noncore square matrices miscellany a noncore square matrix is one whose unique core nilpotent decomposition does not exclusively consist of the core part if the core part is idempotent the matrix itself is called quasiidempotent this paper makes a brief survey of these and other noncore matrices and their properties

bi sobolev mappings and elliptic equations in the plane abstract suppose that f u v is a homeomorphism in the plane of the sobolev class w loc 1 1 such that its inverse is of the same sobolev class we prove that u and v have the same set of critical points as an application we show that u and v are distributional solutions to the same non trivial degenerate elliptic equation in divergence form we study similar properties also in higher dimensions

direct pole placement adaptive tracking of non minimum phase systems in this paper we propose a direct pole placement adaptive tracking scheme for non minimum phase open loop stable linear plants with time delays this controller utilizes the internal model principle to eliminate steady state tracking error for signals with known distinct frequencies the controller order depends only on the number of frequencies in the reference input but not on the order of the plant it is shown that with sufficiently small loop gain the controller can guarantee stable closed loop and asymptotic tracking

a robust direct adaptive controller this paper proposes a new direct adaptive control algorithm which is robust with respect to additive and multiplicative plant unmodeled dynamics the algorithm is designed based on the reduced order plant which is assumed to be minimum phase and of known order and relative degree but is analyzed with respect to the overall plant which due to the unmodeled dynamics may be nonminimum phase and of unknown order and relative degree it is shown that if the unmodeled dynamics are sufficiently small in the low frequency range then the algorithm guarantees boundedness of all signals in the adaptive loop and small residual tracking errors for any bounded initial conditions in the absence of unmodeled dynamics the residual tracking error is shown to be zero

nonnegative idempotent matrices and the minus partial order abstract we describe the structure of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus order

a brief biography and appreciation of calyampudi radhakrishna rao with a bibliography of his books and papers we present a brief biography and appreciation of calyampudi radhakrishna rao to whom this fifth special issue on linear algebra and statistics of linear algebra and its applications is dedicated in celebration of his 75th birthday a bibliography of his books and papers is also included

integral estimates for null lagrangians sobolev spaces of differential forms are studied lp projections onto exact forms are introduced as a tool to obtain integral estimates for null lagrangians new results on compensated compactness are given and mean coercive variational integrals are found existence of minima of certain mean coercive functionals is established

an indirect robust continuous time adaptive controller with minimal modifications abstract the goal of this paper is to examine the robustness of an indirect continuous time adaptive controller parameter projection which is the only modification on the basis of earlier conventional adaptive control algorithms is employed to constrain the estimates inside a prior known region following the somewhat standard analysis of indirect adaptive control systems it is shown that without any restriction on signals in the closed loop system such as persistence of excitation global stability of the system is guaranteed in the presence of unmodelled dynamics and bounded external disturbances furthermore the adaptive controller can still retain the properties of earlier unmodified conventional adaptive controllers when the controlled plant satisfies so called ideal assumptions for implementation no a priori knowledge on the unmodelled dynamics is needed

local regularity for solutions of anisotropic obstacle problems abstract this paper deals with anisotropic obstacle problems for the a harmonic equation div a x u x 0 a local regularity result is given under suitable assumptions

nonlinear cauchy riemann operators in ℝⁿ this paper has arisen from an effort to provide a comprehensive and unifying development of the l p theory of quasiconformal mappings in r n the governing equations for these mappings form nonlinear differential systems of the first order analogous in many respects to the cauchy riemann equations in the complex plane this approach demands that one must work out certain variational integrals involving the jacobian determinant guided by such integrals we introduce two nonlinear differential operators denoted by d and d which act on weakly differentiable deformations f r n of a domain c r n solutions to the so called cauchy riemann equations d f 0 and d f 0 are simply conformal deformations preserving and reversing orientation respectively these operators though genuinely nonlinear possess the important feature of being rank one convex among the many desirable properties we give the fundamental l p estimate d f p a p n d f p in quest of the best constant a p n we are faced with fascinating problems regarding quasiconvexity of some related variational functionals applications to quasiconformal mappings are indicated

on inequality constrained generalized least squares estimation abstract the inequality constrained generalized least squares icgls estimator is considered based on results of the powerful theory of generalized inversion a nice method for obtaining the estimator is given and it is shown how this method leads to closed form expressions the paper is concluded with some remarks on the so called two step estimator

existence results for some nonuniformly elliptic equations with irregular data abstract in this paper we prove the existence of solutions of some nonlinear elliptic equations with degenerate coercivity of the type formula when the datum f belongs to the space lr where r 2 and 0 1

the maximum principle of alexandrov for very weak solutions abstract we extend the classical maximal principle of alexandrov to very weak solutions of the elliptic equation i j 1 n a i j x 2 u x i x j f

unbiased and minimum variance unbiased estimation of estimable functions for fixed linear models with arbitrary covariance structure consider a general linear model for a column vector y of data having e y xa and var y ur2h where a is a vector of unknown parameters and x and h are given matrices that are possibly deficient in rank let b ty where t is any matrix of maximum rank such that th 4 the estimation of a linear function of a by functions of the form c a y where c and a are permitted to depend on b is investigated allowing c gnd a to depend on b expands the class of unbiased estimators in a nontrivial way however it does not add to the class of linear functions of a that are estimable any minimumvariance unbiased estimator is identically for y in the column space of x h equal to the estimator that has minimum variance among strictly linear unbiased estimators

some early statistical contributions to the theory and practice of linear algebra abstract we outline the history of some of the concepts and techniques of linear algebra which are intimately connected with the development of the method of least squares and related fitting procedures our study concentrates on contributions made during the early years of the nineteenth century but it is not entirely restricted to this period

robust model reference adaptive control we propose a new model reference adaptive control algorithm and show that it provides the robust stability of the resulting closed loop adaptive control system with respect to unmodeled plant uncertainties the robustness is achieved by using a relative error signal in combination with a dead zone and a projection in the adaptive law the extra a priori information needed to design the adaptive law are bounds on the plant parameters and an exponential bound on the impulse response of the inverse plant transfer function

robust continuous time adaptive control by parameter projection the problem of adaptive control of a continuous time plant of arbitrary relative degree in the presence of bounded disturbances as well as unmodeled dynamics is addressed the adaptation considered is the usual gradient update law with parameter projection the latter being the only robustness enhancement modification employed it is shown that if the unmodeled dynamics which consists of multiplicative as well as additive system uncertainty is small enough then all the signals in the closed loop system are bounded this shows that extra modifications are not necessary for robustness with respect to bounded disturbances and small unmodeled dynamics in the nominal case where unmodeled dynamics and disturbances are absent the asymptotic error in tracking a given reference signal is zero moreover the performance of the adaptive controller is also robust

an adaptive control algorithm for linear systems having unknown time delay abstract this paper presents an algorithm for adaptive control of linear systems having unknown time delay a rational approximation is used for the delay and it is argued that within this framework the analysis of the properties of the corresponding adaptive control algorithm essentially reduces to the issue of robustness to unmodelled dynamics a global convergence theory is presented using recent methods for analyzing robustness properties of adaptive control algorithms the analysis is supported by simulation results

t minima and application to the convergence of some integral functionals with infinite energy minima abstract we consider integral functionals j as in 1 we study the calderon zygmund theory for infinite energy minima u t minima of j and the stability of the t minima with respect to the convergence in 22 with v w 0 1 p

on very weak solutions to a class of double obstacle problems abstract we prove the uniqueness of very weak solutions by hodge decomposition and obtain the existence by an approximation argument moreover we prove that the solution quasiminimizes the r dirichlet integral in k r and finally the stability of very weak solutions is established for varying obstacle functions

gauss markov and weighted least squares estimation under a general growth curve model abstract gauss markov estimator of x 1 bx 2 under a general growth curve model y x 1 bx 2 v 2 v 1 is given necessary and sufficient conditions for equality between a 1 ya 2 and b 1 yb 2 where a i x i x i w i x i x i w i b i x i x i s i x i x i s i with w i s i being any matrix i 1 2 are derived finally a number of criteria for a 1 ya 2 to coincide with the gauss markov estimator of x 1 bx 2 are established

robustness of adaptive control without deadzones data normalization or persistence of excitation abstract robustness properties have been established for modified adaptive control algorithms the major modifications involve various combinations of normalization relative deadzones persistence of excitation parameter projection and or so called modification results showing global boundedness have used some form of data normalization many results on local properties have depended on persistence of excitation however use of these modifications involve some a priori knowledge of the plant for good performance this may lead to complications in practice here it is shown that some useful local robustness properties hold if only parameter projection is used as a modification of the basic gradient estimator

analysis of clustered data a combined estimating equations approach summary examples of clustered data include data from longitudinal studies and data sampled within groups this paper proposes a regression analysis method for clustered data that optimally weights and combines contrasts of the data through a combination of estimating equations examples of combining between cluster within cluster and longitudinal data contrasts are presented the method results in increased estimation efficiency relative to generalised estimating equations with standard working correlation structures the proposed method also simplifies modelling decisions regarding the true correlation structure of the data and avoids correlation parameter estimation

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

on parallel summability of matrices abstract it is shown that the generalized inverses characterize the parallel sum the almost positive definite a p d matrices introduced by duffin and morley 2 are of two types whose intersection is the class of quasi positive definite matrices mitra and puri 7 the a p d matrices of any one type form a saturated subclass of pairwise parallel summable a p d matrices

changes in the general linear model a unified approach abstract analysis of the general linear model with possibly rank deficient design and dispersion matrices has sometimes generated some confusion and controversy prompting some researchers to discuss it as quite distinct from the case of full rank matrices we show that linear zero functions i e linear functions in observations which have zero expectations for all parameter values provide an intuitive way of developing all the important results in connection with the general linear model thus bridging this imaginary gap we show that the effect of addition or deletion of a set of observations in this model can be clearly understood in statistical terms if viewed through such linear zero functions the effect of adding or dropping a group of parameters is also explained well in this manner several sets of update equations were derived by a host of previous researchers in various special cases of the above set up the results derived here bring out the common underlying principles of these formulae and indeed help simplify most of them these results also provide further insights into recursive residuals design of experiments deletion diagnostics and selection of subset models

extending some results and proofs for the singular linear model abstract this note gives some new forms and shortened proofs for results on the linear model with singular dispersion matrix

on a reverse estimate for hodge decompositions of p laplacian type operators let x p 2 be a p laplacian type operator and consider the hodge decomposition x du d h div h 0 a standard elliptic theory asserts that d q p 1 c du p 1q for all q p 1 there has been considerable recent interest in the validity of the reverse estimate du p 1q c d q p 1 for q p 1 in the regularity study of certain geometrical mappings in this paper we give a relatively new proof of a well known theorem that this reverse estimate holds for all q sufficiently close to the natural power p and also prove that the estimate holds for all q p 1 for certain special weak solutions u

the minus partial order and the shorted matrix the minus partial order was defined by hartwig 6 weakening the conditions of the star partial order of drazin 5 several new properties of the minus partial order are established the minus partial order is used to redefine the shorted matrix 15 17 and to define the infimum a b and the supremum a b of a pair a b of matrices of the same order the definition of the shorted matrix given here is similar to the krein anderson trapp definition of the shorted positive operator

matrix partial orders through generalized inverses unified theory the unified theory presented here covers as special cases the star order of drazin the minus order of hartwig and nambooripad the sharp order and other partial orders introduced by the author

smallness and cancellation in some elliptic systems with measure data abstract in a bounded open subset r n we study dirichlet problems with elliptic systems involving a finite radon measure on r n with values into r n defined by div a x u x d u x in u 0 on where a i x y 1 n j 1 n a i j x y j with 1 n the equation index we prove the existence of a distributional solution u r n obtained as the limit of approximations by assuming i that coefficients a i j are bounded caratheodory functions ii ellipticity of the diagonal coefficients a i j and iii smallness of the quadratic form associated to the off diagonal coefficients a i j i e verifying a r staircase support condition with r 0 such a smallness condition is satisfied for instance in each one of these cases a a i j a j i skew symmetry b a i j is small c a i j may be decomposed into two parts the first enjoying skew symmetry and the second being small in absolute value we give an example that satisfies our hypotheses but does not satisfy assumptions introduced in previous works a brezis s type nonexistence result is also given for general smooth elliptic hyperbolic systems

quasilinear elliptic systems with measure data abstract we study the existence of solutions of quasilinear elliptic systems involving n equations and a measure on the right hand side with the form i 1 n x i 1 n j 1 n a i j x u x j u in u 0 on where 1 n is the equation index is an open bounded subset of r n u r n and is a finite randon measure on r n with values in r n existence of a solution is proved for two different sets of assumptions on a examples are provided that satisfy our conditions but do not satisfy conditions required on previous works on this matter

on the kalman filter with possibly degenerate and correlated errors the kalman filter is a very popular tool for estimation and prediction in the context of a state space model sometimes it is necessary to formulate the state space model in such a way that the model errors are correlated the error dispersion matrix may even be singular in this paper we establish a connection between prediction in the state space model in this general set up and estimation in the general linear model subsequently we use the update equations in the general linear model to derive a generalization of the kalman filter

a robust adaptive controller with minimal modifications for discrete time varying systems the goal of this paper is to show that an indirect adaptive controller with parameter projection as the only modification on the basis of conventional adaptive control algorithms can globally stabilize systems having fast parasitics bounded external disturbances and time varying parameters without any restriction on signals in the closed loop system such as persistence of excitation further the controller can still retain the properties of earlier unmodified conventional adaptive controllers when the controlled plant satisfies so called ideal assumptions or the rates at which the plant parameters change belong to the l sub 1 or l sub 2 space

on some matrix equalities for generalized inverses with applications abstract necessary and sufficient conditions are derived for the matrix equality a pn q to hold where a and n are generalized inverses of matrices some consequences and applications are also given in particular necessary and sufficient conditions are derived for the additive decompositions c a b and a a 1 a k to hold

stability for p laplace type equation in a borderline case abstract we study the dirichlet problem for a p laplacian type operator in the setting of the orlicz zygmund space q log r n q 1 and 0 more precisely our aim is to establish under which assumptions on 0 existence and uniqueness of the solution are assured and to prove continuity and stability of the associated nonlinear operator

design issues in adaptive control an integrated approach to the design of practical adaptive control algorithms is presented many existing ideas are brought together and the effect of various design parameters available to a user is explored the theory is extended by showing how the problem of stabilizability of the estimated model can be overcome by running parallel estimators it is shown how asymptotic tracking of deterministic set points can be achieved in the presence of unmodeled dynamics

on non linear elliptic equation with weight abstract in this paper we prove the existence and uniqueness theorem of the dirichlet problem for certain non linear elliptic operators with muckenhoupt weight

nonnegative group monotone matrices and the minus partial order bapat et al previously described a class of nonnegative matrices dominated by a nonnegative idempotent matrix under the minus partial order in this paper we improve upon that description by first presenting a more general result that gives the precise structure of nonnegative matrices dominated by a group monotone matrix under the minus partial order as a special case we derive the complete class of nonnegative matrices dominated by a nonnegative idempotent matrix that includes the class obtained by bapat et al

the least squares g inverse for the sum of multiple matrices in this paper we give the necessary and sufficient conditions for the set identities a 1 a 2 a k 1 3 a 1 1 3 a 2 1 3 a k 1 3 and a 1 a 2 a k 1 4 a 1 1 4 a 2 1 4 a k 1 4 which concern the least squares g inverse 1 3 inverse and the minimum norm g inverse 1 4 inverse of sum of k matrices

paper instability analysis and improvement of robustness of adaptive control the effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed five possible types of instability mechanisms parameter drift linear instability fast adaptation instability high frequency instability and throughput instability are analyzed using simple examples a procedure is used to construct lyapunov like functions for a modified adaptive controller applied to a dominant plant of relative degree one in the presence of parasitics and disturbances and obtain sufficient conditions under which none of the five types of instability can occur the modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set the size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast the region of attraction becomes the whole space

some further remarks on the singular linear model abstract inspired by a recent article of searle 1994 we consider some specific features of estimation of x in the general linear model y x v where the model matrix x need not have full column rank and the dispersion matrix v can be singular particular attention is paid to the problems related to the invariance of some matrix expressions with respect to the choice of generalized inverses these invariance properties yield interesting matrix algebra while characterizing the best linear unbiased estimator of x

on group inverses and the sharp order abstract in this work the group inverse of a matrix is used to define the order on square matrices of index 1 the order is similar to the order of drazin 2 and the minus order of hartwig 6 10 and nambooripad 17 the order and the order are compared and contrasted many conditions are given which assure the equivalence of the various partial orders studied

a distance between orbits that controls commutator estimates and invertibility of operators abstract a distance between orbit spaces generated by a single element is introduced and it is shown that if an operator is invertible in one orbit it is also invertible in nearby orbits thus proving a version of shneiberg s theorem for orbital methods the same machinery is used to extend the celebrated rochberg weiss commutator theorem to the setting of orbital methods it is shown that these results apply to the real and complex methods of interpolation by proving that these methods can be suitably obtained as orbits of a single element

distributional properties of portfolio weights in this paper we prove several distributional properties for optimal portfolio weights the weights are estimated by replacing the parameters with the sample counterparts all results for finite samples are made assuming normally distributed returns we calculate the exact covariances for the weights obtained by the expected quadratic utility additionally we derive the multivariate density function of the global minimum variance portfolio and the univariate density of the tangency portfolio we obtain the conditional density for the sharpe ratio optimal weights and show that the expectations of the sharpe ratio optimal weights do not exist moreover we determine the asymptotic distributions of the estimated weights assuming that the returns follow a multivariate stationary gaussian process

statistical data analysis and inference prologue c radhakrishna rao s contributions to statistics y dodge nonlinear estimation in linear models f hampel on the maximum number of factors in two construction methods for orthogonal arrays a s hedayat j stufken some geometrical aspects of data analysis and statistics j m oller inferences and test of hypotheses on bayes and empirical bayes two stage allocation procedures for selection problems s s gupta t c liang multi sample functional statistical data analysis e parzen discussing truth or falsity by computing a q value w schaafsma j tolboom b van der meulen linear pivotals and the bayes non bayes compromise g a barnard bayes modified minimax experimental design r v canfield a simple class of tests locally better than the score test t k chandra comparison of tests in the presence of a nuisance parameter r mukerjee tests for redundancy of some variables in multivariate analysis y fujikoshi model choice in the context of simultaneous inference t havranek o soudsky first zeros of empirical characteristic functions and extreme values of gaussian processes j husler parameter and variance estimation comparison of experiments with weighted distributions m j bayarri m h de groot optimal estimation for weighted distributions semi parametric models v p godambe m b rajarshi on efficiency for quasi likelihood and composite quasi likelihood methods c c heyde improved estimators of dispersion of an inverse gaussian distribution n pal b k sinha applications of edgeworth expansions to bootstrap a review g j babu dual poincare type inequalities via the cramer rao and the cauchy schwarz inequalities and related characterizations th cacoullos some optimality results on stein s two stage sampling j k ghosh r mukerjee the cramer frechet rao inequality for sequential estimation in non regular case z govindarajulu i vincze whither delete k jackknifing for smooth statistical functionals p k sen distributions weighted distributions and characterization three useful expressions for expectations involving a wishart matrix and its inverse g p h styan characterizations of distributions via moments of order statistics a survey and comparison of methods g d lin constancy of regression of a polynomial of sample average of positive random variables on their ratios characterizes gamma distribution a kagan probing encountered data meta analysis and weighted distribution methods g p patil c taillie further results on identification when the parameters are partially unknown s el khattabi f streit linear models and matrices weighted least squares estimation in the general gauss markov model j k baksalary s puntanen evaluating pre test predictors of success in linear regression models a cohen h b sackrowitz optimum invariant tests in mixed linear models with two variance components th mathew

nonlinear commutators for the fractional p laplacian and applications we prove a nonlinear commutator estimate concerning the transfer of derivatives onto testfunctions for the fractional p laplacian this implies that solutions to certain degenerate nonlocal equations are higher differentiable also weakly fractional p harmonic functions which a priori are less regular than variational solutions are in fact classical as an application we show that sequences of uniformly bounded frac n s harmonic maps converge strongly outside at most finitely many points

global boundedness of discrete time adaptive control just using estimator projection in this paper we study a discrete time indirect adaptive control algorithm which contains a constrained gradient parameter estimator and a pole assignment control law synthesis module this adaptive control algorithm does not involve modifications like data normalization use of deadzones or injection of persistently excited signals also it requires no a priori knowledge of system modelling errors it is shown that global robustness properties still hold when this simple algorithm is applied to systems with bounded disturbances and arbitrarily small fast parasitic dynamics the problem of indirect decentralized adaptive control of interconnected systems is also considered we use the above adaptive algorithm to design completely decentralized local adaptive controllers for each isolated subsystem by ignoring interactions between subsystems we show that the local controllers designed in this way are robust in the sense that all signals in the closed loop adaptive system are bounded for bounded initial conditions reference inputs disturbances and an arbitrarily small amount of interaction between subsystems and unmodelled dynamics of each subsystem

kantorovich and cauchy schwarz inequalities involving positive semidefinite matrices and efficiency comparisons for a singular linear model abstract matrix kantorovich inequalities involving two positive semidefinite matrices are presented corresponding cauchy schwarz inequalities are discussed some of these are used to compare several efficient and inefficient estimators for a singular linear model

a distance between elliptical distributions based in an embedding into the siegel group this paper describes two different embeddings of the manifolds corresponding to many elliptical probability distributions with the informative geometry into the manifold of positive definite matrices with the siegel metric generalizing a result published previously elsewhere these new general embeddings are applicable to a wide class of elliptical probability distributions in which the normal t student and cauchy are specific examples a lower bound for the rao distance is obtained which is itself a distance and through these embeddings a number of statistical tests of hypothesis are derived

convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations abstract we study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation u t div a x u f x u 0 on a bounded domain subject to dirichlet boundary and to initial conditions the data are supposed to satisfy suitable regularity and growth conditions our approach to the convergence result and decay estimate is based on the ojasiewicz simon gradient inequality which in the case of the semilinear heat equation is known to give optimal decay estimates the abstract results and their applications are discussed also in the framework of orlicz sobolev spaces

a family of matrices the discretized brownian bridge and distance based regression abstract the investigation of a distance based regression model using a one dimensional set of equally spaced points as regressor values and x y as a distance function leads to the study of a family of matrices which is closely related to a discrete analog of the brownian bridge stochastic process we describe its eigenstructure and several properties recovering in particular well known results on tridiagonal toeplitz matrices and related topics

robust adaptive control of proper systems to date all the adaptive control algorithms have been proposed only for strictly proper systems in this paper a scheme is proposed to design an adaptive controller for proper systems to study the robustness of the adaptive controller both additive and multiplicative types of unmodeled dynamics are considered and can also be allowed to be proper or even improper global bounded input bounded output stability is established the achievement of a small in the mean tracking error and perfect tracking rejection of deterministic trajectories disturbances in the absence of system unmodeled dynamics are discussed the results are also verified by simulation studies

hyperelastic deformations of smallest total energy throughout this article mathbb x and mathbb y will be nonempty bounded domains in mathbb r n n geqq 2 the term deformation of mathbb x subset mathbb r n onto mathbb y subset mathbb r n refers to an orientation preserving homeomorphism h mathbb x overset textnormal tiny onto longrightarrow mathbb y in the sobolev class fancyscript w 1 1 mathbb x mathbb y whose inverse f mathbb y overset textnormal tiny onto longrightarrow mathbb x lies in fancyscript w 1 1 mathbb y mathbb x the general law of hyperelasticity asserts that there exists an energy integral mathcal e mathbb x h int mathbb x e x h dh rm dx such that the elastic deformations have the smallest energy we assume here that e is conformally coerced and polyconvex some additional regularity conditions are also imposed under those conditions we establish the existence and global invertibility of the minimizers the key tools in obtaining an extremal deformation h mathbb x overset textnormal tiny onto longrightarrow mathbb y regardless of its boundary values are the free lagrangians finding suitable free lagrangians and using them for a specific stored energy function e is truly a work of art we have done it here for the so called total harmonic energy and a pair of annuli in the plane in fact this challenging problem illustrates rather clearly the strength of the concept of free lagrangians