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Supermodular ordering of Poisson and binomial random vectors by tree-based correlations

Kızıldemir B., Privault N.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 2

385--405

We construct a dependence structure for binomial, Poisson and Gaussian random vectors, based on partially ordered binary trees and sums of independent random variables. Using this construction, we characterize the supermodular ordering of such random vectors via the componentwise ordering of their covariance matrices. For this, we apply Möbius inversion techniques on partially ordered trees, which allow us to connect the Lévy measures of Poisson random vectors on the discrete d-dimensional hypercube to their covariance matrices.

Efficiency Enhancement of Optimal Reduction Method by Strengthening Parallelism of Structural Models Formation Process

Tkachenko S., Soprunyuk O., Tkachenko V., Solomko I.

Machine Dynamics Research

2013

Vol. 37, No. 2

85-90

In the process of complex objects and systems design the graph and multiple models of their structures are widely applied. Representation of structure models in a form of binary trees is the most economical and optimal for a number of databases designing and creating tasks. Parallel reduction is one of the best methods of binary trees formation. The complexity assessment of basic formation procedure of the reduction binary tree for complex objects (especially with significantly irregular structure) results in О(n3), which causes problems in the process of solving the large dimension tasks, where n is the number of system elements. It is possible to reduce the task complexity artificially increasing parallelism of the reduction process. Several studies have revealed the possibility to reduce the task complexity assessment up to О(n2) without any loss of the result quality.

Generating Functions of Embedded Trees and Lattice Paths

Kuba M.

Fundamenta Informaticae

2012

Vol. 117, nr 1-4

215-227

Bouttier, Di Francesco and Guitter introduced a method for solving certain classes of algebraic recurrence relations arising the context of maps and embedded trees. The aim of this note is to apply their method, consisting of a suitable ansatz and (computer assisted) guessing, to three problems, all related to the enumeration of lattice paths. First, we derive the generating function of a family of embedded binary trees, unifying some earlier results in the literature. Second, we show that several enumeration problems concerning so-called simple families of lattice paths can be solved without using the kernel method. Third, we use their method to (re-)derive the length generating function of three vicious walkers and osculating walkers.