The Stochastic Eulerian Tour Problem was introduced in 2008 [1] as a stochastic variant of the well-known Eulerian Tour Problem. In a follow-up paper the same authors investigated some heuristics for solving the Stochastic Eulerian Tour Problem [2]. After a thorough study of these two publications a few issues emerged. In this short research commentary we would like to discuss these issues. For...

A number of researchers studying permutation statistics on the symmetric group Sn have considered pairs (x, Y), where x is an Eulerian statistic and Y is a Mahonian statistic. Of special interest are pairs such as (des, maj), whose joint distribution on Sn is given by Carlitz’s q-Eulerian polynomials. We present a natural Eulerian statistic stc such that the pair (stc, inv) is equally distribut...

We completely characterize the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n) = n! has...

The number of planar Eulerian maps with n edges is well-known to have a simple expression. But what is the number of planar Eulerian orientations with n edges? This problem appears to be difficult. To approach it, we define and count families of subsets and supersets of planar Eulerian orientations, indexed by an integer k, that converge to the set of all planar Eulerian orientations as k incre...

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B(n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2n−1, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B(n...

This study reports the mutual transformations between the third-order Eulerian and Lagrangian solutions for the irrotational progressive gravity waves propagating on the uniform depth. Based on the fundamental principle of kinematics, the instantaneous Eulerian and Lagrangian velocities are identically equal, and using a successive Taylor series expansion to the path and the period of particle ...

A graph G is Eulerian-connected if for any u and v in V (G), G has a spanning (u. v)-trail. A graph G is edge-Eulerian-connected if for any e' and elf in E(G), G has a spanning (e', elf)-trail. For an integer r ~ 0, a graph is called r-Eulerian-connected if for any X s::: E(G) with IXI ~r, and for any u, v E V(G), G has a spanning (u, v)-trail T such that X s::: E(T). The r-edge-Eulerian­ conne...

To assess the impact of dynamical formulation on climate simulations, a semi-Lagrangian and an Eulerian dynamical core have been used for 5-yr climate simulations with the same physical parameterizations. The comparison of the climate simulations is focused on various eddy statistics (the study of time-mean states from the simulations has been published in a previous paper). Significant differe...

In this work, we present a Fully Eulerian framework for fluid-structure interaction (fsi) problems coupling the incompressible Navier-Stokes equations with a hyperelastic solid. The Fully Eulerian framework is a monolithic implicit variational formulation for the coupled problem. In contrast to the well-established Arbitrary Lagrangian Eulerian (ALE) coordinates, the Fully Eulerian framework fo...

We defined the symmetric TSP in lecture 6. The best known approximation algorithm for STSP is the 3/2 approximation algorithm of Christofides. The algorithm is very easy to describe, first choose a minimum cost spanning tree and then add that minimum cost Eulerian augmentation to the tree, that is the minimum cost set of edges to make the tree Eulerian. An undirected graph is Eulerian when ever...