Possibility of Turbulence from a Post-Navier-Stokes Equation

We introduce corrections to the Navier-Stokes equation arising from the transitions between molecular states and the injection of external energy. In the simplest application of the proposed post Navier-Stokes equation, we find a multi-valued velocity field and the immediate possibility of velocity reversal, both features of turbulence

On Matrices, Automata, and Double Counting

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables M, with the same constraint defined by a finite-state automaton A on each row of M and a global cardinality constraint gcc on each column of M. We give two methods for deriving, by double counting, necessary conditions on the cardinality variables of the gcc constraints from the automaton A. The first method yields linear necessary conditions and simple arithmetic constraints. The second method introduces the cardinality automaton, which abstracts the overall behaviour of all the row automata and can be encoded by a set of linear constraints. We evaluate the impact of our methods on a large set of nurse rostering problem instances

Pdf's of Derivatives and Increments for Decaying Burgers Turbulence

A Lagrangian method is used to show that the power-law with a -7/2 exponent in the negative tail of the pdf of the velocity gradient and of velocity increments, predicted by E, Khanin, Mazel and Sinai (1997 Phys. Rev. Lett. 78, 1904) for forced Burgers turbulence, is also present in the unforced case. The theory is extended to the second-order space derivative whose pdf has power-law tails with exponent -2 at both large positive and negative values and to the time derivatives. Pdf's of space and time derivatives have the same (asymptotic) functional forms. This is interpreted in terms of a "random Taylor hypothesis".Comment: LATEX 8 pages, 3 figures, to appear in Phys. Rev.

Automatically Improving Constraint Models in Savile Row through Associative-Commutative Common Subexpression Elimination

When solving a problem using constraint programming, constraint modelling is widely acknowledged as an important and difficult task. Even a constraint modelling expert may explore many models and spend considerable time modelling a single problem. Therefore any automated assistance in the area of constraint modelling is valuable. Common sub-expression elimination (CSE) is a type of constraint reformulation that has proved to be useful on a range of problems. In this paper we demonstrate the value of an extension of CSE called Associative-Commutative CSE (AC-CSE). This technique exploits the properties of associativity and commutativity of binary operators, for example in sum constraints. We present a new algorithm, X-CSE, that is able to choose from a larger palette of common subexpressions than previous approaches. We demonstrate substantial gains in performance using X-CSE. For example on BIBD we observed speed increases of more than 20 times compared to a standard model and that using X-CSE outperforms a sophisticated model from the literature. For Killer Sudoku we found that X-CSE can render some apparently difficult instances almost trivial to solve, and we observe speed increases up to 350 times. For BIBD and Killer Sudoku the common subexpressions are not present in the initial model: an important part of our methodology is reformulations at the preprocessing stage, to create the common subexpressions for X-CSE to exploit. In summary we show that X-CSE, combined with preprocessing and other reformulations, is a powerful technique for automated modelling of problems containing associative and commutative constraints

Anomalous scaling, nonlocality and anisotropy in a model of the passively advected vector field

A model of the passive vector quantity advected by a Gaussian time-decorrelated self-similar velocity field is studied; the effects of pressure and large-scale anisotropy are discussed. The inertial-range behavior of the pair correlation function is described by an infinite family of scaling exponents, which satisfy exact transcendental equations derived explicitly in d dimensions. The exponents are organized in a hierarchical order according to their degree of anisotropy, with the spectrum unbounded from above and the leading exponent coming from the isotropic sector. For the higher-order structure functions, the anomalous scaling behavior is a consequence of the existence in the corresponding operator product expansions of ``dangerous'' composite operators, whose negative critical dimensions determine the exponents. A close formal resemblance of the model with the stirred NS equation reveals itself in the mixing of operators. Using the RG, the anomalous exponents are calculated in the one-loop approximation for the even structure functions up to the twelfth order.Comment: 37 pages, 4 figures, REVTe

Fast finite difference solvers for singular solutions of the elliptic Monge-Amp\`ere equation

The elliptic Monge-Ampere equation is a fully nonlinear Partial Differential Equation which originated in geometric surface theory, and has been applied in dynamic meteorology, elasticity, geometric optics, image processing and image registration. Solutions can be singular, in which case standard numerical approaches fail. In this article we build a finite difference solver for the Monge-Ampere equation, which converges even for singular solutions. Regularity results are used to select a priori between a stable, provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain. This allows singular solutions to be computed using a stable method, and regular solutions to be computed more accurately. The resulting nonlinear equations are then solved by Newton's method. Computational results in two and three dimensions validate the claims of accuracy and solution speed. A computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities.Comment: 23 pages, 4 figures, 4 tables; added arxiv links to references, added coment

Dynamical equations for high-order structure functions, and a comparison of a mean field theory with experiments in three-dimensional turbulence

Two recent publications [V. Yakhot, Phys. Rev. E {\bf 63}, 026307, (2001) and R.J. Hill, J. Fluid Mech. {\bf 434}, 379, (2001)] derive, through two different approaches that have the Navier-Stokes equations as the common starting point, a set of steady-state dynamic equations for structure functions of arbitrary order in hydrodynamic turbulence. These equations are not closed. Yakhot proposed a "mean field theory" to close the equations for locally isotropic turbulence, and obtained scaling exponents of structure functions and an expression for the tails of the probability density function of transverse velocity increments. At high Reynolds numbers, we present some relevant experimental data on pressure and dissipation terms that are needed to provide closure, as well as on aspects predicted by the theory. Comparison between the theory and the data shows varying levels of agreement, and reveals gaps inherent to the implementation of the theory.Comment: 16 pages, 23 figure

Hydrodynamic turbulence and intermittent random fields

In this article, we construct two families of nonsymmetrical multifractal fields. One of these families is used for the modelization of the velocity field of turbulent flows.Comment: 25 Pages; to appear in Communications in Mathematical Physic

Anomalous scaling of a passive scalar in the presence of strong anisotropy

Field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar field, advected by the Gaussian strongly anisotropic velocity field. Inertial-range anomalous scaling behavior is established, and explicit asymptotic expressions for the n-th order structure functions of scalar field are obtained; they are represented by superpositions of power laws with nonuniversal (dependent on the anisotropy parameters) anomalous exponents. In the limit of vanishing anisotropy, the exponents are associated with tensor composite operators built of the scalar gradients, and exhibit a kind of hierarchy related to the degree of anisotropy: the less is the rank, the less is the dimension and, consequently, the more important is the contribution to the inertial-range behavior. The leading terms of the even (odd) structure functions are given by the scalar (vector) operators. For the finite anisotropy, the exponents cannot be associated with individual operators (which are essentially ``mixed'' in renormalization), but the aforementioned hierarchy survives for all the cases studied. The second-order structure function is studied in more detail using the renormalization group and zero-mode techniques.Comment: REVTEX file with EPS figure

Discriminating instance generation for automated constraint model selection

One approach to automated constraint modelling is to generate, and then select from, a set of candidate models. This method is used by the automated modelling system Conjure. To select a preferred model or set of models for a problem class from the candidates Conjure produces, we use a set of training instances drawn from the target class. It is important that the training instances are discriminating. If all models solve a given instance in a trivial amount of time, or if no models solve it in the time available, then the instance is not useful for model selection. This paper addresses the task of generating small sets of discriminating training instances automatically. The instance space is determined by the parameters of the associated problem class. We develop a number of methods of finding parameter configurations that give discriminating training instances, some of them leveraging existing parameter-tuning techniques. Our experimental results confirm the success of our approach in reducing a large set of input models to a small set that we can expect to perform well for the given problem class