autocorrelation based fuzzy clustering of time series the traditional approaches to clustering a set of time series are generally applicable if there is a fixed underlying structure to the time series so that each will belong to one cluster or the other however time series often display dynamic behaviour in their evolution over time this dynamic behaviour should be taken into account when attempting to cluster time series for instance during a certain period a time series might belong to a certain cluster afterwards its dynamics might be closer to that of another cluster in this case the traditional clustering approaches are unlikely to find and represent the underlying structure in the given time series this switch from one time state to another which is typically vague can be naturally treated following a fuzzy approach this paper proposes a fuzzy clustering approach based on the autocorrelation functions of time series in which each time series is not assigned exclusively to only one cluster but it is allowed to belong to different clusters with various membership degrees

some remarks on quasiconvexity and rank one convexity we explore some necessary conditions for quasiconvexity in an attempt to show that rank one convexity does not imply quasiconvexity when the target space for deformations is two dimensional an interesting construction is presented showing how rank one directions may fit with each other making the task harder than in higher dimensions

mappings of least dirichlet energy and their hopf differentials the paper is concerned with mappings h colon mathbb x begin array ll rm onto longrightarrow end array mathbb y between planar domains having least dirichlet energy the existence and uniqueness up to a conformal change of variables in mathbb x of the energy minimal mappings is established within the class overline fancyscript h 2 mathbb x mathbb y of strong limits of homeomorphisms in the sobolev space fancyscript w 1 2 mathbb x mathbb y a result of considerable interest in the mathematical models of nonlinear elasticity the inner variation of the independent variable in mathbb x leads to the hopf differential h z overline h bar z rm d z otimes rm d z and its trajectories for a pair of doubly connected domains in which mathbb x has finite conformal modulus we establish the following principle a mapping h in overline fancyscript h 2 mathbb x mathbb y is energy minimal if and only if its hopf differential is analytic in mathbb x and real along partial mathbb x in general the energy minimal mappings may not be injective in which case one observes the occurrence of slits in mathbb x cognate with cracks slits are triggered by points of concavity of mathbb y they originate from partial mathbb x and advance along vertical trajectories of the hopf differential toward mathbb x where they eventually terminate so no crosscuts are created

a numerical method for detecting singular minimizers of multidimensional problems in nonlinear elasticity in this paper we describe and analyse a numerical method that detects singular minimizers and avoids the lavrentiev phenomenon for three dimensional problems in nonlinear elasticity this method extends to three dimensions the corresponding one dimensional method of ball and knowles

fractional equivalent lagrangian densities for a fractional higher order equation in this communication we show that the equivalent lagrangian densities elds of a fractional higher order nonlinear schrodinger equation with stable soliton like solutions can be related in a hitherto unknown way this new relationship is described in terms of a new fractional operator that includes both left and right sided fractional derivatives using this operator it is possible to generate new elds that contain different fractional parts in addition to the already known elds which only differ by a sum of first order partial derivatives of two arbitrary functions

alternative c means clustering algorithms abstract in this paper we propose a new metric to replace the euclidean norm in c means clustering procedures on the basis of the robust statistic and the influence function we claim that the proposed new metric is more robust than the euclidean norm we then create two new clustering methods called the alternative hard c means ahcm and alternative fuzzy c means afcm clustering algorithms these alternative types of c means clustering have more robustness than c means clustering numerical results show that ahcm has better performance than hcm and afcm is better than fcm we recommend afcm for use in cluster analysis recently this afcm algorithm has successfully been used in segmenting the magnetic resonance image of ophthalmology to differentiate the abnormal tissues from the normal tissues

cluster differences scaling with a within clusters loss component and a fuzzy successive approximation strategy to avoid local minima cluster differences scaling is a method for partitioning a set of objects into classes and simultaneously finding a low dimensional spatial representation ofk cluster points to model a given square table of dissimilarities amongn stimuli or objects the least squares loss function of cluster differences scaling originally defined only on the residuals of pairs of objects that are allocated to different clusters is extended with a loss component for pairs that are allocated to the same cluster it is shown that this extension makes the method equivalent to multidimensional scaling with cluster constraints on the coordinates a decomposition of the sum of squared dissimilarities into contributions from several sources of variation is described including the appropriate degrees of freedom for each source after developing a convergent algorithm for fitting the cluster differences model it is argued that the individual objects and the cluster locations can be jointly displayed in a configuration obtained as a by product of the optimization finally the paper introduces a fuzzy version of the loss function which can be used in a successive approximation strategy for avoiding local minima a simulation study demonstrates that this strategy significantly outperforms two other well known initialization strategies and that it has a success rate of 92 out of 100 in attaining the global minimum

fuzzy clustering for data time arrays with inlier and outlier time trajectories in many knowledge discovery and data mining tasks fuzzy clustering is one of the most common tools for data partitioning in this paper dynamic fuzzy clustering models for classifying a set of multivariate time trajectories time series sequences are developed in particular by adopting an exploratory approach based on a geometric algebraic formulation of the data time array different kinds of dynamic fuzzy clustering models based on cross sectional and longitudinal aspects are suggested furthermore a modified version of the previous clustering models that can be seen as a generalization of these models is proposed by utilizing these models we can obtain beneficial effects in the clustering process when anomalous trajectories trajectories with anomalous positions and slopes are present in the dataset in fact the models are suitable for detecting structures of time trajectories with anomalous patterns that are not uniformly distributed over the structure s domains and are characterized by strange slopes in these models the disruptive effect of the anomalous trajectories is neutralized and smoothed and the information on the influence of individual time trajectories on the detected groups is given furthermore some remarks on dynamic three way extensions of a few robust fuzzy clustering models for two way data are suggested demonstrative examples are shown and a comparison assessment based on artificial multivariate time varying data is carried out

garch based robust clustering of time series in this paper we propose different robust fuzzy clustering models for classifying heteroskedastic volatility time series following the so called model based approach to time series clustering and using a partitioning around medoids procedure the proposed models are based on a garch parametric modeling of the time series i e the unconditional volatility and the time varying volatility garch representation of the time series we first suggest a timid robustification of the fuzzy clustering then we propose three robust fuzzy clustering models belonging to the so called metric noise and trimmed approaches respectively each model neutralizes the negative effects of the outliers in the clustering process in a different manner in particular the first robust model based on the metric approach achieves its robustness with respect to outliers by taking into account a robust distance measure the second based on the noise approach achieves its robustness by introducing a noise cluster represented by a noise prototype the third based on the trimmed approach achieves its robustness by trimming away a certain fraction of outlying time series the usefulness and effectiveness of the proposed clustering models is illustrated by means of a simulation study and two applications in finance and economics

natural variational principles on riemannian structures in the calculus of variations the euler lagrange operator e l refers to the differential operator derived from the first variation of the fundamental or action integral solutions of the differential equations e l 0 are thus the critical points of the fundamental integral conversely given a differential operator say t it is natural to ask if t can be identified as the euler lagrange operator derived from some lagrangian l this is the inverse problem to the calculus of variations if the lagrangian is invariant under the action of a lie group g then it is well known that the corresponding euler lagrange equations are also invariant under g thus the following equivariant inverse problem to the calculus of variations may be stated if t is a differential operator which is i invariant under the action of a lie group g and ii the euler lagrange operator of some lagrangian then when is t the euler lagrange operator of a g invariant lagrangian in this paper we pose the equivariant inverse problem for differential operators defined by natural tensors on riemannian structures by using the machinery of the classical tensor calculus and techniques from classical invariant theory we will show that the obstructions to the construction of natural lagrangians are in a one to one correspondence with the conformally invariant characteristic forms discovered by chern and simons 7 the literature on the inverse problem to the calculus of variations is by now an extensive one an abbreviated review of the problem s history commences with h helmholtz who in 1887 derived necessary conditions for t to be an euler lagrange operator in the case where the differential equations t 0 constitute a system of second order ordinary differential equations shortly thereafter see havas 13 for the relevant references the local sufficiency of these necessary conditions was established and the extension of helmholtz s results to more general classes of differential operators achieved necessary conditions for a differential operator t to be locally an euler lagrange operator will be referred to in this paper as the helmholtz conditions on t

robustness properties of k means and trimmed k means abstract the generalized k means method is based on the minimization of the discrepancy between a random variable or a sample of this random variable and a set with k points measured through a penalty function as in the m estimators setting k 1 a penalty function with unbounded derivative naturally leads to nonrobust generalized k means however surprisingly the lack of robustness extends also to the case of bounded that is generalized k means do not inherit the robustness properties of the m estimator from which they came attempting to robustify the generalized k means method the generalized trimmed k means method arises from combining k means idea with a so called impartial trimming procedure in this article study generalized k means and generalized trimmed k means performance from the viewpoint of hampel s robustness criteria that is we investigate the influence function breakdown point and qualitative robustness confirming the superiority provided by the trimming we inc

quantile autocovariances a powerful tool for hard and soft partitional clustering of time series abstract a key issue in cluster analysis is determining a proper dissimilarity measure between two data objects and many pairwise dissimilarities have been proposed to deal with time series assuming that the clustering purpose is to group series according to the underlying dependence structures a detailed study of the behavior in clustering of a dissimilarity based on comparing estimated quantile autocovariance functions qaf is carried out quantile autocovariances provide information about the serial dependence structure that other conventional features are not able to capture which suggests great potential to perform clustering of series the asymptotic behavior of the sample quantile autocovariances is studied and an algorithm to determine optimal combinations of lags and pairs of quantile levels to perform clustering is introduced the proposed metric is used to perform hard and soft partitioning based clustering first a broad simulation study examines the behavior of the proposed metric in crisp clustering with the pam procedure a novel fuzzy c medoids algorithm based on the qaf dissimilarity is then proposed and compared with other fuzzy procedures in a new simulation study conducted to cluster fuzzy scenarios involving ar and garch models in all cases the qaf based procedures outperform or are highly competitive with a range of dissimilarities reported in the literature particularly exhibiting high capability to cluster conditionally heteroskedastic time series and robustness to the distributional form of the errors two specific applications involving air quality data and financial time series illustrate the usefulness of the proposed procedures

invertibility and weak continuity of the determinant for the modelling of cavitation and fracture in nonlinear elasticity in this paper we present and analyze a variational model in nonlinear elasticity that allows for cavitation and fracture the main idea in unifying the theories of cavitation and fracture is to regard both cavities and cracks as phenomena of the creation of a new surface accordingly we define a functional that measures the area of the created surface this functional has relationships with the theory of cartesian currents we show that the boundedness of that functional implies sequential weak continuity of the determinant of the deformation gradient and that the weak limit of one to one almost everywhere deformations is also one to one almost everywhere we then use these results to obtain the existence of minimizers of variational models that incorporate elastic energy and this created surface energy taking into account orientation preserving and non interpenetration conditions

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

dispersive shock waves in the kadomtsev petviashvili and two dimensional benjamin ono equations abstract dispersive shock waves dsws in the kadomtsev petviashvili kp equation and two dimensional benjamin ono 2dbo equation are considered using step like initial data along a parabolic front employing a parabolic similarity reduction exactly reduces the study of such dsws in two space one time 2 1 dimensions to finding dsw solutions of 1 1 dimensional equations with this ansatz the kp and 2dbo equations can be exactly reduced to the cylindrical korteweg de vries ckdv and cylindrical benjamin ono cbo equations respectively whitham modulation equations which describe dsw evolution in the ckdv and cbo equations are derived and riemann type variables are introduced dsws obtained from the numerical solutions of the corresponding whitham systems and direct numerical simulations of the ckdv and cbo equations are compared with very good agreement obtained in turn dsws obtained from direct numerical simulations of the kp and 2dbo equations are compared with the ckdv and cbo equations again with good agreement it is concluded that the 2 1 dsw behavior along self similar parabolic fronts can be effectively described by the dsw solutions of the reduced 1 1 dimensional equations

variational principles for second order quasi linear scalar equations abstract the problem of constructing variational principles for a given second order quasi linear partial differential equation is considered in particular we address the problem of finding a first order function f whose product with the given differential operator is the euler lagrange operator derived from some lagrangian two sets of equations for such a function f are obtained necessary and sufficient conditions for the integration of the first set are established in general and these lead to a considerable simplification of the second set in certain special cases such as the case when the operator is elliptic the problem is completely solved the utility of our results is illustrated by a variety of examples

wavelets based clustering of multivariate time series crisp and fuzzy clustering methods based on a combination of univariate and multivariate wavelet features are considered for the clustering of multivariate time series the performance of each of these methods is evaluated for stationary and variance nonstationary multivariate time series with different error correlation structures the main outcomes of the simulation studies are are as follows the superior performance of this approach for both the crisp and fuzzy cluster methods compared to some of the other approaches for clustering multivariate time series the very good performance of the fuzzy relational method overall to cluster longer time series when all of them do not appear to group exclusively into well separated clusters we consider an application to multivariate greenhouse gases time series and show that the crisp and fuzzy clustering methods considered are well validated

a validity measure for fuzzy clustering the authors present a fuzzy validity criterion based on a validity function which identifies compact and separate fuzzy c partitions without assumptions as to the number of substructures inherent in the data this function depends on the data set geometric distance measure distance between cluster centroids and more importantly on the fuzzy partition generated by any fuzzy algorithm used the function is mathematically justified via its relationship to a well defined hard clustering validity function the separation index for which the condition of uniqueness has already been established the performance of this validity function compares favorably to that of several others the application of this validity function to color image segmentation in a computer color vision system for recognition of ic wafer defects which are otherwise impossible to detect using gray scale image processing is discussed

the equivariant inverse problem and the maxwell equations in affirmative the equivariant inverse problem for maxwell type euler lagrange expressions is solved this allows the proof of the uniqueness of the maxwell equations

determinantal ideals and the capelli identities abstract recent work in the formal calculus of variations and also in riemannian geometry and general relativity have relied upon certain important previously established properties of determinant ideals in characteristic zero this note presents direct elementary proofs of these properties by utilizing the capelli identities of classical invariant theory

whitham modulation theory for 2 1 dimensional equations of kadomtsev petviashvili type whitham modulation theory for certain two dimensional evolution equations of kadomtsev petviashvili kp type is presented three specific examples are considered in detail the kp equation the two dimensional benjamin ono 2dbo equation and a modified kp m2kp equation a unified derivation is also provided in the case of the m2kp equation the corresponding whitham modulation system exhibits features different from the other two the approach presented here does not require integrability of the original evolution equation indeed while the kp equation is known to be a completely integrable equation the 2dbo equation and the m2kp equation are not known to be integrable in each of the cases considered the whitham modulation system obtained consists of five first order quasilinear partial differential equations the riemann problem i e the analogue of the gurevich pitaevskii problem for the one dimensional reduction of the m2kp equation is studied for the m2kp equation the system of modulation equations is used to analyze the linear stability of traveling wave solutions

the scattering transform for the benjamin ono equation in the small dispersion limit abstract using exact formulae for the scattering data of the benjamin ono equation valid for general rational potentials recently obtained in miller and wetzel 17 we rigorously analyze the scattering data in the small dispersion limit in particular we deduce precise asymptotic formulae for the reflection coefficient the location of the eigenvalues and their density and the asymptotic dependence of the phase constant associated with each eigenvalue on the eigenvalue itself our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering and they also provide new details not previously reported in the literature

time series clustering with arma mixtures clustering problems are central to many knowledge discovery and data mining tasks however most existing clustering methods can only work with flxed dimensional representations of data patterns in this paper we study the clustering of data patterns that are represented as sequences or time series possibly of difierent lengths we propose a model based approach to this problem using mixtures of autoregressive moving average arma models we derive an expectation maximization em algorithm for learning the mixing coe cients as well as the parameters of the component models to address the model selection problem we use the bayesian information criterion bic to determine the number of clusters in the data experiments are conducted on a number of simulated and real datasets results from the experiments show that our method compares favorably with other methods proposed previously by others for similar time series clustering tasks

quasiconvexity at the boundary positivity of the second variation and elastic stability for one dimensional problems of the calculus of variations there is a well developed theory relating the positivity of the second variation at a critical point i e linearized stability to strong relative minima these results largely due to weierstrass are usually proved by the method of fields of extremals as is described in bolza 1904 and morrey 1966 or by an argument involving a careful use of taylor series as in hestenes 1966 chapter 3 14

fuzzy c ordered means clustering fuzzy clustering helps to find natural vague boundaries in data the fuzzy c means method is one of the most popular clustering methods based on minimization of a criterion function however one of the greatest disadvantages of this method is its sensitivity to the presence of noise and outliers in data this paper introduces a new robust fuzzy clustering method named fuzzy c ordered means fcom clustering this method uses both the huber s m estimators and the yager s owa operators to obtain its robustness the proposed method is compared to many other ones e g the fuzzy c means fcm the possibilistic clustering pc the fuzzy noise clustering method ncm the l p norm clustering l p fcm 0 p 1 the l 1 norm clustering l 1 fcm the fuzzy clustering with polynomial fuzzifier pfcm and the e insensitive fuzzy c means fcm to this end experiments on synthetic data with outliers have been performed as well as on data with heavy tailed and overlapping groups of points in background noise

higher order quasiconvexity reduces to quasiconvexity in this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions precisely it is proved that a smooth strictly 2 quasiconvex function with p growth at infinity p 1 is the restriction to symmetric matrices of a 1 quasiconvex function with the same growth as a consequence lower semicontinuity results for second order variational problems are deduced as corollaries of well known first order theorems

comparison and classification of stationary multivariate time series abstract this paper presents procedures to compare and classify stationary multivariate time series the classification procedure is based on the p value of a test of hypothesis that is performed for every pair of series under consideration the test of hypothesis is based on the difference between vector autoregressive parameter estimates of the series simulation studies show that the test of hypothesis and the classification procedure perform fairly well for series of reasonable length

a note on the gustafson kessel and adaptive fuzzy clustering algorithms in this letter we show that the gustafson kessel g k algorithm 1979 and the original adaptive fuzzy clustering afc algorithm can be thought of as special cases of a more general algorithm our analysis shows that the g k algorithm is better suited for ellipsoidal clusters of equal volume whereas the original afc algorithm is better suited for linear clusters we also discuss a new variation of these algorithms which can be used to improve the results of the g k and afc algorithms in some cases

clustering for sparsely sampled functional data we develop a flexible model based procedure for clustering functional data the technique can be applied to all types of curve data but is particularly useful when individuals are observed at a sparse set of time points in addition to producing final cluster assignments the procedure generates predictions and confidence intervals for missing portions of curves our approach also provides many useful tools for evaluating the resulting models clustering can be assessed visually via low dimensional representations of the curves and the regions of greatest separation between clusters can be determined using a discriminant function finally we extend the model to handle multiple functional and finite dimensional covariates and show how it can be applied to standard finite dimensional clustering problems involving missing data

addendum to sharp inequalities that generalize the divergence theorem an extension of the notion of quasi convexity the paper sharp inequalities that generalize the divergence theorem an extension of the notion of quasi convexity published in proc r soc a 2013 469 20130075 doi 10 1098 rspa 2013 0075 is c

comparing fuzzy probabilistic and possibilistic partitions when clustering produces more than one candidate to partition a finite set of objects o there are two approaches to validation i e selection of a best partition and implicitly a best value for c which is the number of clusters in o first we may use an internal index which evaluates each partition separately second we may compare pairs of candidates with each other or with a reference partition that purports to represent the true cluster structure in the objects this paper generalizes many of the classical indices that have been used with outputs of crisp clustering algorithms so that they are applicable for candidate partitions of any type i e crisp or soft with soft comprising the fuzzy probabilistic and possibilistic cases space prevents inclusion of all of the possible generalizations that can be realized this way here we concentrate on the rand index and its modifications we compare our fuzzy rand index with those of campello hullermeier and rifqi and brouwer and show that our extension of the rand index is o n while the other three are all o n2 numerical examples are given to illustrate various facets of the new indices in particular we show that our indices can be used even when the partitions are probabilistic or possibilistic and that our method of generalization is valid for any index that depends only on the entries of the classical i e four pair types contingency table for this problem

weighted dynamic time warping for time series classification dynamic time warping dtw which finds the minimum path by providing non linear alignments between two time series has been widely used as a distance measure for time series classification and clustering however dtw does not account for the relative importance regarding the phase difference between a reference point and a testing point this may lead to misclassification especially in applications where the shape similarity between two sequences is a major consideration for an accurate recognition therefore we propose a novel distance measure called a weighted dtw wdtw which is a penalty based dtw our approach penalizes points with higher phase difference between a reference point and a testing point in order to prevent minimum distance distortion caused by outliers the rationale underlying the proposed distance measure is demonstrated with some illustrative examples a new weight function called the modified logistic weight function mlwf is also proposed to systematically assign weights as a function of the phase difference between a reference point and a testing point by applying different weights to adjacent points the proposed algorithm can enhance the detection of similarity between two time series we show that some popular distance measures such as dtw and euclidean distance are special cases of our proposed wdtw measure we extend the proposed idea to other variants of dtw such as derivative dynamic time warping ddtw and propose the weighted version of ddtw we have compared the performances of our proposed procedures with other popular approaches using public data sets available through the ucr time series data mining archive for both time series classification and clustering problems the experimental results indicate that the proposed approaches can achieve improved accuracy for time series classification and clustering problems

null lagrangians weak continuity and variational problems of arbitrary order abstract we consider the problem of minimizing integral functionals of the form i u f x k u x dx where r p u r and k u denotes the set of all partial derivatives of u with orders k the method is based on a characterization of null lagrangians l ku depending only on derivatives of order k applications to elasticity and other theories of mechanics are given

generalized autoregressive conditional heteroskedasticity a natural generalization of the arch autoregressive conditional heteroskedastic process introduced in engle 1982 to allow for past conditional variances in the current conditional variance equation is proposed stationarity conditions and autocorrelation structure for this new class of parametric models are derived maximum likelihood estimation and testing are also considered finally an empirical example relating to the uncertainty of the inflation rate is presented

on the generation of dispersive shock waves abstract we review various methods for the analysis of initial value problems for integrable dispersive equations in the weak dispersion or semiclassical regime some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains so called dispersive shock waves as the result of shock formation in a limiting dispersionless system they also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves revealing fascinating universal wave patterns

a coherence based approach for the pattern recognition of time series abstract a pattern recognition approach based on the frequency domain measure of squared coherence is a useful approach to identify linearly related groupings of time series over different periods of time it is considered in an application to identify similar patterns of the yearly rates of change in the gross domestic product gdp of twenty two highly developed countries in an econophysics context the approach is also tested in simulation studies using linearly related time series and it is shown to have a very good success rate of correct pattern matching

a meshing strategy for a quadratic iso parametric fem in cavitation computation in nonlinear elasticity abstract the approximation properties of a quadratic iso parametric finite element method for a typical cavitation problem in nonlinear elasticity in two dimensions are analyzed more precisely 1 the finite element interpolation errors are established in terms of the mesh parameters 2 a mesh distribution strategy based on an error equi distribution principle is given 3 the convergence of finite element cavity solutions is proved numerical experiments show that in fact the optimal convergence rate can be achieved by the numerical cavity solutions

fuzzy clustering of time series in the frequency domain traditional and fuzzy cluster analyses are applicable to variables whose values are uncorrelated hence in order to cluster time series data which are usually serially correlated one needs to extract features from the time series the values of which are uncorrelated the periodogram which is an estimator of the spectral density function of a time series is a feature that can be used in the cluster analysis of time series because its ordinates are uncorrelated additionally the normalized periodogram and the logarithm of the normalized periodogram are also features that can be used in this paper we consider a fuzzy clustering approach for time series based on the estimated cepstrum the cepstrum is the spectrum of the logarithm of the spectral density function we show in our simulation studies for the typical generating processes that have been considered fuzzy clustering based on the cepstral coefficients performs very well compared to when it is based on other features

on the zero dispersion limit of the benjamin ono cauchy problem for positive initial data we study the cauchy initial value problem for the benjamin ono equation in the zero dispersion limit and we establish the existence of this limit in a certain weak sense by developing an appropriate analogue of the method invented by lax and levermore to analyze the corresponding limit for the korteweg de vries equation 2010 wiley periodicals inc

non lipschitz minimizers of smooth uniformly convex functionals we construct non lipschitz minimizers of smooth uniformly convex functionals of type i u f du x dx our method is based on the use of null lagrangians

the equivariant inverse problem and the uniqueness of the yang mills equations the equivariant inverse problem for yang mills type euler lagrange expressions is solved in the affirmative this leads to a proof of the uniqueness of the yang mills equations