This article considers a family of functionals J to be maximized over the planar convex sets K for which the perimeter and Steiner point have been fixed. Assuming that J is the integral of a positive quadratic expression in the support function h and its derivative, the maximizer is always either a triangle or a line segment (which can be considered as a collapsed triangle). Among the concrete consequences of the main theorem is the fact that, given any convex body K1 of finite perimeter, the set in this class that is farthest away in the sense of the L2 distance is always a line segment. The same property is proved for the Hausdorff distance.

Let X be the random variable that counts the number of triangles in the binomial random graph G(n, p). We show that for some positive constant c, the probability that X deviates from its expectation by at least λVar(X)1/2 is at most e−cλ2, provided p = o(1), λ = ω() and λ ≤ (n3p3 + n4p5)1/6.

Consider a finite irreducible Markov chain on state space S with transition matrix M and stationary distribution π. Let R be the diagonal matrix of return times, Rii = 1/πi. Given distributions σ, τ and k ∈ S, the exit frequency xk(σ, τ) denotes the expected number of times a random walk exits state k before an optimal stopping rule from σ to τ halts the walk. For a target distribution τ, we define Xτ as the n × n matrix given by (Xτ)ij = xj(i, τ), where i also denotes the singleton distribution on state i.

The dual Markov chain with transition matrix = RM⊤R−1 is called the reverse chain. We prove that Markov chain duality extends to matrices of exit frequencies. Specifically, for each target distribution τ, we associate a unique dual distribution τ*. Let denote the matrix of exit frequencies from singletons to τ* on the reverse chain. We show that , where b is a non-negative constant vector (depending on τ). We explore this exit frequency duality and further illuminate the relationship between stopping rules on the original chain and reverse chain.

Consider a randomly oriented graph G = (V, E) and let a, s and b be three distinct vertices in V. We study the correlation between the events {a → s} and {s → b}. We show that, counter-intuitively, when G is the complete graph Kn, n ≥ 5, then the correlation is positive. (It is negative for n = 3 and zero for n = 4.) We briefly discuss and pose problems for the same question on other graphs.

We prove two results about the asymptotic formula for The first result is for x7/12+ε≤h≤x and h/(log x)B≤Q≤h, where ε,B>0 are arbitrary constants. For the second result we assume that the Generalized Riemann Hypothesis holds and we obtain a stronger error term and a better uniformity on h.

The inspection game has been widely used as a model for crime deterrence. In this article, we shall present a broad review of the literature and an original result. The original result concerns the interaction between a single, long-run inspector and a population of short-run, boundedly rational would-be criminals. This approach gives rise to an optimal control problem that can be solved explicitly. We show that a forward-looking inspector obtains the same payoff he would obtain if he could commit to a probability of inspection. The consequences of this result for the economics of crime deterrence are also explored.

Using a local construction from a previous paper, we exhibit a numerical invariant, the differential Swan conductor, for an isocrystal on a variety over a perfect field of positive characteristic overconvergent along a boundary divisor; this leads to an analogous construction for certain p-adic and l-adic representations of the étale fundamental group of a variety. We then demonstrate some variational properties of this definition for overconvergent isocrystals, paying special attention to the case of surfaces.

We study naturality properties of the transverse invariant in knot Floer homology under contact (+1)-surgery. This can be used as a calculational tool for the transverse invariant. As a consequence, we show that the Eliashberg–Chekanov twist knots En are not transversely simple for n odd and n > 3.

A new approach is proposed for the description of thin viscous flows near the edges of a solid surface. For a steady flow, the lubrication approximation and the no-slip condition are assumed to be valid on most of the surface, except for relatively small neighbourhoods of the edges, where a universality principle is postulated: the behaviour of the liquid in these regions is universally determined by flux, external conditions and material properties. The resulting mathematical model is formulated as an ordinary differential equation involving the height of the liquid film and the flux as unknowns, and analytical results are outlined. The form of the universal functions which describe the behaviour in the edge regions is also discussed, obtaining conditions of compatibility with lubrication theory for small fluxes. Finally, an ordinary differential equation is introduced for the description of intermediate asymptotic profiles of a liquid film which flows off a bounded solid surface.