a generalization of joint diagonalization criteria for source separation in the field of blind source separation joint diagonalization based approaches constitute an important framework leading to useful algorithms such as the popular joint approximate diagonalization of eigenmatrices jade and simultaneous third order tensor diagonalization stotd algorithms however they are often restricted to the case of cumulants of order four in this paper we extend the results leading to jade and stotd to cumulants of any order greater than or equal to three by exhibiting a new family of contrast functions that constitutes then a unified framework for the above known results this also leads us to generalize some links between contrast functions and joint diagonalization criteria on which these algorithms are based in turn one contrast of the new family allows us to show that a function previously proposed as a separation criterion is also a contrast moreover for the two generalized jade and stotd contrasts the analytical optimal solution in the case of two sources is derived and shown to keep the same simple expression whatever the cumulant order finally some computer simulations illustrate the potential advantage one can take by considering statistics of different orders for the joint diagonalization of cumulant matrices

quasi optimal convergence rate of an adaptive discontinuous galerkin method we analyze an adaptive discontinuous finite element method adfem for symmetric second order linear elliptic operators the method is formulated on nonconforming meshes made of simplices or quadrilaterals with any polynomial degree and in any dimension geq2 we prove that the adfem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops we design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent the geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the adfem we show that the adfem and the afem on nonconforming meshes yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

on the stability of some groups of formal diffeomorphisms by the birkhoff decomposition abstract let g be the group of one parameter identity tangent diffeomorphisms on the line whose coefficients are formal laurent series in the parameter e with a pole of finite order at 0 it is well known that the birkhoff decomposition can be defined in such a group we investigate the stability of the birkhoff decomposition in subgroups of g and give a formula for this decomposition these results are strongly related to renormalization in quantum field theory since it was proved by a connes and d kreimer that after dimensional regularization the unrenormalized effective coupling constants are the image by a formal identity tangent diffeomorphism of the coupling constants of the theory in the massless 6 3 theory this diffeomorphism is in g and its birkhoff decomposition gives directly the bare coupling constants and the renormalized coupling constants

an exponential formula for polynomial vector fields ii lie series exponential substituition and rooted trees abstract grobner s lie series and the exponential formula provide different explicit formulas for the flow generated by a finite dimensional polynomial vector field the present paper gives 1 a generalization of the lie series in case of non commuting variables called exponential substituition 2 a structural understanding of the three formulas and their mutual relationships in terms of rooted trees and 3 as a byproduct new results on the enumeration coding and statistics of different kinds of rooted trees

guaranteed and robust discontinuous galerkin a posteriori error estimates for convection diffusion reaction problems we propose and study a posteriori error estimates for convection diffusion reaction problems with inhomogeneous and anisotropic diffusion approximated by weighted interior penalty discontinuous galerkin methods our twofold objective is to derive estimates without undetermined constants and to analyze carefully the robustness of the estimates in singularly perturbed regimes due to dominant convection or reaction we first derive locally computable estimates for the error measured in the energy semi norm these estimates are evaluated using h div w conforming diffusive and convective flux reconstructions thereby extending the previous work on pure diffusion problems the resulting estimates are semi robust in the sense that local lower error bounds can be derived using suitable cutoff functions of the local peclet and damkohler numbers fully robust estimates are obtained for the error measured in an augmented norm consisting of the energy semi norm a dual norm of the skew symmetric part of the differential operator and a suitable contribution of the interelement jumps of the discrete solution numerical experiments are presented to illustrate the theoretical results

moduli of stability for germs of homogeneous vectorfields on r3 abstract in this paper we will show that there are an uncountable number of homogeneous quadratic vectorfields on r 3 so that no two of these vectorfields are topologically equivalent

combinatorics of rooted trees and hopf algebras we begin by considering the graded vector space with a basis consisting of rooted trees with grading given by the count of non root vertices we define two linear operators on this vector space the growth and pruning operators which respectively raise and lower grading their commutator is the operator that multiplies a rooted tree by its number of vertices and each operator naturally associates a multiplicity to each pair of rooted trees by using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint and obtain several results about the associated multiplicities now the symmetric algebra on the vector space of rooted trees after a degree shift can be endowed with a coproduct to make a hopf algebra this was defined by kreimer in connection with renormalization we extend the growth and pruning operators as well as the inner product mentioned above to kreimer s hopf algebra on the other hand the vector space of rooted trees itself can be given a noncommutative multiplication with an appropriate coproduct this leads to the hopf algebra of grossman and larson we show that the inner product on rooted trees leads to an isomorphism of the grossman larson hopf algebra with the graded dual of kreimer s hopf algebra correcting an earlier result of panaite

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

some perturbation results for a normalized non orthogonal joint diagonalization problem abstract non orthogonal joint diagonalization nojd of a given real symmetric matrix set a a j j 0 p is to find a nonsingular matrix w such that w a j w for j 0 1 p are all as diagonal as possible if the columns of the solution w are all required to be unit length we call such nojd problem as the normalized nojd nnojd problem in this paper we discuss the perturbation theory for nnojd as an optimization problem based on the perturbation analysis of general constrained optimization problem given in 16 we obtain an upper bound for the distance between an approximated solution of the perturbed optimal problem and the set of exact joint diagonalizers as corollaries a perturbation bound and an error bound are also given numerical examples validate the bounds

an equilibrated a posteriori error estimator for the interior penalty discontinuous galerkin method interior penalty discontinuous galerkin ipdg methods for second order elliptic boundary value problems have been derived from a mixed variational formulation of the problem numerical flux functions across interelement boundaries play an important role in that theory residual type a posteriori error estimators for ipdg methods have been derived and analyzed by many authors including the convergence analysis of the resulting adaptive schemes typically the effectivity indices deteriorate with increasing polynomial order of the ipdg methods the situation is more favorable for a posteriori error estimators derived by means of the so called hypercircle method equilibrated fluxes are obtained by using a mixed method and an extension operator for brezzi douglas marini elements and this can be done in the same way for all the discontinuous galerkin methods that fit into a known unified framework the hypercircle method immediately provides the reliability of the estimator whereas its efficiency can be

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

fully computable error bounds for discontinuous galerkin finite element approximations on meshes with an arbitrary number of levels of hanging nodes we obtain fully computable a posteriori error bounds on the broken energy seminorm and discontinuous galerkin norm dg norm of the error in first order symmetric interior penalty galerkin sipg nonsymmetric interior penalty galerkin nipg and incomplete interior penalty galerkin iipg finite element approximations of a linear second order elliptic problem on meshes containing an arbitrary number of levels of hanging nodes and comprised of triangular elements the estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and dg norm of the error these estimators are also shown to provide a lower bound for the broken energy seminorm and dg norm of the error up to a constant and higher order data oscillation terms we also obtain an explicit computable bound for the value of the interior penalty parameter needed to ensure the existence of the discontinuous galerkin finite element approximation for all versions of the method

convergence of adaptive discontinuous galerkin approximations of second order elliptic problems a residual type a posteriori error estimator is introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem with dirichlet neumann type boundary conditions an adaptive algorithm using this estimator together with specific marking and refinement strategies is constructed and shown to achieve any specified error level in the energy norm in a finite number of cycles the convergence rate is in effect linear with a guaranteed error reduction at every cycle results of numerical experiments are presented

equilibrated residual error estimates are p robust abstract equilibrated residual error estimators applied to high order finite elements are analyzed the estimators provide always a true upper bound for the energy error we prove that also the efficiency estimate is robust with respect to the polynomial degrees a complete theory is provided for tensor product elements in the case of simplicial elements the theorem was first based on a conjecture but the conjecture was quite recently proved by costabel and mcintosh m costabel a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains irmar preprint 08 40 rennes august 2008 numerical evidence for the p robustness is provided

quasi optimal convergence rate for an adaptive finite element method we analyze the simplest and most standard adaptive finite element method afem with any polynomial degree for general second order linear symmetric elliptic operators as is customary in practice the afem marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property we prove that the afem is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops this geometric decay is instrumental to derive the optimal cardinality of the afem we show that the afem yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity

convergence analysis of an adaptive interior penalty discontinuous galerkin method we study the convergence of an adaptive interior penalty discontinuous galerkin ipdg method for a two dimensional model second order elliptic boundary value problem based on a residual type a posteriori error estimator we prove that after each refinement step of the adaptive scheme we achieve a guaranteed reduction of the global discretization error in the mesh dependent energy norm associated with the ipdg method in contrast to recent work on adaptive ipdg methods o karakashian and f pascal convergence of adaptive discontinuous galerkin approximations of second order elliptic problems preprint university of tennessee knoxville tn 2007 the convergence analysis does not require multiple interior nodes for refined elements of the triangulation in fact it will be shown that bisection of the elements is sufficient the main ingredients of the proof of the error reduction property are the reliability and a perturbed discrete local efficiency of the estimator a bulk criterion that takes care of a proper selection of edges and elements for refinement and a perturbed galerkin orthogonality property with respect to the energy inner product the results of numerical experiments are given to illustrate the performance of the adaptive method

a posteriori error analysis for locally conservative mixed methods in this work we present a theoretical analysis for a residual type error estimator for locally conservative mixed methods this estimator was first introduced by braess and verfurth for the raviart thomas mixed finite element method working in mesh dependent norms we improve and extend their results to cover any locally conservative mixed method under minimal assumptions in particular avoiding the saturation assumption made by braess and verfurth our analysis also takes into account discontinuous coefficients with possibly large jumps across interelement boundaries the main results are applied to the p1 nonconforming finite element method and the interior penalty discontinuous galerkin method as well as the mixed finite element method

a posteriori error estimates for a discontinuous galerkin approximation of second order elliptic problems several a posteriori error estimators are introduced and analyzed for a discontinuous galerkin formulation of a model second order elliptic problem in addition to residual type estimators we introduce some estimators that are couched in the ideas and techniques of domain decomposition results of numerical experiments are presented

a posteriori error estimation for lowest order raviart thomas mixed finite elements for the lowest order raviart thomas mixed finite element we derive an a posteriori error estimator that provides actual guaranteed computable upper bounds on the error in the flux variable regardless of jumps in the material coefficients across interfaces moreover the estimator is efficient in that it provides a local lower bound on the error up to a constant that is independent of the solution and the local mesh size the estimator may be evaluated at virtually no additional cost compared to the evaluation of the finite element approximation itself

a blind source separation technique using second order statistics separation of sources consists of recovering a set of signals of which only instantaneous linear mixtures are observed in many situations no a priori information on the mixing matrix is available the linear mixture should be blindly processed this typically occurs in narrowband array processing applications when the array manifold is unknown or distorted this paper introduces a new source separation technique exploiting the time coherence of the source signals in contrast with other previously reported techniques the proposed approach relies only on stationary second order statistics that are based on a joint diagonalization of a set of covariance matrices asymptotic performance analysis of this method is carried out some numerical simulations are provided to illustrate the effectiveness of the proposed method

hopf algebraic structure of families of trees in this paper we describe hopf algebras which are associated with certain families of trees these hopf algebras originally arose in a natural fashion one of the authors 5 was investigating data structures based on trees which could be used to efficiently compute certain differential operators given data structures such as trees which can be multiplied and which act as higherorder derivations on an algebra one expects to find a hopf algebra of some sort we were pleased to find that not only was there a hopf algebra associated with these data structures but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees previous work applying hopf algebras to combinatorial objects such as 10 13 or 14 has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets rather than on trees themselves

quadratic optimization for simultaneous matrix diagonalization simultaneous diagonalization of a set of matrices is a technique that has numerous applications in statistical signal processing and multivariate statistics although objective functions in a least squares sense can be easily formulated their minimization is not trivial because constraints and fourth order terms are usually involved most known optimization algorithms are therefore subject to certain restrictions on the class of problems orthogonal transformations sets of symmetric hermitian or positive definite matrices to name a few in this paper we present a new algorithm called qdiag that splits the overall optimization problem into a sequence of simpler second order subproblems there are no restrictions imposed on the transformation matrix which may be nonorthogonal indefinite or even rectangular and there are no restrictions regarding the symmetry and definiteness of the matrices to be diagonalized except for one of them we apply the new method to second order blind source separation and show that the algorithm converges fast and reliably it allows for an implementation with a complexity independent of the number of matrices and therefore is particularly suitable for problems dealing with large sets of matrices

a bijection between rooted trees and fermionic fock states grafting and growth operators in fock space and fermionic operators for rooted trees we show that fermionic fock states in the occupation number representation can be indexed uniquely by rooted trees our main ingredients in this construction are the matula numbers the fundamental theorem of arithmetic and a relabeling of fermionic quantum states by natural numbers as a byproduct of the correspondence mentioned above we realize the grafting and the growth operators comprising important constructions in the context of hopf algebras in the fermionic fock space also we show how to construct fermionic creation and annihilation operators in the context of rooted trees new representations of the solutions of combinatorial dyson schwinger equations and of the antipode in the connes kreimer hopf algebra of rooted trees related to the occupation number picture are presented

perturbation analysis for matrix joint block diagonalization abstract the matrix joint block diagonalization problem jbdp of a given matrix set a a i i 1 m is about finding a nonsingular matrix w such that all w t a i w are block diagonal it includes the matrix joint diagonalization problem jdp as a special case for which all w t a i w are required diagonal generically such a matrix w may not exist but there are practical applications such as multidimensional independent component analysis mica for which it does exist under the ideal situation i e no noise is presented however in practice noises do get in and as a consequence the matrix set is only approximately block diagonalizable i e one can only make all w t a i w nearly block diagonal at best where w is an approximation to w obtained usually by computation the main goal of this paper is to develop a perturbation theory for jbdp to address among others the question how accurate this w is previously such a theory for jdp has been discussed but no effort had been attempted for jbdp because in large part there is no quantitative way to describe solution uniqueness of jbdp until 2017 when cai and liu 2017 9 successfully obtained a necessary and sufficient uniqueness condition based on the condition in this article we will establish a perturbation theory for jbdp our main contributions include an error bound for the approximate block diagonalizer w and a backward error analysis for jbdp numerical tests are presented to validate the theoretical results

general dyson schwinger equations and systems we classify combinatorial dyson schwinger equations giving a hopf subalgebra of the hopf algebra of feynman graphs of the considered quantum field theory we first treat single equations with an arbitrary eventually infinite number of insertion operators we distinguish two cases in the first one the hopf subalgebra generated by the solution is isomorphic to the faa di bruno hopf algebra or to the hopf algebra of symmetric functions in the second case we obtain the dual of the enveloping algebra of a particular associative algebra seen as a lie algebra we also treat systems with an arbitrary finite number of equations with an arbitrary number of insertion operators with at least one of degree 1 in each equation

an exponential formula for polynomial vector fields abstract the well known matrix exponentiation for linear first order systems of ode is generalised to polynomial vector fields the substitution process of our exponential formula is closely related to the usual iteration of polynomial mappings

solving nonlinear equations from higher order derivations in linear stages for all polynomials a r of degree less than or equal to d and all higher order derivations e e s e 1 and f f s f 1 where 1 s and 1 i m of length s less or equal to r the reason for wanting such derivations e1 em is simple given several derivations f1 fm it is useful to have derivations e1 em which are good local approximations to the fi and which are easy to compute with notice that since the right hand sides of equations are known while the left hand sides involve the unspecified coefficients of the polynomial functions the equations are equivalent to a system of nonlinear algebraic equations involving the coefficients it is the purpose of this paper to give an algorithm which solves such a system when a solution exists in fact as the title suggests this algorithm will work by solving a sequence of r linear systems

non orthogonal joint diagonalization in the least squares sense with application in blind source separation approximate joint diagonalization of a set of matrices is an essential tool in many blind source separation bss algorithms a common measure of the attained diagonalization of the set is the weighted least squares wls criterion however most well known algorithms are restricted to finding an orthogonal diagonalizing matrix relying on a whitening phase for the nonorthogonal factor often such an approach implies unbalanced weighting which can result in degraded performance we propose an iterative alternating directions algorithm for minimizing the wls criterion with respect to a general not necessarily orthogonal diagonalizing matrix under some mild assumptions we prove weak convergence in the sense that the norm of parameters update is guaranteed to fall below any arbitrarily small threshold within a finite number of iterations we distinguish between hermitian and symmetrical problems using bss simulations results we demonstrate the improvement in estimating the mixing matrix resulting from the relaxation of the orthogonality restriction