a domain decomposition method based on weighted interior penalties for advection diffusion reaction problems we propose a domain decomposition method for advection diffusion reaction equations based on nitsche s transmission conditions the advection dominated case is stabilized using a continuous interior penalty approach based on the jumps in the gradient over element boundaries we prove the convergence of the finite element solutions of the discrete problem to the exact solution and propose a parallelizable iterative method the convergence of the resulting domain decomposition method is proved and this result holds true uniformly with respect to the diffusion parameter the numerical scheme that we propose here can thus be applied straightforwardly to diffusion dominated advection dominated and hyperbolic problems some numerical examples are presented in different flow regimes showing the influence of the stabilization parameter on the performance of the iterative method and we compare our method with some other domain decomposition techniques for advection diffusion equations

local discontinuous galerkin methods for the stokes system in this paper we introduce and analyze local discontinuous galerkin methods for the stokes system for a class of shape regular meshes with hanging nodes we derive a priori estimates for the l2 norm of the errors in the velocities and the pressure we show that optimal order estimates are obtained when polynomials of degree k are used for each component of the velocity and polynomials of degree k 1 for the pressure for any k ge1 we also consider the case in which all the unknowns are approximated with polynomials of degree k and show that although the orders of convergence remain the same the method is more efficient numerical experiments verifying these facts are displayed

a computable error bound for matrix functionals abstract many problems in applied mathematics require the evaluation of matrix functionals of the form f a u t f a u where a is a large symmetric matrix and u is a vector golub and collaborators have described how approximations of such functionals can be computed inexpensively by using the lanczos algorithm the present note shows that error bounds for these approximations can be computed essentially for free when bounds for derivatives of f on an interval containing the spectrum of a are available

symbolic dynamics analysis of nonstationary data from a model of a magnetic system with solitons the probability distribution of a complexity measure is used to characterize chaotic states an estimator of the algorithmic complexity of a time series of symbolic words is calculated within a fixed length time window which sweeps through the time series analyzed the words are derived through a symbolic dynamics scheme applied in an m dimensional delay coordinate space time intervals instead of the variables of the system are used the chaotic states of a model of a magnetic domain wall are characterized better by the methods presented than with the use of fractal dimensions and new intermittent states of the system were easily identified using an artificial nonstationary signal composed of different chaotic states of the bloch wall as a test for chaos chaos intermittency we demonstrate that the method developed is suitable for the detection and characterization of intermittency it is also shown that nonstationarity in the form of a slow monotonic drift in the control parameter may extend the stability range of periodic states of the spatially extended system studied a trackinglike phenomenon

a strict error bound with separated contributions of the discretization and of the iterative solver in non overlapping domain decomposition methods this paper deals with the estimation of the distance between the solution of a static linear mechanic problem and its approximation by the finite element method solved with a non overlapping domain decomposition method feti or bdd we propose a new strict upper bound of the error which separates the contribution of the iterative solver and the contribution of the discretization numerical assessments show that the bound is sharp and enables us to define an objective stopping criterion for the iterative solver

an analysis of the practical dpg method we give a complete error analysis of the discontinuous petrov galerkin dpg method accounting for all the approximations made in its practical implementation specifi cally we consider the dpg method that uses a trial space consisting of polynomials of degree p on each mesh element earlier works showed that there is a trial to test operator t which when applied to the trial space defines a test space that guarantees stability in dpg formulations this operator t is local it can be applied element by element however an infinite dimensional problem on each mesh element needed to be solved to apply t in prac tical computations t is approximated using polynomials of some degree r p on each mesh element we show that this approximation maintains optimal convergence rates provided that r 2 p n where n is the space dimension two or more for the laplace equation we also prove a similar result for the dpg method for linear elasticity remarks on the conditioning of the stiffness matrix in dpg methods are also included

statistical properties of the moving average price in dollar yen exchange rates we introduce a weighted moving average analysis for the tick by tick data of yen dollar exchange rates the weights are determined automatically for given data by applying the yule walker formula for autoregressive model although the data are non stationary the resulting moving average gives a quite nice property that the deviation around the moving average becomes a white noise the weights decay exponentially with time scale less than 2 min implying that dealers are watching only very recent market state

isogeometric residual minimization igrm for non stationary stokes and navier stokes problems abstract we show that it is possible to obtain a linear computational cost fem based solver for non stationary stokes and navier stokes equations our method employs a technique developed by guermond and minev 2011 which consists of singular perturbation plus a splitting scheme while the time integration schemes are implicit we use finite elements to discretize the spatial counterparts at each time step we solve a pde having weak derivatives in one direction only which allows for the linear computational cost at the expense of handling strong second order derivatives of the previous time step solution on the right hand side of these pdes this motivates the use of smooth functions such as b splines for high reynolds numbers some of these pdes become unstable to deal robustly with these instabilities we propose to use a residual minimization technique we test our method on problems having manufactured solutions as well as on the cavity flow problem

anti gaussian quadrature formulas an anti gaussian quadrature formula is an n 1 point formula of degree 2n 1 which integrates polynomials of degree up to 2n 1 with an error equal in magnitude but of opposite sign to that of the n point gaussian formula its intended application is to estimate the error incurred in gaussian integration by halving the difference between the results obtained from the two formulas we show that an anti gaussian formula has positive weights and that its nodes are in the integration interval and are interlaced by those of the corresponding gaussian formula similar results for gaussian formulas with respect to a positive weight are given except that for some weight functions at most two of the nodes may be outside the integration interval the anti gaussian formula has only interior nodes in many cases when the kronrod extension does not and is as easy to compute as the n 1 point gaussian formula

analysis of preconditioners for saddle point problems mixed finite element formulations give rise to large sparse block linear systems of equations the solution of which is often sought via a preconditioned iterative technique in this work we present a general analysis of block preconditioners based on the stability conditions inherited from the formulation of the finite element method the babuska brezzi or inf sup conditions the analysis is motivated by the notions of norm equivalence and field of values equivalence of matrices in particular we give sufficient conditions for diagonal and triangular block preconditioners to be norm and field of values equivalent to the system matrix

solving hermitian positive definite systems using indefinite incomplete factorizations incomplete ldl factorizations sometimes produce an indefinite preconditioner even when the input matrix is hermitian positive definite the two most popular iterative solvers for symmetric systems cg and minres cannot use such preconditioners they require a positive definite preconditioner one approach that has been extensively studied to address this problem is to force positive definiteness by modifying the factorization process we explore a different approach use the incomplete factorization with a krylov method that can accept an indefinite preconditioner the conventional wisdom has been that long recurrence methods like gmres or alternatively non optimal short recurrence methods like symmetric qmr and bicgstab must be used if the preconditioner is indefinite we explore the performance of these methods when used with an incomplete factorization but also explore a less known krylov method called pcg odir that is both optimal and uses a short recurrence and can use an indefinite preconditioner furthermore we propose another optimal short recurrence method called ip minres that can use an indefinite preconditioner and a variant of pcg odir which we call ip cg that is more numerically stable and usually requires fewer iterations

a stopping criterion for the conjugate gradient algorithm in a finite element method framework the conjugate gradient method has always been successfully used in solving the symmetric and positive definite systems obtained by the finite element approximation of self adjoint elliptic partial differential equations taking into account recent results 13 19 20 22 which make it possible to approximate the energy norm of the error during the conjugate gradient iterative process we adapt the stopping criterion introduced in 3 moreover we show that the use of efficient preconditioners does not require to change the energy norm used by the stopping criterion finally we present the results of several numerical tests that experimentally validate the effectiveness of our stopping criterion

the dpg method for the stokes problem we discuss well posedness and convergence theory for the dpg method applied to a general system of linear partial differential equations pdes and specialize the results to the classical stokes problem the stokes problem is an iconic troublemaker for standard bubnov galerkin methods if discretizations are not carefully designed they may exhibit non convergence or locking by contrast dpg does not require us to treat the stokes problem in any special manner we illustrate and confirm our theoretical convergence estimates with numerical experiments

an iterative method with error estimators abstract iterative methods for the solution of linear systems of equations produce a sequence of approximate solutions in many applications it is desirable to be able to compute estimates of the norm of the error in the approximate solutions generated and terminate the iterations when the estimates are sufficiently small this paper presents a new iterative method based on the lanczos process for the solution of linear systems of equations with a symmetric matrix the method is designed to allow the computation of estimates of the euclidean norm of the error in the computed approximate solutions these estimates are determined by evaluating certain gauss anti gauss or gauss radau quadrature rules

discrete least squares finite element methods abstract a finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators 1 the differential operator defining the spatial boundary value problem and 2 a riesz map on the test space the resulting linear system is overdetermined two different approaches for solving the system are suggested although others are discussed 1 solving the associated normal equation with linear solvers for symmetric positive definite systems e g cholesky factorization and 2 solving the overdetermined system with orthogonalization algorithms e g qr factorization the finite element assembly algorithm for each of these approaches is described in detail the normal equation approach is usually faster for direct solvers and requires less storage the second approach reduces the condition number of the system by a power of two and is less sensitive to round off error the rectangular stiffness matrix of second approach is demonstrated to have condition number o h 1 for a variety of formulations of poisson s equation the stiffness matrix from the normal equation approach is demonstrated to be related to the monolithic stiffness matrices of least squares finite element methods and it is proved that the two are identical in some cases an example with poisson s equation indicates that the solutions of these two different linear systems can be nearly indistinguishable if round off error is not an issue and rapidly converge to each other the orthogonalization approach is suggested to be beneficial for problems which induce poorly conditioned linear systems experiments with poisson s equation in single precision arithmetic as well as the linear acoustics problem near resonance in double precision arithmetic verify this conclusion the methodology described here is an outgrowth of the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2015 the strength of dpg is highlighted throughout however the majority of theory presented is more general extensions to constrained minimization principles are also considered throughout but are not analyzed in experiments

a geometric multigrid preconditioning strategy for dpg system matrices abstract the discontinuous petrov galerkin dpg methodology of demkowicz and gopalakrishnan 2010 2011 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes a key question that has not yet been answered in general though there are some results for poisson e g is how best to precondition the dpg system matrix so that iterative solvers may be used to allow solution of large scale problems in this paper we detail a strategy for preconditioning the dpg system matrix using geometric multigrid which we have implemented as part of camellia roberts 2014 2016 and demonstrate through numerical experiments its effectiveness in the context of several variational formulations we observe that in some of our experiments the behavior of the preconditioner is closely tied to the discrete test space enrichment we include experiments involving adaptive meshes with hanging nodes for lid driven cavity flow demonstrating that the preconditioners can be applied in the context of challenging problems we also include a scalability study demonstrating that the approach and our implementation scales well to many mpi ranks

matrices moments and rational quadrature abstract many problems in science and engineering require the evaluation of functionals of the form f u a u t f a u where a is a large symmetric matrix u a vector and f a nonlinear function a popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the lanczos procedure applied to a with initial vector u and then evaluating pairs of gauss and gauss radau quadrature rules associated with the tridiagonal matrix determined by the lanczos procedure the present paper extends this approach to allow the use of rational gauss quadrature rules

analysis of the dpg method for the poisson equation we give an error analysis of the recently developed dpg method applied to solve the poisson equation and a convection diffusion problem we prove that the method is quasioptimal error estimates in terms of both the mesh size h and the polynomial degree p for various element shapes can be derived from our results results of extensive numerical experiments are also presented

error estimates for the solution of linear systems let ax b be a system of linear equations where a is a square and nonsingular matrix when solving it by a direct or an iterative method an approximate solution x is obtained x denotes the current iterate in the case of an iterative method in this paper we propose estimates for the norm of the error e x x these estimates are valid for any nonsingular matrix and for approximations x of x obtained by any method and thus they also implicitly take into account the effect of rounding errors

a locking free hp dpg method for linear elasticity with symmetric stresses we present two new methods for linear elasticity that simultaneously yield stress and displacement approximations of optimal accuracy in both the mesh size h and polynomial degree p this is achieved within the recently developed discontinuous petrov galerkin dpg framework in this framework both the stress and the displacement approximations are discontinuous across element interfaces we study locking free convergence properties and the interrelationships between the two dpg methods

correlation patterns of nikkei index constituents towards a mean field model abstract an analysis of minute tick data from the japanese stock index market is reported for a three year period of 2000 7 4 2003 6 30 correlation patterns and principal component distributions were determined for 180 constituents of the nikkei 225 index excluding the effects of after hours trading and constituent revisions the first principal component describes about 30 of the total variance in constituent log returns subject to slow decrease with the size of the correlation window suggesting that a small number of physical parameters may describe the internal dynamics of the index allowing for an adiabatic representation of index dynamics and a self consistent mean field model of its constituents finally it is shown that the introduction of a time gap into minute tick data significantly improves the correlations of the price weighed index with its constituents even when such gap inserts are strictly penalized this phenomenon corresponds to a heterogenous response time of index constituents to the adiabatic collective motion and also demonstrates the inhomogeneous nature of equidistant time ticks in financial trading

robust dpg method for convection dominated diffusion problems we propose and analyze a discontinuous petrov galerkin dpg method for convection dominated diffusion problems that provides robust l 2 error estimates for the field variables which are quasi optimal in the energy norm a key feature of the method is to construct test functions defined by a variational formulation with bilinear form test norm specifically designed for the goal of robustness a main theoretical ingredient is a stability analysis of the adjoint problem numerical experiments underline our theoretical results and in particular confirm robustness of the dpg method for well chosen test norms

a discontinuous petrov galerkin methodology for adaptive solutions to the incompressible navier stokes equations the discontinuous petrov galerkin methodology with optimal test functions dpg of demkowicz and gopalakrishnan 18 20 guarantees the optimality of the solution in an energy norm and provides several features facilitating adaptive schemes whereas bubnov galerkin methods use identical trial and test spaces petrov galerkin methods allow these function spaces to differ in dpg test functions are computed on the fly and are chosen to realize the supremum in the inf sup condition the method is equivalent to a minimum residual method for well posed problems with sufficiently regular solutions dpg can be shown to converge at optimal rates the inf sup constants governing the convergence are mesh independent and of the same order as those governing the continuous problem 48 dpg also provides an accurate mechanism for measuring the error and this can be used to drive adaptive mesh refinements we employ dpg to solve the steady incompressible navier stokes equations in two dimensions building on previous work on the stokes equations and focusing particularly on the usefulness of the approach for automatic adaptivity starting from a coarse mesh we apply our approach to a manufactured solution due to kovasznay as well as the lid driven cavity flow backward facing step and flow past a cylinder problems

from functional analysis to iterative methods we examine condition numbers preconditioners and iterative methods for finite element discretizations of coercive pdes in the context of the fundamental solvability result the lax milgram lemma working in this hilbert space context is justified because finite element operators are restrictions of infinite dimensional hilbert space operators to finite dimensional subspaces moreover useful insight is gained as to the relationship between hilbert space and matrix condition numbers and translating hilbert space fixed point iterations into matrix computations provides new ways of motivating and explaining some classic iteration schemes in this framework the simplest preconditioner for an operator from a hilbert space into its dual is the riesz isomorphism simple analysis gives spectral bounds and iteration counts bounded independent of the finite element subspaces chosen moreover the abstraction allows us to consider not only riesz map preconditioning for convection diffusion equations in h 1 but also operators on other hilbert spaces such as planar elasticity in left h 1 right 2

stopping criteria for adaptive finite element solvers we consider a family of practical stopping criteria for linear solvers for adaptive finite element methods for symmetric elliptic problems a contraction property between two consecutive levels of refinement of the adaptive algorithm is shown when a family of smallness criteria for the corresponding linear solver residuals are assumed on each level or refinement more importantly based on known and new results for the estimation of the residuals of the conjugate gradient method we show that the smallness criteria give rise to practical stopping criteria for the iterations of the linear solver which guarantees that the inexact adaptive algorithm converges a series of numerical experiments highlights the practicality of the theoretical developments

a posteriori error estimates including algebraic error and stopping criteria for iterative solvers for the finite volume discretization of a second order elliptic model problem we derive a posteriori error estimates which take into account an inexact solution of the associated linear algebraic system we show that the algebraic error can be bounded by constructing an equilibrated raviart thomas nedelec discrete vector field whose divergence is given by a proper weighting of the residual vector next claiming that the discretization error and the algebraic one should be in balance we construct stopping criteria for iterative algebraic solvers an attention is paid in particular to the conjugate gradient method which minimizes the energy norm of the algebraic error using this convenient balance we also prove the efficiency of our a posteriori estimates i e we show that they also represent a lower bound up to a generic constant for the overall energy error a local version of this result is also stated this makes our approach suitable for adaptive mesh refinement which also takes into account the algebraic error numerical experiments illustrate the proposed estimates and construction of efficient stopping criteria for algebraic iterative solvers

stopping criteria for iterations in finite element methods this work extends the results of arioli 1 2 on stopping criteria for iterative solution methods for linear finite element problems to the case of nonsymmetric positive definite problems we show that the residual measured in the norm induced by the symmetric part of the inverse of the system matrix is relevant to convergence in a finite element context we then use krylov solvers to provide alternative ways of calculating or estimating this quantity and present numerical experiments which validate our criteria

a class of discontinuous petrov galerkin methods part i the transport equation considering a simple model transport problem we present a new finite element method while the new method fits in the class of discontinuous galerkin dg methods it differs from standard dg and streamline diffusion methods in that it uses a space of discontinuous trial functions tailored for stability the new method unlike the older approaches yields optimal estimates for the primal variable in both the element size h and polynomial degree p and outperforms the standard upwind dg method

iterative substructuring preconditioners for mortar element methods in two dimensions the mortar methods are based on domain decomposition and they allow for the coupling of different variational approximations in different subdomains the resulting methods are nonconforming but still yield optimal approximations in this paper we will discuss iterative substructuring algorithms for the algebraic systems arising from the discretization of symmetric second order elliptic equations in two dimensions both spectral and finite element methods for geometrically conforming as well as nonconforming domain decompositions are studied in each case we obtain a polylogarithmic bound on the condition number of the preconditioned matrix

adaptive petrov galerkin methods for first order transport equations we propose stable variational formulations for certain linear unsymmetric operators with first order transport equations in bounded domains serving as the primary focus of this paper the central objective is to develop for such classes adaptive solution concepts with provable error reduction to adaptively resolve anisotropic solution features such as propagating singularities the presently proposed variational formulations allow in particular the employment of trial spaces spanned by directional representation systems since such systems typically given as frames are known to be stable only in l 2 special emphasis is placed on l 2 stable formulations the proposed stability concept is based on perturbations of certain ideal test spaces in petrov galerkin formulations see also l f demkowicz and j gopalakrishnan comput methods appl mech engrg 199 2010 pp 1558 1572 l demkowicz and j gopalakrishnan numer methods partial differential equations 27 2011 pp 70 105 j

residual and backward error bounds in minimum residual krylov subspace methods minimum residual norm iterative methods for solving linear systems ax b can be viewed as and are often implemented as sequences of least squares problems involving krylov subspaces of increasing dimensions the minimum residual method minres c paige and m saunders siam j numer anal 12 1975 pp 617 629 and generalized minimum residual method gmres y saad and m schultz siam j sci statist comput 7 1986 pp 856 869 represent typical examples in c paige and z strakos bounds for the least squares distance using scaled total least squares numer math to appear revealing upper and lower bounds on the residual norm of any linear least squares ls problem were derived in terms of the total least squares tls correction of the corresponding scaled tls problem in this paper theoretical results of c paige and z strakos bounds for the least squares distance using scaled total least squares numer math to appear are extended to the gmres context the bounds that are developed are important in theory but they also have fundamental practical implications for the finite precision behavior of the modified gram schmidt implementation of gmres and perhaps for other minimum norm methods

superlinear convergence of krylov subspace methods for self adjoint problems in hilbert space the conjugate gradient and minimum residual methods for self adjoint problems in hilbert space are considered linear and superlinear convergence results with respect to both q and r rates are reviewed new results on ell step q superlinear and r superlinear convergence for the minimum residual method are provided and examples are considered to underscore the relevance of a hilbert space theory

fluctuation scaling of quotation activities in the foreign exchange market abstract we study the scaling behavior of quotation activities for various currency pairs in the foreign exchange market the components centrality is estimated from multiple time series and visualized as a currency pair network the power law relationship between a mean of quotation activity and its standard deviation for each currency pair is found the scaling exponent and the ratio between common and specific fluctuations increase with the length of the observation time window t the result means that although for t 1 min the market dynamics are governed by specific processes and at a longer time scale t 100 min the common information flow becomes more important we point out that quotation activities are not independently poissonian for t 1 min and temporally or mutually correlated activities of quotations can happen even at this time scale a stochastic model for the foreign exchange market based on a bipartite graph representation is proposed

analysis of high resolution foreign exchange data of usd jpy for 13 years we analyze high resolution foreign exchange data consisting of 20 million data points of usd jpy for 13 years to report firm statistical laws in distributions and correlations of exchange rate fluctuations a conditional probability density analysis clearly shows the existence of trend following movements at time scale of 8 ticks about 1 minute

first order system least squares for second order partial differential equations part i this paper develops a least squares functional that arises from recasting general second order uniformly elliptic partial differential equations in n 2 or 3 dimensions as a system of first order equations in part i z cai r d lazarov t manteuffel and s mccormick siam j numer anal 31 1994 pp 1785 1799 a similar functional was developed and shown to be elliptic in the h divv times h 1 norm and to yield optimal convergence for finite element subspaces of h divv times h 1 in this paper the functional is modified by adding a compatible constraint and imposing additional boundary conditions on the first order system the resulting functional is proved to be elliptic in the h 1 n 1 norm this immediately implies optimal error estimates for finite element approximation by standard subspaces of h 1 n 1 another direct consequence of this ellipticity is that multiplicative and additive multigrid algorithms applied to the resulting discrete functionals are optimally convergent as an alternative to perturbation based approaches the least squares approach developed here applies directly to convection diffusion reaction equations in a unified way and also admits a fast multigrid solver historically a missing ingredient in least squares methodology

estimating and improving the signal to noise ratio of time series by symbolic dynamics we investigate the effect of symbolic encoding applied to times series consisting of some deterministic signal and additive noise as well as time series given by a deterministic signal with randomly distributed initial conditions as a model of event related brain potentials we introduce an estimator of the signal to noise ratio snr of the system by means of time averages of running complexity measures such as shannon and r enyi entropies and prove its asymptotical equivalence with the linear snr in the case of shannon entropies of symbol distributions a snr improvement factor is defined exhibiting a maximum for intermediate values of noise amplitude in analogy to stochastic resonance phenomena we demonstrate that the maximum of the snr improvement factor can be shifted toward smaller noise amplitudes by using higher order r enyi entropies instead of the shannon entropy for a further improvement of the snr a half wave encoding of noisy time series is introduced finally we discuss the effect of noisy phases on the linear snr as well as on the snr defined by symbolic dynamics it is shown that longer symbol sequences yield an improvement of the latter

symbolic dynamics of event related brain potentials we apply symbolic dynamics techniques such as word statistics and measures of complexity to nonstationary and noisy multivariate time series of electroencephalograms eeg in order to estimate event related brain potentials erp their significance against surrogate data as well as between different experimental conditions is tested these methods are validated by simulations using stochastic dynamical systems with time dependent control parameters and compared with traditional erp analysis techniques continuous eeg data are cut into epochs according to stimuli events presented to the subjects these ensembles of time series can be considered as ensembles of trajectories given by some dynamical systems we employ a statistical mechanics approach motivated by the frobenius perron equation and apply it to coarse grained symbolic descriptions of the dynamics we develop time dependent measures of complexity founded on running cylinder sets and show that these quantities are able to distinguish simulated data obtained by different control parameters as well as experimental data between different experimental conditions as a first finding our approach restores the well known erp components and it reveals additionally qualitative changes in the eeg that cannot be detected by means of the traditional techniques we criticize the prerequisites of the traditional approach to erp analysis and propose to consider erp instead in terms of dynamical system theory and information theory

frequency analysis of tick quotes on the foreign exchange market and agent based modeling a spectral distance approach abstract high frequency financial data of the foreign exchange market eur chf eur gbp eur jpy eur nok eur sek eur usd nzd usd usd cad usd chf usd jpy usd nok and usd sek are analyzed by utilizing the kullback leibler divergence between two normalized spectrograms of the tick frequency and the generalized jensen shannon divergence among them the temporal structure variations of the similarity between currency pairs is detected and characterized a simple agent based model in which n market participants exchange m currency pairs is proposed the equation for the tick frequency is approximately derived theoretically based on the analysis of this model the spectral distance of the tick frequency is associated with the similarity of the behavior perception and decision of the market participants in exchanging these currency pairs