Denote by x a random innnite path in the graph of Pascal's triangle (left and right turns are selected independently with xed probabilities) and by d n (x) the binomial coeecient at the n'th level along the path x. Then for a dense G set of in the unit interval, fd n (x)g is almost surely dense but not uniformly distributed modulo 1.

We discuss a parallel library of efficient algorithms for model reduction of largescale systems with state-space dimension up to O(104). We survey the numerical algorithms underlying the implementation of the chosen model reduction methods. The approach considered here is based on state-space truncation of the system matrices and includes absolute and relative error methods for both stable and ...

Denote by x a random infinite path in the graph of Pascal’s triangle (left and right turns are selected independently with fixed probabilities) and by dn(x) the binomial coefficient at the n’th level along the path x. Then for a dense Gδ set of θ in the unit interval, {dn(x)θ} is almost surely dense but not uniformly distributed modulo 1.

In this paper, we propose a symbolic-numerical algorithm for collision-free placement and motion of an object avoiding collisions with obstacles. The algorithm is based on the combination of configuration space and energy approaches. According to the configuration space approach, the position and orientation of the geometric object to be moved or placed is represented as an individual point in ...

The boundary problem for graphs like Pascal’s but with general multiplicities of edges is related to a ‘backward’ problem of moments of the Hausdorff type.

The partition function p and Ramanujan’s τ -function are related by similar generating functions and consequently by similar recursions involving the sum-ofdivisors function σ(n). These recursions are formally similar to Newton’s symmetric function relations, and one of our goals in this paper is to point out that the GirardWaring solution of Newton’s relations yields a formula for τ(n) as a su...

Abstract: We consider a square matrix Aǫ whose entries have first order asymptotics of the form (Aǫ)i j ∼ ai j ǫAi j when ǫ goes to 0, for some ai j ∈ C and Ai j ∈ R. We show that under a non-degeneracy condition, the order of magnitudes of the different eigenvalues ofAǫ are given by min-plus eigenvalues of min-plus Schur complements built from A = (Ai j ), or equivalently by generalized minima...

The Strahler number or Horton-Strahler number of a tree, originally introduced in geophysics, has a surprisingly rich theory. We sketch some milestones in its history, and its connection to arithmetic expressions, graph traversing, decision problems for context-free languages, Parikh’s theorem, and Newton’s procedure for approximating zeros of differentiable functions.

Boundary representation models reconstructed from 3D range data suffer from various inaccuracies caused by noise in the data and the model building software. The quality of such models can be improved in a beautification step, which finds geometric regularities approximately present in the model and tries to impose a consistent subset of these regularities on the model. A framework for beautifi...

We are used to imagining Pascal’s triangle as extending forever downwards from a vertex located at the top. But it is interesting to see it as occupying the first quadrant of the plane with it’s vertex at (0, 0). Imagine further that the plane is made of graph paper — that is, that we have embedded into it the Cayley graph of Z × Z with respect to the standard generating set. If we place the en...