The eigenvalue spacing of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed arc of the unit circle and offer an approach toward other asymptotics. For the case of all unipotent matrices, the proof gives a probabilistic interpretation to identities of Macdonald from symmetric funct...

Implicit modeling with skeleton-based primitives has been limited up to now to planar skeletons elements, since no closed-form solution was found for convolution along more complex curves. We show that warping techniques can be adapted to efficiently generate convolution-like implicit primitives of varying radius along helices, a useful 3D skeleton found in a number of natural shapes. Depending...

in R, where t ranges over the unit circle S = R/2πZ. The curve Uk(t) is locally neighborly: as long as t1, . . . , tk lie in an open arc of S of a certain length φk > 0, the convex hull of the points Uk (t1) , . . . , Uk (tk) is a face of Bk. We characterize the maximum possible length φk, proving, in particular, that φk > π/2 for all k and that the limit of φk is π/2 as k grows. This allows us...

In Euclidean geometry the vertices P of those angles ∠APB of size α that pass through the endpoints A, B of a given segment trace the arc of a circle. In hyperbolic geometry on the other hand a set of equivalently defined points P determines a different kind of curve. In this paper the most basic property of the curve, its convexity, is established. No straight-forward proof could be found. The...

The aim of edge modification problems is to change the edge set of a given graph as little as possible in order to satisfy a certain property. Edge modification problems in graphs have a lot of applications in different areas, and many polynomial-time algorithms and NP-completeness proofs for this kind of problems are known. In this work we prove new NP-completeness results for these problems i...

We investigate the “hot spots” property for the survival time probability of Brownian motion with killing and reflection in planar convex domains whose boundary consists of two curves, one of which is an arc of a circle, intersecting at acute angles. This leads to the “hot spots” property for the mixed Dirichlet–Neumann eigenvalue problem in the domain with Neumann conditions on one of the curv...