Showing papers on "Eulerian path published in 1999"

Sorting Permutations by Reversals and Eulerian Cycle Decompositions

Abstract: We analyze the strong relationship among three combinatorial problems, namely, the problem of sorting a permutation by the minimum number of reversals (MIN-SBR), the problem of finding the maximum number of edge-disjoint alternating cycles in a breakpoint graph associated with a given permutation (MAX-ACD), and the problem of partitioning the edge set of an Eulerian graph into the maximum number of cycles (MAX-ECD). We first illustrate a nice characterization of breakpoint graphs, which leads to a linear-time algorithm for their recognition. This characterization is used to prove that MAX-ECD and MAX-ACD are equivalent, showing the latter to be NP-hard. We then describe a transformation from MAX-ACD to MIN-SBR, which is therefore shown to be NP-hard as well, answering an outstanding question which has been open for some years. Finally, we derive the worst-case performance of a well-known lower bound for MIN-SBR, obtained by solving MAX-ACD, discussing its implications on approximation algorithms for MIN-SBR.

An Eulerian Limited-Area Atmospheric Transport Model

Abstract: A limited-area, offline, Eulerian atmospheric transport model has been developed. The model is based on a terrain-following vertical coordinate and a mass-conserving, positive definite advection scheme with small phase and amplitude errors. The objective has been to develop a flexible, all-purpose offline model. The model includes modules for emission input, vertical turbulent diffusion, and deposition processes. The model can handle an arbitrary number of chemical components and provides a framework for inclusion of modules describing physical and chemical transformation processes between different components. Idealized test cases, as well as simulations of the atmospheric distribution of 222Rn, demonstrate the ability of the model to meet the requirements of mass conservation and positiveness and to produce realistic simulations of a simple atmospheric tracer.

A Finite Element Method Study of Eulerian Droplets Impingement Models

Abstract: To compute droplet impingement on airfoils, an Eulerian model for air flows containing water droplets is proposed as an alternative to the traditional Lagrangian particle tracking approach. Appropriate boundary conditions are presented for the droplets equations, with a stability analysis of the solution near the airfoil surface. Several finite element formulations are proposed to solve the droplets equations, based on conservative and non-conservative forms and using different stabilization terms. Numerical results on single and multi-elements airfoils for droplets of mean volume diameter, as well as for a Langmuir distribution of diameters, are presented and validated against measured values

Formulations and hardness of multiple sorting by reversals

Abstract: We consider two generalizations of signed Sorting Bg Reversals (SBR), both aimed at formalizing the problem of reconstructing the evolutionary history of a set of species. In particular, we address Multiple SBR, calling for a signed permutation at minimum reversal distance from a given set of signed permutations, and Dee SBR, calling for a tree with the minimum number of edges spanning a given set of nodes in the complete graph where each node corresponds to a signed permutation and there is an edge between each pair of signed permutations one reversal away from each other. We describe a graph-theoretic relaxation of MSBR, which is the counterpart of the so-called alternating-cycle decomposition relaxation for SBR., illustrating a convenient mathematical formulation for this relaxation. Moreover, we use this relaxation to show that, even if the number of given permutations equals 3, MSBR is NP-hard, and hence so is nee SBR. In fact, we show that the two problems are APX-hard, i.e. they do not have a polynomial-time approximation scheme unless P=NP. Finally, we mention known Zapproximation algorithms for two general problems which generalize MSBR and Tree SBR, respectively. To our knowledge, this work is the f?rst one discussing the complexity of MBSR (and Tree SBR), as well as potential solution approaches to the problem based on the use of a tight relaxation.

A Randomized Fully Polynomial Time Approximation Scheme for the All-Terminal Network Reliability Problem

Abstract: The classic all-terminal network reliability problem posits a graph, each of whose edges fails independently with some given probability. The goal is to determine the probability that the network becomes disconnected due to edge failures. This problem has obvious applications in the design of communication networks. Since the problem is $\SP$-complete and thus believed hard to solve exactly, a great deal of research has been devoted to estimating the failure probability. In this paper, we give a fully polynomial randomized approximation scheme that, given any n-vertex graph with specified failure probabilities, computes in time polynomial in n and $1/\epsilon$ an estimate for the failure probability that is accurate to within a relative error of $1\pm\epsilon$ with high probability. We also give a deterministic polynomial approximation scheme for the case of small failure probabilities. Some extensions to evaluating probabilities of k-connectivity, strong connectivity in directed Eulerian graphs and r-way disconnection, and to evaluating the Tutte polynomial are also described.

A particle-gridless hybrid method for incompressible flows

Abstract: A particle–gridless hybrid method for the analysis of incompressible flows is presented. The numerical scheme consists of Lagrangian and Eulerian phases as in an arbitrary Lagrangian–Eulerian (ALE) method, where a new-time physical property at an arbitrary position is determined by introducing an artificial velocity. For the Lagrangian calculation, the moving-particle semi-implicit (MPS) method is used. Diffusion and pressure gradient terms of the Navier–Stokes equation are calculated using the particle interaction models of the MPS method. As an incompressible condition, divergence of velocity is used while the particle number density is kept constant in the MPS method. For the Eulerian calculation, an accurate and stable convection scheme is developed. This convection scheme is based on a flow directional local grid so that it can be applied to multi-dimensional convection problems easily. A two-dimensional pure convection problem is calculated and a more accurate and stable solution is obtained compared with other schemes. The particle–gridless hybrid method is applied to the analysis of sloshing problems. The amplitude and period of sloshing are predicted accurately by the present method. The range of the occurrence of self-induced sloshing predicted by the present method shows good agreement with the experimental data. Calculations have succeeded even for the higher injection velocity range, where the grid method fails to simulate. Copyright © 1999 John Wiley & Sons, Ltd.

Enumerative theory of maps

Abstract: Preface. 1. Preliminaries. 1.1. Maps. 1.2. Polynomials on maps. 1.3. Enufunctions. 1.4. Polysum functions. 1.5. The Lagrangian inversion. 1.6. The shadow functional. 1.7. Asymptotic estimation. 1.8. Notes. 2. Outerplanar Maps. 2.1. Plane trees. 2.2. Wintersweets. 2.3. Unicyclic maps. 2.4. General outerplanar maps. 2.5. Notes. 3. Triangulations. 3.1. Outerplanar triangulations. 3.2. Planar triangulations. 3.3. Triangulations on the disc. 3.4. Triangulations on the projective plane. 3.5. Triangulations on the torus. 3.6. Notes. 4. Cubic Maps. 4.1. Planar cubic maps. 4.2. Bipartite cubic maps. 4.3. Cubic Hamiltonian maps. 4.4. Cubic maps on surfaces. 4.5. Notes. 5. Eulerian Maps. 5.1. Planar Eulerian maps. 5.2. Tutte formula. 5.3. Planar Eulerian triangulations. 5.4. Regular Eulerian maps. 5.5. Notes. 6. Nonseparable Maps. 6.1. Outerplanar nonseparable maps. 6.2. Eulerian nonseparable maps. 6.3. Planar nonseparable maps. 6.4. Nonseparable maps on the surfaces. 6.5. Notes. 7. Simple Maps. 7.1. Loopless maps. 7.2. Loopless Eulerian maps. 7.3. General simple maps. 7.4. Simple bipartite maps. 7.5. Notes. 8. General Maps. 8.1. General planar maps. 8.2. Planar c-nets. 8.3. Convex polyhedra. 8.4. Quadrangulations via c-nets. 8.5. General maps on surfaces. 8.6. Notes. 9. Chrosum Equations. 9.1. Tree equations. 9.2. Outerplanar equations. 9.3. General equations. 9.4. Triangulation equations. 9.5. Well-definedness. 9.6. Notes. 10. Polysum Equations. 10.1. Polysum for bitrees. 10.2. Outerplanar polysums. 10.3. General polysums. 10.4. Nonseparable polysums. 10.5. Notes. 11. Chromatic Solutions. 11.1. General solutions. 11.2. Cubic triangles. 11.3. Invariants. 11.4. Four color solutions. 11.5. Notes. 12. Stochastic Behaviors. 12.1. Asymptotics for outerplanar maps. 12.2. The average of tree-rooted maps. 12.3. Hamiltonian circuits per map. 12.4. The asymmetry of maps. 12.5. Asymptotics via equations. 12.6. Notes. Bibliography. Index.

Studying Self-Organized Criticality with Exactly Solved Models

Abstract: These lecture-notes are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models The abelian group structure of the algebra of particle addition operators, the burning test for recurrent states, equivalence to the spanning trees problem are described The exact solution of the directed version of the model in any dimension, and determination of the exponents for avalanche distribution are explained The model's equivalence to Scheidegger's model of river basins, Takayasu's aggregation model and the voter model is discussed For the undirected case, the exact solution in 1-dimension and on the Bethe lattice is briefly described Known results about the two dimensional case are summarized Generalization to the abelian distributed processors model is discussed, with the Eulerian walkers model and Manna's stochastic sandpile model as examples I conclude by listing some still-unsolved problems

Reversible molecular dynamics for rigid bodies and hybrid Monte Carlo

Abstract: A time-reversible molecular dynamics algorithm is presented for rigid bodies in the quarternion representation. The algorithm is developed on the basis of the Trotter factorization scheme, and its structure is similar to that of the velocity Verlet algorithm. When the rigid body is an asymmetric top, its computationally inconvenient Eulerian equation of motion is integrated by combining the computationally convenient solutions to the Eulerian equations of motion for two symmetric tops. It is shown that a larger time step is allowed in the time-reversible algorithm than in the Gear predictor–corrector algorithm. The efficiency of the hybrid Monte Carlo method for a molecular system is also examined using the time-reversible molecular dynamics algorithm in the quarternion representation.

Accurate Determination of the Lagrangian Bias for the Dark Matter Halos

Abstract: We use a new method, the cross-power spectrum between the linear density field and the halo number density field, to measure the Lagrangian bias for dark matter halos. The method has several important advantages over the conventional correlation function analysis. By applying this method to a set of high-resolution simulations of 2563 particles, we have accurately determined the Lagrangian bias, over 4 mag in halo mass, for four scale-free models with the index n=-0.5, -1.0, -1.5, and -2.0 and three typical cold dark matter models. Our result for massive halos with M≥M* (M* is a characteristic nonlinear mass) is in very good agreement with the analytical formula of Mo & White for the Lagrangian bias, but the analytical formula significantly underestimates the Lagrangian clustering for the less massive halos, M

Turbulent dispersion in a non-homogeneous field

Abstract: Dispersion of fluid particles in non-homogeneous turbulence was studied for fully developed flow in a channel. A point source at a distance of 40 wall units from the wall is considered. Data obtained by carrying out experiments in a direct numerical simulation (DNS) are used to test a stochastic model which utilized a modified Langevin equation. All of the parameters, with the exception of the time scales, are obtained from Eulerian statistics. Good agreement is obtained by making simple assumptions about the spatial variation of the time scales.

The Euler-Poincare Equations in Geophysical Fluid Dynamics

Abstract: Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: (1) Euler-Poincar\'e equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained; (2) an abstract Kelvin-Noether theorem is derived for such systems. By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincar\'{e} form and discuss their corresponding Kelvin-Noether theorems and potential vorticity conservation laws.

Non Asymptotic Properties of Transport and Mixing

Abstract: We study relative dispersion of passive scalar in non-ideal cases, i.e. in situations in which asymptotic techniques cannot be applied; typically when the characteristic length scale of the Eulerian velocity field is not much smaller than the domain size. Of course, in such a situation usual asymptotic quantities (the diffusion coefficients) do not give any relevant information about the transport mechanisms. On the other hand, we shall show that the Finite Size Lyapunov Exponent, originally introduced for the predictability problem, appears to be rather powerful in approaching the non-asymptotic transport properties. This technique is applied in a series of numerical experiments in simple flows with chaotic behaviors, in experimental data analysis of drifter and to study relative dispersion in fully developed turbulence.

Rotation Of Trajectories In Lagrangian Stochastic Models Of Turbulent Dispersion

Abstract: We present a new measure for the rotation of Lagrangian trajectories in turbulence that simplifies and generalises that suggested by Wilson and Flesch ( Boundary-Layer Meteorol. 84, 411–426). The new measure is the cross product of the velocity and acceleration and is directly related to the area, rather than the angle, swept out by the velocity vector. It makes it possible to derive a simple but exact kinematic expression for the mean rotation of the velocity vector and to partition this expression into terms that are closed in terms of Eulerian velocity moments up to second order and unclosed terms. The unclosed terms arise from the interaction of the fluctuating part of the velocity and the rate of change of the fluctuating velocity.

Taylor expansions in powers of time of Lagrangian and Eulerian two-point two-time velocity correlations in turbulence

Abstract: A method is developed for generating the Taylor expansions in powers of the time difference of the Lagrangian and Eulerian two-point two-time velocity correlations in turbulence. The expansions are based on the Taylor series of the Eulerian and Lagrangian velocity fields subject to given dynamics along with initial and boundary conditions. The lowest few coefficients in the expansions enable us to construct approximations to the correlations. An application of the method to turbulence obeying the Navier-Stokes dynamics yields approximations, particularly Pade approximations that agree well with direct numerical simulations of homogeneous isotropic turbulence at moderate Reynolds numbers. The ratios of the second-order to the zeroth-order coefficients of the Taylor series of the Lagrangian and Eulerian correlations give, respectively, the estimates for the Lagrangian and Eulerian micro time scales τL and τE. An analysis of a high resolution (5123 grid points) direct numerical simulation database at large Reynolds number suggests the scalings τL∝k−2/3 and τE∝k−1 for wave numbers k in the inertial subrange. The role of flow structures in turbulence in determining the time scales is also discussed.

Pair dispersion in synthetic fully developed turbulence.

Abstract: The Lagrangian statistics of relative dispersion in fully developed turbulence is numerically investigated. A scaling range spanning many decades is achieved by generating a two-dimensional velocity field by means of a stochastic process with prescribed statistics and of a dynamical model (shell model) with fluctuating characteristic times. When the velocity field obeys Kolmogorov similarity, the Lagrangian statistics is self similar and agrees with Richardson's predictions [Proc. R. Soc. London Ser. A 110, 709 (1926)]. For intermittent velocity fields the scaling laws for the Lagrangian statistics are found to depend on the Eulerian intermittency in agreement with the multifractal description. As a consequence of the Kolmogorov law the Richardson law for the variance of pair separation is, however, not affected by intermittency corrections. Moreover, Lagrangian exponents do not depend on the particular Eulerian dynamics. A method of data analysis, based on fixed scale statistics rather than usual fixed time statistics, is shown to give much wider scaling range, and should be preferred for the analysis of experimental data.

Developments of the arbitrary lagrangian-eulerian method in non-linear solid mechanics applications to forming processes

Abstract: The numerical simulation of forming processes can be performed using the updated Lagrangian method. However, in general in the case of large deformations distortion of the mesh occurs. As a result the calculation becomes inaccurate or it may even crash. Hence a complete remeshing for an element mesh with a new mesh topology is necessary. The Arbitrary Lagrangian{ Eulerian (ALE) method can be used to reduce mesh distortion in order to prevent complete remeshings. In this thesis we investigate the applicability of the ALE method and try to improve several of its aspects.

A combinatorial proof of the log-concavity of the numbers of permutations with $k$ runs

Abstract: We combinatorially prove that the number $R(n,k)$ of permutations of length $n$ having $k$ runs is a log-concave sequence in $k$, for all $n$. We also give a new combinatorial proof for the log-concavity of the Eulerian numbers.

Lagrangian theory of structure formation in pressure-supported cosmological fluids

Abstract: The Lagrangian theory of structure formation in cos- mological fluids, restricted to the matter model "dust", provides successful models of large-scale structure in the Universe in the laminar regime, i.e., where the fluid flow is single-streamed and "dust"-shells are smooth. Beyond the epoch of shell-crossing a qualitatively different behavior is expected, since in general anisotropic stresses powered by multi-stream forces arise in col- lisionless matter. In this paper we provide the basic framework for the modeling of pressure-supported fluids, restricting atten- tion to isotropic stresses and to the cases where pressure can be given as a function of the density. We derive the governing set of Lagrangian evolution equations and study the resulting system using Lagrangian perturbation theory. We discuss the first-order equations and compare them to the Eulerian theory of gravita- tional instability, as well as to the case of plane-symmetric col- lapse. We obtain a construction rule that allows to derive first- order solutions of the Lagrangian theory from known first-order solutions of the Eulerian theory and so extend Zel'dovich's ex- trapolation idea into the multi-streamed regime. These solutions can be used to generalize current structure formation models in the spirit of the "adhesion approximation".

The solution of two-dimensional free-surface problems using automatic mesh generation

Abstract: A new method is described for the iterative solution of two-dimensional free-surface problems, with arbitrary initial geometries, in which the interior of the domain is represented by an unstructured, triangular Eulerian mesh and the free surface is represented directly by the piecewise-quadratic edges of the isoparametric quadratic-velocity, linear-pressure Taylor–Hood elements. At each time step, the motion of the free surface is computed explicitly using the current velocity field and, once the new free-surface location has been found, the interior nodes of the mesh are repositioned using a continuous deformation model that preserves the original connectivity. In the event that the interior of the domain must be completely remeshed, a standard Delaunay triangulation algorithm is used, which leaves the initial boundary discretisation unchanged. The algorithm is validated via the benchmark viscous flow problem of the coalescence of two infinite cylinders of equal radius, in which the motion is due entirely to the action of capillary forces on the free surface. This problem has been selected for a variety of reasons: the initial and final (steady state) geometries differ considerably; in the passage from the former to the latter, large free-surface curvatures—requiring accurate modelling—are encountered; an analytical solution is known for the location of the free surface; there exists a large body of literature on alternative numerical simulations. A novel feature of the present work is its geometric generality and robustness; it does not require a priori knowledge of either the evolving domain geometry or the solution contained therein. Copyright © 1999 John Wiley & Sons, Ltd.

Using residence time for the extraction of recirculation regions

Abstract: This paper introduces the concept of residence-time, from the Eulerian view point, in a rigorous manner. The equations for various flow regimes are derived and a numerical solver is introduced based on Lax-Wendroff integration. An implementation is discussed that allows the coupling of this solver to any explicit CFD code. Examples of this concept are shown for extracting recirculation regions by segregating old fluid from fluid that has not been in the simulation for much time. The comparison of iso-surfaces generated using this procedure and separation surfaces are examine-d.

Flag vectors of Eulerian partially ordered sets

Abstract: The closed cone of flag vectors of Eulerian partially ordered sets is studied. It is completely determined up through rank seven. Half-Eulerian posets are defined. Certain limit posets of Billera and Hetyei are half-Eulerian; they give rise to extreme rays of the cone for Eulerian posets. A new family of linear inequalities valid for flag vectors of Eulerian posets is given.