Pattern‐avoiding permutations and Brownian excursion part I: Shapes and fluctuations

Permutations that avoid given patterns are among the most classical objects in combinatorics and have strong connections to many fields of mathematics, computer science and biology. In this paper we study the scaling limits of a random permutation avoiding a pattern of length 3 and their relations to Brownian excursion. Exploring this connection to Brownian excursion allows us to strengthen the recent results of Madras and Pehlivan [25] and Miner and Pak [29] as well as to understand many of the interesting phenomena that had previously gone unexplained. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 394–419, 2017     

Hierarchical Archimedean copulas through multivariate compound distributions

In this paper, we propose a new hierarchical Archimedean copula construction based on multivariate compound distributions. This new imbrication technique is derived via the construction of a multivariate exponential mixture distribution through compounding. The absence of nesting and marginal conditions, contrarily to the nested Archimedean copulas approach, leads to major advantages, such as a flexible range of possible combinations in the choice of distributions, the existence of explicit formulas for the distribution of the sum, and computational ease in high dimensions. A balance between flexibility and parsimony is targeted. After presenting the construction technique, properties of the proposed copulas are investigated and illustrative examples are given. A detailed comparison with other construction methodologies of hierarchical Archimedean copulas is provided. Risk aggregation under this newly proposed dependence structure is also examined.     

Mathematical modeling of two problems of wave dynamics in heterogeneous media

Two problems of heterogeneous media mechanics are investigated in the paper. The first one, concerned with the shock wave/dust layer interaction, is solved within the framework of the equilibrium model of heterogeneous media mechanics. The second problem deals with the simulation of a Riemann problem for a stratified layer of solid particles.This paper is based on work that was presented at the 20th International Colloquium on the Dynamics of Explosions and Reactive Systems, Montreal, Canada, July 31–August 5, 2005     

Rheological behavior of concentrated suspensions as affected by the dynamics of the mixing process

The rheological characterizations of concentrated suspensions are generally carried out assuming “well-mixed” suspensions. However, the variation of the concentration distributions of the ingredients of the formulation, i.e., the “goodness of mixing”, the size and shape distributions of the particle clusters and the rheological behavior of the suspension all depend on the thermo-mechanical history that the suspension is exposed to during the mixing process. Here, various experimental tools are used for the characterization of the degree of mixedness (concentration distributions) of various ingredients along with the characterization of rheological material functions, wall slip behavior and the maximum packing fraction of a graphite/elastomer suspension. The degree of mixedness values of the ingredients of the suspensions processed using batch and continuous processes and under differing operating conditions were characterized quantitatively using wide-angle X-ray diffraction and thermo gravimetric analysis and were elucidated under the light of the electrical properties of the suspension as affected by the mixing process. Upon achieving better homogeneity of the graphite particles and the binder and decreases in the size and breadth of the size distributions of particle clusters (as inferred from electrical measurements and maximum packing fraction values), the elasticity (storage modulus) and the shear viscosity (magnitude of the complex viscosity from small-amplitude oscillatory shear and shear viscosity from steady torsional and capillary rheometry) of the suspension decreased significantly and the wall slip velocity values increased. These findings demonstrate the intimate relationships that exist between the rheological behavior of concentrated suspensions and the thermo-mechanical history that they are exposed to during the processing stage and suggest that the preparation conditions for suspensions should be carefully selected and well documented to achieve reproducible characterization of rheological material functions.     

The Möbius function of separable and decomposable permutations

We give a recursive formula for the Möbius function of an interval [σ,π] in the poset of permutations ordered by pattern containment in the case where π is a decomposable permutation, that is, consists of two blocks where the first one contains all the letters 1,2,…,k for some k. This leads to many special cases of more explicit formulas. It also gives rise to a computationally efficient formula for the Möbius function in the case where σ and π are separable permutations. A permutation is separable if it can be generated from the permutation 1 by successive sums and skew sums or, equivalently, if it avoids the patterns 2413 and 3142.We also show that the Möbius function in the poset of separable permutations admits a combinatorial interpretation in terms of normal embeddings among permutations. A consequence of this interpretation is that the Möbius function of an interval [σ,π] of separable permutations is bounded by the number of occurrences of σ as a pattern in π. Another consequence is that for any separable permutation π the Möbius function of (1,π) is either 0, 1 or −1.     

Explicit ruin formulas for models with dependence among risks

We show that a simple mixing idea allows one to establish a number of explicit formulas for ruin probabilities and related quantities in collective risk models with dependence among claim sizes and among claim inter-occurrence times. Examples include compound Poisson risk models with completely monotone marginal claim size distributions that are dependent according to Archimedean survival copulas as well as renewal risk models with dependent inter-occurrence times.     

Influence of a combustion-driven oscillation on global mixing in the flame from a refinery flare

An assessment of the influence of strong combustion-driven oscillations on mixing rates and visible radiation in the flame from a full-scale refinery flare is reported. Importantly, the oscillations were generated naturally, with no external forcing, and at a high Reynolds number of 4 × 10⁶. These conditions differentiate this study from those of previous investigations, which all involved some external forcing and were at a Re too low to ensure fully turbulent flow within the flame. A frame-by-frame analysis of video footage, providing good resolution of the instantaneous edge of each flame, was used to assess flame dimensions, and so to determine a global residence time. Since the flames are in the fast-chemistry regime, the visual imagers can be used to determine a global mixing rate. The analysis reveals a consistent picture that the combustion-driven oscillations do not result in a significant change to the global mixing rate, but do increase the visible radiation. This is in contrast to previous investigations, using externally forced jets, where forcing at the preferred mode has been found to increase mixing rates and reduce radiation.     

Two-component Brownian coagulation: Monte Carlo simulation and process characterization

The compositional distribution within aggregates of a given size is essential to the functionality of composite aggregates that are usually enlarged by rapid Brownian coagulation. There is no analytical solution for the process of such two-component systems. Monte Carlo method is an effective numerical approach for two-component coagulation. In this paper, the differentially weighted Monte Carlo method is used to investigate two-component Brownian coagulation, respectively, in the continuum regime, the free-molecular regime and the transition regime. It is found that (1) for Brownian coagulation in the continuum regime and in the free-molecular regime, the mono-variate compositional distribution, i.e., the number density distribution function of one component amount (in the form of volume of the component in aggregates) satisfies self-preserving form the same as particle size distribution in mono-component Brownian coagulation; (2) however, for Brownian coagulation in the transition regime the mono-variate compositional distribution cannot reach self-similarity; and (3) the bivariate compositional distribution, i.e., the combined number density distribution function of two component amounts in the three regimes satisfies a semi self-preserving form. Moreover, other new features inherent to aggregative mixing are also demonstrated; e.g., the degree of mixing between components, which is largely controlled by the initial compositional mass fraction, improves as aggregate size increases.