A novel Eulerian Gaussian beam method was developed in [8] to compute the Schrödinger equation efficiently in the semiclassical regime. In this paper, we introduce an efficient semi-Eulerian implementation of this method. The new algorithm inherits the essence of the Eulerian Gaussian beam method where the Hessian is computed through the derivatives of the complexified level set functions inste...

Let G be an eulerian digraph with a fixed edge coloring (not necessarily a proper edge coloring). A compatible circuit of G is an eulerian circuit such that every two consecutive edges in the circuit have different colors. We characterize the existence of compatible circuits for directed graphs avoiding certain vertices of outdegree three. Our result is analogous to a result of Kotzig for compa...

Dilute liquid sprays can be modeled at the mesoscale using a kinetic equation, namely the Williams-Boltzmann equation, containing terms for spatial transport, evaporation and fluid drag. The most common method for simulating the Williams-Boltzmann equation uses Lagrangian particle tracking wherein a finite ensemble of numerical “parcels” provides a statistical estimate of the joint surface area...

An improved treatment of advection is essential for atmospheric transport and chemistry models. Eulerian treatments are generally plagued with instabilities, unrealistic negative constituent values, diffusion, and dispersion errors. A higher-order Eulerian model improves one error at significant cost but magnifies another error. The cost of semi-Lagrangian models is too high for many applicatio...

We present a general Lagrangian formalism that allows the treatment of vorticity. We give solutions for the rotational perturbations up to the third-order in a flat background universe. We show how the primordial vorticity affects the evolution of the density fluctuation in high-density regions. The study of the galaxy formation based on gravitational instability has not revealed the origin of ...

We show that the problem of counting the number of Eulerian circuits in an undirected graph is complete for the class #P. The method employed is mod-p reduction from counting Eulerian orientations.

Let G be a simple connected graph all of whose vertices have even degrees. An Eulerian circuit in G is a closed walk (see, for example, [2]) which uses every edge of G exactly once. Two Eulerian circuits are called equivalent if one is a cyclic permutation of the other. It is clear that the size of such an equivalence class equals the number of edges of graph G. Let EC(G) denote the number of e...

The (q, r)-Eulerian polynomials are the (maj−exc, fix, exc) enumerative polynomials of permutations. Using Shareshian and Wachs’ exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical q-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han’s three-variable statistic (inv−lec, pix, lec). We also prove a new ...

1.1. Eulerian and Lagrangian coordinates. Let us begin with Eulerian and Lagrangian coordinates. The Eulerian coordinate (x, t) is the physical space plus time. The Eulerian description of the flow is to describe the flow using quantities as a function of a spatial location x and time t, e.g. the flow velocity u(x, t). This can be visualized by sitting on the bank of a river and watching the wa...