random rotations of the wiener path let w h be an abstract wiener space and letr w be a strongly measurable random variable with values in the set of isometries onh suppose that rh is smooth in the sobolev sense and that it is a quasi nilpotent operator onh for everyh h it is shown that r w h is again a gaussian 0 h 2 random variable consequently if e i i w is a complete orthonormal basis ofh then tilde w sum nolimits i delta r w e i e i defines a measure preserving transformation a rotation onw it is also shown that if for some strongly measurable operator valued onh random variabler r w k h is 0 h 2 gaussian for allk h h thenr is an isometry and rh is quasi nilpotent for allh h the relation between the stochastic calculi for these wiener pathsw and tilde w as well as the conditions of the inverbibility of the map w to tilde w are discussed and the problem of the absolute continuity of the image of the wiener measure under euclidean motion on the wiener space i e w to tilde w composed with a shift is studied

the vanishing of l2 harmonic one forms on based path spaces abstract we prove the triviality of the first l 2 cohomology class of based path spaces of riemannian manifolds furnished with brownian motion measure and the consequent vanishing of l 2 harmonic one forms we give explicit formulae for closed and co closed one forms expressed as differentials of functions and co differentials of l 2 two forms respectively these are considered as extended clark ocone formulae a feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one forms to the exterior differentiation operator used to construct the de rham complex and the self adjoint laplacian on l 2 one forms this laplacian is shown to have a spectral gap

partition search for non binary constraint satisfaction previous algorithms for unrestricted constraint satisfaction use reduction search which inferentially removes values from domains in order to prune the backtrack search tree this paper introduces partition search which uses an efficient join mechanism instead of removing values from domains analytical prediction of quantitative performance of partition search appears to be intractable and evaluation therefore has to be by experimental comparison with reduction search algorithms that represent the state of the art instead of working only with available reduction search algorithms this paper introduces enhancements such as semijoin reduction preprocessing using bloom filtering

de rham hodge kodaira operator on loop groups we consider a based loop group le g over a compact lie groupg endowed with its pinned wiener measure the law of the brownian bridge ong and we shall calculate the ricci curvature for differentialn forms over le g a type of bochner weitzenbock formula for general differentialn forms or shigekawa identity will be established

stochastic calculus with anticipating integrands we study the stochastic integral defined by skorohod in 24 of a possibly anticipating integrand as a function of its upper limit and establish an extended ito formula we also introduce an extension of stratonovich s integral and establish the associated chain rule in all the results the adaptedness of the integrand is replaced by a certain smoothness requirement

anl2estimate for riemannian anticipative stochastic integrals abstract we define a metric and a markovian connection on the path space of a riemannian manifold which are different from those introduced in cm and prove a corresponding weitzenbock formula an l 2 inequality for the divergence is obtained as a consequence

computational framework for coupling compressible and low mach number codes a hybrid multicode computational method is developed that allows combining compressible and low mach number codes into a single computational solver the proposed methodology can be used for integrated simulations of multicomponent engineering problems the unsteady coupling between the two codes is performed via exchanging time dependent state information through the interfaces using overlapping meshes the proposed coupling procedure is validated on laminar steady and unsteady test problems both constant density and variable density regimes of the low mach number code are investigated results of the test cases performed in frameworks of one way coupling and two way coupling are documented several methods of implementation of interface conditions for the compressible code are compared by looking at the levels of dilatation as well as numerical errors the injection method when all the variables are simply transformed from one code to another through an interpolation procedure is chosen for its superior stability accuracy and simplicity of implementation numerical results for the coupled calculations obtained with the injection method show good agreement with both standalone calculations and analytical solutions

an overlapping schwarz method for spectral element solution of the incompressible navier stokes equations efficient solution of the navier stokes equations in complex domains is dependent upon the availability of fast solvers for sparse linear systems for unsteady incompressible flows the pressure operator is the leading contributor to stiffness as the characteristic propagation speed is infinite in the context of operator splitting formulations it is the pressure solve which is the most computationally challenging despite its elliptic origins we examine several preconditioners for the consistent l2poisson operator arising in the pn pn 2spectral element formulation of the incompressible navier stokes equations we develop a finite element based additive schwarz preconditioner using overlapping subdomains plus a coarse grid projection operator which is applied directly to the pressure on the interior gauss points for large two dimensional problems this approach can yield as much as a fivefold reduction in simulation time over previously employed methods based upon deflation

from semidiscrete to fully discrete stability of runge kutta schemes by the energy method the integration of semidiscrete approximations for time dependent problems is encountered in a variety of applications the runge kutta rk methods are widely used to integrate the ode systems which arise in this context resulting in large ode systems called methods of lines these methods of lines are governed by possibly ill conditioned systems with a growing dimension consequently the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading instead we present here a stability analysis of rk methods for well posed semidiscrete approximations based on a general energy method we review the stability question for such rk approximations and highlight its intricate dependence on the growing dimension of the problem in particular we prove the strong stability of general fully discrete rk methods governed by coercive approximations we conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection diffusion equations with variable coefficients a straightforward implementation of our results verify the strong stability of rk methods for local finite difference schemes as well as global spectral approximations since our approach is based on the energy method which is carried in the physical space and since it avoids the von neumann analysis which is carried in the dual fourier space we are able to easily adapt additional extensions due to nonperiodic boundary conditions general geometries etc

on the power of structural decompositions of graph based representations of constraint problems the constraint satisfaction problem csp is a central issue of research in artificial intelligence due to its intractability many efforts have been made in order to identify tractable classes of csp instances and in fact deep and useful results have already been achieved in particular this paper focuses on structural decomposition methods which are essentially meant to look for near acyclicity properties of the graphs or hypergraphs that encode the structure of the constraints interactions in general constraint scopes comprise an arbitrary number of variables and thus this structure may be naturally encoded via hypergraphs however in many practical applications decomposition methods are applied over suitable graph representations of the possibly non binary csp instances at hand despite the great interest in such binary approaches a formal analysis of their power in terms of their ability of identifying islands of tractability was missing in the literature the aim of this paper is precisely to fill this gap by studying the relationships among binary structural methods and by providing a clear picture of the tractable fragments of csp that can be specified with respect to each of these decomposition approaches when they are applied to binary representations of non binary csp instances in particular various long standing questions about primal dual and incidence graph encodings are answered the picture is then completed by comparing methods on binary encodings with methods specifically conceived for non binary constraints

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

weak levi civita connection for the damped metric on the riemannian path space and vanishing of ricci tensor in adapted differential geometry abstract we shall establish in the context of adapted differential geometry on the path space p m o m a weitzenbock formula which generalizes that in a b cruzeiro and p malliavin j funct anal 177 2000 219 253 without hypothesis on the ricci tensor the renormalized ricci tensor will be vanished the connection introduced in a b cruzeiro and s fang 1997 j funct anal 143 400 414 will play a central role

a stable and accurate partitioned algorithm for conjugate heat transfer abstract we describe a new partitioned approach for solving conjugate heat transfer cht problems where the governing temperature equations in different material domains are time stepped in an implicit manner but where the interface coupling is explicit the new approach called the champ scheme conjugate heat transfer advanced multi domain partitioned is based on a discretization of the interface coupling conditions using a generalized robin mixed condition the weights in the robin condition are determined from the optimization of a condition derived from a local stability analysis of the coupling scheme the interface treatment combines ideas from optimized schwarz methods for domain decomposition problems together with the interface jump conditions and additional compatibility jump conditions derived from the governing equations for many problems i e for a wide range of material properties grid spacings and time steps the champ algorithm is stable and second order accurate using no sub time step iterations i e a single implicit solve of the temperature equation in each domain in extreme cases e g very fine grids with very large time steps it may be necessary to perform one or more sub iterations each sub iteration generally increases the range of stability substantially and thus one sub iteration is likely sufficient for the vast majority of practical problems the champ algorithm is developed first for a model problem and analyzed using normal mode theory the theory provides a mechanism for choosing optimal parameters in the mixed interface condition a comparison is made to the classical dirichlet neumann dn method and where applicable to the optimized schwarz os domain decomposition method for problems with different thermal conductivities and diffusivities the champ algorithm outperforms the dn scheme for domain decomposition problems with uniform conductivities and diffusivities the champ algorithm performs better than the typical os scheme with one grid cell overlap the champ scheme is also developed for general curvilinear grids and cht examples are presented using composite overset grids that confirm the theory and demonstrate the effectiveness of the approach

a theoretical evaluation of selected backtracking algorithms abstract in recent years many new backtracking algorithms for solving constraint satisfaction problems have been proposed the algorithms are usually evaluated by empirical testing this method however has its limitations our paper adopts a different purely theoretical approach which is based on characterizations of the sets of search tree nodes visited by the backtracking algorithms a notion of inconsistency between instantiations and variables is introduced and is shown to be a useful tool for characterizing such well known concepts as backtrack backjump and domain annihilation the characterizations enable us to a prove the correctness of the algorithms and b partially order the algorithms according to two standard performance measures the number of nodes visited and the number of consistency checks performed among other results we prove the correctness of backjumping and conflict directed backjumping and show that forward checking never visits more nodes than backjumping our approach leads us also to propose a modification to two hybrid backtracking algorithms backmarking with backjumping bmj and backmarking with conflict directed backjumping bm cbj so that they always perform fewer consistency checks than the original algorithms

multiplicative schwarz methods for parabolic problems based on domain decomposition we give a multiplicative schwarz domain decomposition method for semi linear parabolic problems we consider the relation between the convergence rate and discretization parameters including the diameter of the sub domain we give the error estimate which tells us that the convergence of the approximate solution is independent of the iteration number at each time level finally we give a numerical example

stability analysis of numerical boundary conditions in domain decomposition algorithms this paper discusses numerical stability of a class of non overlapping domain decomposition algorithms inherent shortcomings associated with different methods for proving stability are pointed out von neumann analysis yields only a necessary condition for stability because it does not consider the overall effect of the boundary conditions between subdomains matrix analysis produces also a necessary condition for stability since the matrices of coefficients associated with the algorithms are not symmetric the gkso on the other hand produces necessary and sufficient conditions for stability however it is difficult to apply systematically this analysis to the algorithms

preconditioned spectral multi domain discretization of the incompressible navier stokes equations in this paper we present a preconditioned multi domain algorithm applied to the elliptic kernels arising from the spectral collocation of the incompressible navier stokes equations in three space dimensions with one homogeneous direction the technique based on the iterative solution of the schur complement matrix allows for efficient numerical solution of the operators in complex geometries consisting of a collection of non overlapping rectangular subdomains the method is shown to be nearly optimal in terms of condition number behavior in a double path of refinement strategy i e whenever both the number of legendre modes and the number of subdomains are significantly increased it is thus well suited for engineering applications in the fields of direct numerical simulation and large eddy simulation of turbulence

stability of interfaces with mesh refinement a study is conducted of the stability of mesh refinement in space and time for several different interface equations and finite difference approximations first a root condition which implies stability for the initial boundary value problem for this type of interface is derived from the root condition the stability of several interface equations is proved using the maximum principle in some cases the final verification steps can be done analytically in other cases a simple computer program has been written to check the condition for values of a parameter along the boundary of the unit circle using this method stability for lax wendroff with all the interface conditions considered and for leapfrog with interpolation interface conditions when the fine and coarse grids overlap is proved

a spectrally accurate method for overlapping grid solution of incompressible navier stokes equations an overlapping mesh methodology that is spectrally accurate in space and up to third order accurate in time is developed for solution of unsteady incompressible flow equations in three dimensional domains the ability to decompose a global domain into separate but overlapping subdomains eases mesh generation procedures and increases flexibility of modeling flows with complex geometries the methodology employs implicit spectral element discretization of equations in each subdomain and explicit treatment of subdomain interfaces with spectrally accurate spatial interpolation and high order accurate temporal extrapolation and requires few if any iterations yet maintains the global accuracy and stability of the underlying flow solver the overlapping mesh methodology is thoroughly validated using two dimensional and three dimensional benchmark problems in laminar and turbulent flows the spatial and temporal convergence is documented and is in agreement with the nominal order of accuracy of the solver the influence of long integration times as well as inflow outflow global boundary conditions on the performance of the overlapping grid solver is assessed in a turbulent benchmark of fully developed turbulent pipe flow the turbulent statistics with the overlapping grids is validated against published available experimental and other computation data scaling tests are presented that show near linear strong scaling even for moderately large processor counts a novel overlapping grid method for incompressible flow equations is developed the overlapping grid method is based on spectral element discretization the method maintains global spectral accuracy with polynomial refinement an explicit interface extrapolation scheme for temporal coupling is proposed the scheme allows for high order temporal accuracy with only a few iterations

adaptive local overlapping grid methods for parabolic systems in two space dimensions abstract adaptive mesh refinement techniques are described for two dimensional systems of parabolic partial differential equations solutions are calculated using galerkin s method with a piecewise bilinear basis in space and backward euler integration in time a posteriori estimates of the local discretization error of piecewise bilinear finite element solutions are obtained by a p refinement technique these error estimates are used to control a local h refinement strategy where finer grids are recursively introduced in regions where a prescribed tolerance is exceeded fine grids at a given level of refinement may overlap each other and independent solutions are generated on each of them a version of the schwarz alternating principle is used to coordinate solutions between overlapping fine grids computational results demonstrating the performance of the adaptive procedure on linear and nonlinear problems and apparent convergence of the error estimate for linear heat conduction problem and uniform global refinement is presented

an overset grid method for 3d unsteady incompressible flows a composite grid numerical method is developed for simulating unsteady three dimensional 3d incompressible flows in complex geometries the governing equations are solved using a second order accurate finite volume method based on the dual time stepping artificial compressibility approach overset grids are employed to discretize arbitrarily complex geometries and a new interface algorithm is developed to facilitate communication between neighboring grids the algorithm is inspired by the necessary and sufficient conditions for satisfying global mass conservation in a composite domain and is simple to implement in 3d numerical experiments show that the new interpolation scheme is superior to straightforward trilinear interpolation of all flow variables as it minimizes non physical spurious oscillations in the overlap region is less sensitive to grid refinement and greatly enhances the computational efficiency of the iterative algorithm the advantages of the new method are especially pronounced when adjacent overset subdomains are discretized with different spatial resolutions the potential of the method as a powerful technique for simulating complex engineering flows is demonstrated by applying it to calculate vortex shedding from a circular cylinder mounted between two endplates and flow in a rectangular channel with two wall mounted obstacles

a chimera method based on a dirichlet neumann robin coupling for the navier stokes equations we present a chimera method for the numerical solution of incompressible flows past objects in relative motion the chimera method is implemented as an iteration by subdomain method based on dirichlet neumann robin coupling the dd method we propose is not only geometric but also algorithmic for the solution on each subdomain is obtained on separate processes and the exchange of information between the subdomains is carried out by a master code this strategy is very flexible as it requires almost no modification to the original numerical code therefore only the master code has to be adapted to the numerical codes and the strategies used on each subdomain as a basic flow solver we a use stabilized finite element method

an unconditionally stable parallel difference scheme for parabolic equations an unconditionally stable difference scheme with intrinsic parallelism is constructed in this paper taking the parabolic equation u t u x x as an example the main ideas are described as follows we decompose the domain w into some overlapping subdomains take values of the last time layer as values of the time layer on inner boundary points of subdomains solve it with the fully implicit scheme inside each subdomain and then take correspondent values of its neighbor subdomains as their values for inner boundary points of each subdomain and mean of its neighbor subdomain and itself at overlapping points its unconditional stability and convergence are proved though its truncation error is o 1 we will still prove that the estimate order is improved to o t h

an l 2 theory for differential forms on path spaces i an l 2 theory of differential forms is proposed for the banach manifold of continuous paths on a riemannian manifold m furnished with its brownian motion measure differentiation must be restricted to certain hilbert space directions the h tangent vectors to obtain a closed exterior differential operator the relevant spaces of differential forms the h forms are perturbed by the curvature of m a hodge decomposition is given for l 2 h one forms and the structure of h two forms is described the dual operator d is analysed in terms of a natural connection on the h tangent spaces malliavin calculus is a basic tool c 2007 elsevier inc all rights reserved

a domain decomposition method for incompressible viscous flow a method for using domain decomposition to solve the equations of incompressible viscous flow is presented the method is described in detail and test results are given for two test problems a notable feature of the method is that the incompressibility constraint is never explicitly imposed the domain decomposition uses finite difference and spectral methods on overlapping domains with second order accurate interpolation of the velocity relating the solutions on the different domains the method is shown to be globally second order accurate by the test results

domain filtering consistencies for non binary constraints in non binary constraint satisfaction problems the study of local consistencies that only prune values from domains has so far been largely limited to generalized arc consistency or weaker local consistency properties this is in contrast with binary constraints where numerous such domain filtering consistencies have been proposed in this paper we present a detailed theoretical algorithmic and empirical study of domain filtering consistencies for non binary problems we study three domain filtering consistencies that are inspired by corresponding variable based domain filtering consistencies for binary problems these consistencies are stronger than generalized arc consistency but weaker than pairwise consistency which is a strong consistency that removes tuples from constraint relations among other theoretical results and contrary to expectations we prove that these new consistencies do not reduce to the variable based definitions of their counterparts on binary constraints we propose a number of algorithms to achieve the three consistencies one of these algorithms has a time complexity comparable to that for generalized arc consistency despite performing more pruning experiments demonstrate that our new consistencies are promising as they can be more efficient than generalized arc consistency on certain non binary problems

tree clustering for constraint networks research note abstract the paper offers a systematic way of regrouping constraints into hierarchical structures capable of supporting search without backtracking the method involves the formation and preprocessing of an acyclic database that permits a large variety of queries and local perturbations to be processed swiftly either by sequential backtrack free procedures or by distributed constraint propagation processes

composite overlapping meshes for the solution of partial differential equations abstract we discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them a composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet continuity conditions interpolation are imposed at the overlapping boundaries the principal advantage of composite grids is in the generation of grids for regions with complicated geometries the grid construction program cmpgrd is used to create composite grids with any number of component grids we describe some techniques for the solution of elliptic and time dependent pdes on composite meshes applications to the solution of the compressible navier stokes equations are presented

renormalized differential geometry on path space structural equation curvature abstract the theory of integration in infinite dimensions is in some sense the backbone of probability theory on this backbone the stochastic calculus of variations has given rise to the flesh of differential calculus its first step is the construction at each point of the probability space of a cameron martin like tangent space in which the desired differential calculus can be developed this construction proceeds along the lines of first order differential geometry in this paper we address the following questions what could be the meaning of curvature of the probability space how and why how can curvatures be defined and computed why could a second order differential geometry be relevant to stochastic analysis we try to answer these questions for the probability space associated to the brownian motion of a compact riemannian manifold why a basicenergy identity for anticipative stochastic integralswill be obtained as a byproduct of our computation of curvature how there are essentially four bottlenecks in the development of differential geometry on wiener riemann manifolds i the difficulty of finding an atlas of local charts such that the changes of charts preserve the class of the wiener like measures together with their associated cameron martin like tangent spaces ii the difficulty of finding cylindrical approximations preserving the natural geometrical objects iii the difficulty of renormalizing the divergent series to which the summation operations of finite dimensional differential geometry give rise in the non intrinsic context of local charts iv the nonavailability of the computational procedures analogous to the local coordinates systems of the classical differential geometry in the context of path space the ito filtration provides a much richer structure than that available in the framework of an abstract wiener riemann manifold our work is a systematic attempt to replace the machinery of local charts with amethodology of moving frames in our context stochastic parallel transport provides a canonical moving frame on the path space the concept of a cylindrical approximation has to be reshaped in our new situation into some geometric limit theorems establishing that the riemannian geometric objects of the cylindrical approximations induce by a limiting procedure geometric objects on the path space those limit theorems are reminiscent of the classical theorems which say that a stratonovich sde is the limit of an appropriate sequence of ode the canonical coordinate system provided by the moving frame will make it possible to proceed to the needed renormalizations byintrinsicstochastic integrals in this context the anticipative stochastic integral theory of nualart and pardoux will play a decisive role finally the moving frame will provide aneffective algorithm of computation for this differential geometry in infinite dimensions in our study we encounter a new type of renormalization thehypoelliptic renormalization which corresponds to the fact that the bracket of smooth vector fields taking their values in the cameron martin space can get out of this cameron martin space this hypoelliptic problem induces the nonrenormalizability of some geometrical objects it leads also to a concept oftangent processesto probability spaces extending that based on the cameron martin theorem for tangent processes a formula of integration by parts still holds furthermore the tangent processes form a lie algebra under the bracket on the other hand tangent processes cannot be stochastically integrated this operation is well defined only for cameron martin type vector fields

stability analysis of interface temporal discretization in grid overlapping methods we investigate the stability of a temporal discretization of interface terms in grid overlapping methods a matrix stability analysis is performed on a model problem of the one dimensional diffusion equation on overlapping grids the scheme stability is first analyzed theoretically and a proof of the unconditional stability of the first order interface extrapolation scheme with the first and second order time integration for any overlap size is presented for the higher order schemes we obtain explicit estimates of the spectral radius of the corresponding discrete matrix operator and document the values of the stability threshold depending on the number of grid points and the size of overlap the influence of iterations on stability properties is also investigated numerical experiments are then presented relating the obtained stability bounds to the observed numerical values semidiscrete analysis confirms the derived scaling for the stability bounds

analysis of a fractional step method on overset grids a fractional step method for solving the incompressible navier stokes equations on overset grids is derived as a matrix factorization of the spatially and temporally discretized system of equations the algorithm is applied to several test problems using second order accurate finite volume flux differencing on staggered grid systems and a hybrid implicit explicit time advancement scheme spatial order of accuracy is shown to depend on the behavior of the overset grid overlap during grid refinement the temporal order of accuracy of the time advancement algorithm on a single grid is maintained on the overset grid

vanishing of one dimensional l 2 cohomologies of loop groups abstract let g be a simply connected compact lie group let l e g be the based loop group with the base point e which is the identity element let e be the pinned brownian motion measure on l e g and let l 2 1 t l e g e d p 1 t l e g e 1 p 2 be a closed 1 form on l e g using results in rough path analysis we prove that there exists a measurable function f on l e g such that d f moreover we prove that dim ker 0 for the hodge kodaira type operator acting on 1 forms on l e g

multiplicative schwarz methods for parabolic problems the class of multiplicative schwarz methods originated from the classical schwarz alternating method it has been shown to be one of the most powerful methods for solving finite element or finite d

a finite difference domain decomposition algorithm for numerical solution of the heat equation a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented in this procedure interface values between subdomains are found by an explicit finite difference formula once these values are calculated interior values are determined by backward differencing in time a natural extension of this method allows for the use of different time steps in different subdomains maximum norm error estimates for these procedures are derived which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters

ito maps and analysis on path spaces we consider versions of malliavin calculus on path spaces of compact manifolds with diffusion measures defining gross sobolev spaces of differentiable functions and proving their intertwining with solution maps mathcal i of certain stochastic differential equations this is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator d and markov uniqueness of the associated dirichlet form a continuity result for the divergence operator by kree and kree is extended to this situation the regularity of conditional expectations of smooth functionals of classical wiener space given mathcal i is considered and shown to have strong implications for these questions a major role is played by the possibly sub riemannian connections induced by stochastic differential equations damped markovian connections are used for the covariant derivatives

binary vs non binary constraints there are two well known transformations from non binary constraints to binary constraints applicable to constraint satisfaction problems csps with finite domains the dual transformation and the hidden variable transformation we perform a detailed formal comparison of these two transformations our comparison focuses on two backtracking algorithms that maintain a local consistency property at each node in their search tree the forward checking and maintaining arc consistency algorithms we first compare local consistency techniques such as arc consistency in terms of their inferential power when they are applied to the original non binary formulation and to each of its binary transformations for example we prove that enforcing arc consistency on the original formulation is equivalent to enforcing it on the hidden transformation we then extend these results to the two backtracking algorithms we are able to give either a theoretical bound on how much one formulation is better than another or examples that show such a bound does not exist for example we prove that the performance of the forward checking algorithm applied to the hidden transformation of a problem is within a polynomial bound of the performance of the same algorithm applied to the dual transformation of the problem our results can be used to help decide if applying one of these transformations to all or part of a constraint satisfaction model would be beneficial