We study the issues of the reconstruction and stability of the inverse nodal problem for the one-dimensional p-Laplacian eigenvalue problem. A key step is the application of a modified Prüfer substitution to derive a detailed asymptotic expansion for the eigenvalues and nodal lengths. Two associated Ambarzumyan problems are also solved.

A method of expressing a sum of money as a decimal of a £ to 3 places has long been known. When further decimals are necessary, they may be got by the following simple rule, which appears to be new: multiply the last two decimals found at any stage of the process (after the first 3 have been obtained) by 4, take the digit in the ten's place of the product, append it to the decimals already found as the next decimal, and repeat the process; with the proviso that should the 4-product end with 48, 68, or 88 the digit in the ten's place is to be taken as 5, 7, or 9 respectively.

Let M be a compact connected oriented surface with exactly one boundary component. A pseudo-immersion of M into the plane is a smooth map such that it has only fold singularities and is an orientation-preserving immersion near the boundary of M. For a pseudo-immersion we study the relation between the winding number of the boundary curve and the number of the singular set components.

The object of the present paper is merely to exhibit the methods employed by the ancient Greek geometers in their solution of this celebrated problem. A critical discussion of these methods, of the origin of the problem, and perhaps also an account of more recent researches and a notice of the literature connected therewith may form the subject of a subsequent paper.

A differential inequality technique is used to determine a lower bound on the blow-up time for solutions to the heat equation subject to a nonlinear boundary condition when blow-up of the solution does occur. In addition, a sufficient condition which implies that blow-up does occur is determined.

This article presents uniform random generators of plane partitions according to size (the number of cubes in the 3D interpretation). Combining a bijection of Pak with the method of Boltzmann sampling, we obtain random samplers that are slightly superlinear: the complexity is O(n(ln n)3) in approximate-size sampling and O(n4/3) in exact-size sampling (under a real-arithmetic computation model). To our knowledge, these are the first polynomial-time samplers for plane partitions according to size (there exist polynomial-time samplers of another type, which draw plane partitions that fit inside a fixed bounding box). The same principles yield efficient samplers for (a × b)-boxed plane partitions (plane partitions with two dimensions bounded), and for skew plane partitions. The random samplers allow us to perform simulations and observe limit shapes and frozen boundaries, which have been analysed recently by Cerf and Kenyon for plane partitions, and by Okounkov and Reshetikhin for skew plane partitions.

We investigate the rank of random (symmetric) sparse matrices. Our main finding is that with high probability, any dependency that occurs in such a matrix is formed by a set of few rows that contains an overwhelming number of zeros. This allows us to obtain an exact estimate for the co-rank.