On directed analogues of expander and hyperfinite graph sequences

We introduce and study analogues of expander and hyperfinite graph sequences in the context of directed acyclic graphs, which we call ‘extender’ and ‘hypershallow’ graph sequences, respectively. Our main result is a probabilistic construction of non-hypershallow graph sequences.

Randomized Construction of Complexes with Large Diameter

We consider the question of the largest possible combinatorial diameter among pure dimensional and strongly connected $$(d-1)$$ ( d - 1 ) -dimensional simplicial complexes on n vertices, denoted $$H_s(n, d)$$ H s ( n , d ) . Using a probabilistic construction we give a new lower bound on $$H_s(n, d)$$ H s ( n , d ) that is within an $$O(d^2)$$ O ( d 2 ) factor of the upper bound. This improves on the previously best known lower bound which was within a factor of $$e^{\varTheta (d)}$$ e Θ ( d ) of the upper bound. We also make a similar improvement in the case of pseudomanifolds.

Improved Ramsey-type results for comparability graphs

Several discrete geometry problems are equivalent to estimating the size of the largest homogeneous sets in graphs that happen to be the union of few comparability graphs. An important observation for such results is that if G is an n-vertex graph that is the union of r comparability (or more generally, perfect) graphs, then either G or its complement contains a clique of size $n^{1/(r+1)}$ .This bound is known to be tight for $r=1$ . The question whether it is optimal for $r\ge 2$ was studied by Dumitrescu and Tóth. We prove that it is essentially best possible for $r=2$ , as well: we introduce a probabilistic construction of two comparability graphs on n vertices, whose union contains no clique or independent set of size $n^{1/3+o(1)}$ .Using similar ideas, we can also construct a graph G that is the union of r comparability graphs, and neither G nor its complement contain a complete bipartite graph with parts of size $cn/{(log n)^r}$ . With this, we improve a result of Fox and Pach.

GENERATING A PROBABILISTIC CONSTRUCTION SCHEDULE

Introduction: The design stage and preparations for construction include the development of construction schedules needed to justify the duration of construction works. Methods: Based on probabilistic scheduling, a multitude of solutions can be generated for each implementation roadmap (progress chart). These decisions can be defined as optimistic, most probable or likely, and pessimistic. Rational roadmaps are selected in accordance with benchmarking. Simple and discounted payback periods are used as frequently applied criteria included in the system of evaluating the economic effectiveness of investment projects. Based on identifying the given indicators of project evaluation, a method of designing probabilistic construction progress charts has been developed; the latter serve as the basis for devising respective organizational-technological solutions. Results: The design of optimistic, pessimistic, and most probable construction roadmaps (schedules or progress charts) enables the use of a developed model for probabilistic prognostication of future production risks affecting the delay of construction completion.

Gaussian multiplicative chaos and Liouville quantum gravity

The purpose of this chapter is to explain the probabilistic construction of Polyakov’s Liouville quantum gravity using the theory of Gaussian multiplicative chaos. In particular, this chapter contains a detailed description of the so-called Liouville measures of the theory and their conjectured relation to the scaling limit of large planar maps properly embedded in the sphere. This chapter is rather short and requires no prior knowledge on the topic.