Surprisingly, this property appears to have been unnoticed in the literature to date.

We consider the behaviour of buckling-driven thin-film blisterings using Von Karman's plate theory. Our results are asymptotic in the thinness and emphasize the incorporation of in-plane displacements in the model, a factor often ignored in the literature. Our work indicates that the inclusion of these displacements has a profound effect on the nature of solutions. We compare different constraints on displacements by estimating the leading order of corresponding energies. Our results strongly suggest that branching and other fine-scale structures arise in thin-film blistering, as has been observed in experiments.

The large-time asymptotic behaviour of real-valued solutions of the pure initial-value problem for Burgers' equation ut + uuxuxx = 0, is studied. The initial data satisfy u0(x) ~ nx as |x|, where nR. There are two constants of the motion that affect the large-time behaviour:Hopf considered the case n = 0 (i.e. u0L1(R)), and the casesufficiently small was considered by Dix. Here we completely remove that smallness condition. When n < 1, we find an explicit function U(), depending only onand n, such thatuniformly in . When n1, there are two different functions U() that simultaneously attract the quantity t12u(t12, t), and each one wins in its own range of . Thus we give an asymptotic description of the solution in different regions and compute its decay rate in L. Sharp error estimates are proved.

Letbe an arbitrary non-empty bounded Lipschitz domain in RMRN. Given> 0, squeezeby the factorin the y-direction to obtain the squeezed domain := {(x, y) | (x, y)}. Letandbe positive constants. Consider the following semilinear damped wave equation on ,where is the exterior normal vector field on and G is an appropriate nonlinearity, which ensures that (W) generates a (local) flow ̃ on X := H1()L2(). We show that there is a closed subspace X0 of X and a flow ̃0 on X0 that is the limit flow of the family ̃,> 0. We show that, as 0, the family ̃ converges in some singular sense to ̃ and establish a technical singular asymptotic compactness property. As a corollary, we obtain an upper-semicontinuity result for global attractors of the family ̃, 0, generalizing results obtained previously by Hale and Raugel for domains that are ordinate sets of a positive function.

The results obtained here are also applied in our paper On a general Conley index continuation principle for singular perturbation problems to establish a singular Conley index continuation principle for damped wave equations on thin domains.