The $k$-Cut Model in Deterministic and Random Trees

The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.

Embedding Small Digraphs and Permutations in Binary Trees and Split Trees

We investigate the number of permutations that occur in random labellings of trees. This is a generalisation of the number of subpermutations occurring in a random permutation. It also generalises some recent results on the number of inversions in randomly labelled trees (Cai et al. in Combin Probab Comput 28(3):335–364, 2019). We consider complete binary trees as well as random split trees a large class of random trees of logarithmic height introduced by Devroye (SIAM J Comput 28(2):409–432, 1998. 10.1137/s0097539795283954). Split trees consist of nodes (bags) which can contain balls and are generated by a random trickle down process of balls through the nodes. For complete binary trees we show that asymptotically the cumulants of the number of occurrences of a fixed permutation in the random node labelling have explicit formulas. Our other main theorem is to show that for a random split tree, with probability tending to one as the number of balls increases, the cumulants of the number of occurrences are asymptotically an explicit parameter of the split tree. For the proof of the second theorem we show some results on the number of embeddings of digraphs into split trees which may be of independent interest.

Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

Almost Giant Clusters for Percolation on Large Trees with Logarithmic Heights

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

LANG'S CONJECTURE AND SHARP HEIGHT ESTIMATES FOR THE ELLIPTIC CURVES y2 = x3 + ax

For elliptic curves given by the equation Ea : y2 = x3 + ax, we establish the best-possible version of Lang's conjecture on the lower bound for the canonical height of non-torsion rational points along with best-possible upper and lower bounds for the difference between the canonical and logarithmic height.

Seven-Sensor Fast-Response Probe for Full-Scale Wind Turbine Flowfield Measurements

The unsteady wind profile in the atmospheric boundary layer upstream of a modern wind turbine is measured. The measurements are accomplished using a novel measurement approach that is comprised of an autonomous uninhabited aerial vehicle (UAV) that is equipped with a seven-sensor fast-response aerodynamic probe (F7S-UAV). The autonomous UAV enables high spatial resolution (∼6.3% of rotor diameter) measurements, which hitherto have not been accomplished around full-scale wind turbines. The F7S-UAV probe developed at ETH Zurich is the key-enabling technology for the measurements. The time-averaged wind profile from the F7S-UAV probe is found to be in very good agreement to an independently measured profile using the UAV. This time-averaged profile, which is measured in moderately complex terrain, differs by as much as 30% from the wind profile that is extrapolated from a logarithmic height formula; therefore, the limited utility of extrapolated profiles, which are commonly used in site assessments, is made evident. The time-varying wind profiles show that at a given height, the velocity fluctuations can be as much as 44% of the time-averaged velocity, therefore indicating that there are substantial loads that may impact the fatigue life of the wind turbine’s components. Furthermore, the shear in the velocity profile also subjects the fixed pitch blade to varying incidences and loading. Analysis of the associated velocity triangles indicates that the sectional lift coefficient at midspan of this modern turbine would vary by 12% in the measured time-averaged wind profile. These variations must be accounted in the structural design of the blades. Thus, the measurements of the unsteady wind profile accomplished with this novel measurement system demonstrate that it is a cost effective complement to the suite of available site assessment measurement tools.

SKIP QUADTREES: DYNAMIC DATA STRUCTURES FOR MULTIDIMENSIONAL POINT SETS

We present a new multi-dimensional data structure, which we call the skip quadtree (for point data in R2) or the skip octree (for point data in Rd, with constant d > 2). Our data structure combines the best features of two well-known data structures, in that it has the well-defined “box”-shaped regions of region quadtrees and the logarithmic-height search and update hierarchical structure of skip lists. Indeed, the bottom level of our structure is exactly a region quadtree (or octree for higher dimensional data). We describe efficient algorithms for inserting and deleting points in a skip quadtree, as well as fast methods for performing point location, approximate range, and approximate nearest neighbor queries.

HEIGHTS AND LOGARITHMIC gcd ON ALGEBRAIC CURVES

Let F(x,y) be an irreducible polynomial over ℚ, satisfying F(0,0) = 0. Skolem proved that the integral solutions of F(x,y) = 0 with fixed gcd are bounded [13] and Walsh gave an explicit bound in terms of d = gcd (x,y) and F [16]. Assuming that (0,0) is a non-singular point of the plane curve F(x,y) = 0, we extend this result to algebraic solution, and obtain an asymptotic equality instead of inequality. We show that for any algebraic solution (α,β), the quotient h(α)/ log d is approximatively equal to degyF and the quotient h(β)/ log d to deg x F; here h(·) is the absolute logarithmic height and d is the (properly defined) "greatest common divisor" of α and β.