time independent invariants of motion for the quadratic system a hamiltonian method is developed to obtain first integrals of the form p qmu and p qmu rnu for a general system of two dimensional autonomous ordinary differential equations with quadratic terms where p q and r are linear or quadratic polynomials and mu nu are two real numbers it is found that there is an intimate relationship between the polynomials and the equilibrium points of the system which is useful for determining the existence of periodic orbits and asymptotic behaviour

an information maximization approach to blind separation and blind deconvolution we derive a new self organizing learning algorithm that maximizes the information transferred in a network of nonlinear units the algorithm does not assume any knowledge of the input distributions and is defined here for the zero noise limit under these conditions information maximization has extra properties not found in the linear case linsker 1989 the nonlinearities in the transfer function are able to pick up higher order moments of the input distributions and perform something akin to true redundancy reduction between units in the output representation this enables the network to separate statistically independent components in the inputs a higher order generalization of principal components analysis we apply the network to the source separation or cocktail party problem successfully separating unknown mixtures of up to 10 speakers we also show that a variant on the network architecture is able to perform blind deconvolution cancellation of unknown echoes and reverberation in a speech signal finally we derive dependencies of information transfer on time delays we suggest that information maximization provides a unifying framework for problems in blind signal processing

quasipolynomial generalization of lotka volterra mappings in recent years it has been shown that lotka volterra mappings constitute a valuable tool from both the theoretical and the applied points of view with developments in very diverse fields such as physics population dynamics chemistry and economy the purpose of this work is to demonstrate that many of the most important ideas and algebraic methods that constitute the basis of the quasipolynomial formalism originally conceived for the analysis of ordinary differential equations can be extended into the mapping domain the extension of the formalism into the discrete time context is remarkable as far as the quasipolynomial methodology had never been shown to be applicable beyond the differential case it will be demonstrated that lotka volterra mappings play a central role in the quasipolynomial formalism for the discrete time case moreover the extension of the formalism into the discrete time domain allows a significant generalization of lotka volterra mappings as well as a whole transfer of algebraic methods into the discrete time context the result is a novel and more general conceptual framework for the understanding of lotka volterra mappings as well as a new range of possibilities that become open not only for the theoretical analysis of lotka volterra mappings and their generalizations but also for the development of new applications

lie theory of multiplicative tensors we study tensors on lie groupoids suitably compatible with the groupoid structure called multiplicative our main result gives a complete description of these objects only in terms of infinitesimal data special cases include the infinitesimal counterparts of multiplicative forms multivector fields and holomorphic structures obtained through a unifying and conceptual method we also give a full treatment of multiplicative vector valued forms particularly nijenhuis operators and related structures

integration of holomorphic lie algebroids we prove that a holomorphic lie algebroid is integrable if and only if its underlying real lie algebroid is integrable thus the integrability criteria of crainic fernandes theorem 4 1 in crainic fernandes in ann math 2 157 2003 do also apply in the holomorphic context without any modification as a consequence we prove that a holomorphic poisson manifold is integrable if and only if its real part or imaginary part is integrable as a real poisson manifold

reduction of courant algebroids and generalized complex structures we present a theory of reduction for courant algebroids as well as dirac structures generalized complex and generalized kahler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction the enhanced symmetry group of a courant algebroid leads us to define extended actions and a generalized notion of moment map key examples of generalized kahler reduced spaces include new explicit bi hermitian metrics on cp2

generalized complex geometry generalized complex geometry encompasses complex and symplectic ge ometry as its extremal special cases we explore the basic properties of this geometry including its enhanced symmetry group elliptic deforma tion theory relation to poisson geometry and local structure theory we also dene and study generalized complex branes which interpolate be tween at bundles on lagrangian submanifolds and holomorphic bundles on complex submanifolds

a geometrical method towards first integrals for dynamical systems we develop a method based on darboux s and liouville s works to find first integrals and or invariant manifolds for a physically relevant class of dynamical systems without making any assumption on these elements forms we apply it to three dynamical systems lotka volterra lorenz and rikitake

families of invariants of the motion for the lotka volterra equations the linear polynomials family the modified carleman embedding method already introduced by the authors to find first integrals invariants of the motion of polynomial form to the lotka volterra system is described in detail and its efficiency to treat the n dimensional system proved using this method an extensive investigation is performed for polynomials of the first degree which allow a classification of the integrals in three families for some systems possessing one invariant it is possible to find a second invariant using rescaling methods they represent very restrictive solutions implying that there exists a great number of conditions among the equation s coefficients to satisfy a proof is given that the volterra invariants can be deduced as a limit finally the interesting properties of the solutions of these systems are studied in detail

multiplicative forms at the infinitesimal level we describe arbitrary multiplicative differential forms on lie groupoids infinitesimally i e in terms of lie algebroid data this description is based on the study of linear differential forms on lie algebroids and encompasses many known integration results related to poisson geometry we also revisit multiplicative multivector fields and their infinitesimal counterparts drawing a parallel between the two theories

adaptive blind source separation for virtually any source probability density function blind source separation bss aims to recover a set of statistically independent source signals from a set of linear mixtures of the same sources in the noiseless real mixture two source two sensor scenario once the observations are whitened decorrelated and normalized only a givens rotation matrix remains to be identified in order to achieve the source separation in this paper an adaptive estimator of the angle that characterizes such a rotation is derived it is shown to converge to a stable valid separation solution with the only condition that the sum of source kurtosis be distinct from zero an asymptotic performance analysis is carried out resulting in a closed form expression for the asymptotic probability density function of the proposed estimator it is shown how the estimator can be incorporated into a complete adaptive source separation system by combining it with an adaptive prewhitening strategy and how it can be useful in a general bss scenario of more than two signals by means of a pairwise approach a variety of simulations assess the accuracy of the asymptotic results display the properties of the estimator such as its robust fast convergence and compare this on line bss implementation with other adaptive bss procedures

a fast fixed point algorithm for independent component analysis we introduce a novel fast algorithm for independent component analysis which can be used for blind source separation and feature extraction we show how a neural network learning rule can be transformed into a fixedpoint iteration which provides an algorithm that is very simple does not depend on any user defined parameters and is fast to converge to the most accurate solution allowed by the data the algorithm finds one at a time all nongaussian independent components regardless of their probability distributions the computations can be performed in either batch mode or a semiadaptive manner the convergence of the algorithm is rigorously proved and the convergence speed is shown to be cubic some comparisons to gradient based algorithms are made showing that the new algorithm is usually 10 to 100 times faster sometimes giving the solution in just a few iterations

lie algebroid structures on double vector bundles and representation theory of lie algebroids abstract a vb algebroid is essentially defined as a lie algebroid object in the category of vector bundles there is a one to one correspondence between vb algebroids and certain flat lie algebroid superconnections up to a natural notion of equivalence in this setting we are able to construct characteristic classes which in special cases reproduce characteristic classes constructed by crainic and fernandes we give a complete classification of regular vb algebroids and in the process we obtain another characteristic class of lie algebroids that does not appear in the ordinary representation theory of lie algebroids

adaptive online learning algorithms for blind separation maximum entropy and minimum mutual information there are two major approaches for blind separation maximum entropy me and minimum mutual information mmi both can be implemented by the stochastic gradient descent method for obtaining the demixing matrix the mi is the contrast function for blind separation the entropy is not to justify the me the relation between me and mmi is first elucidated by calculating the first derivative of the entropy and proving that the mean subtraction is necessary in applying the me and at the solution points determined by the mi the me will not update the demixing matrix in the directions of increasing the cross talking second the natural gradient instead of the ordinary gradient is introduced to obtain efficient algorithms because the parameter space is a riemannian space consisting of matrices the mutual information is calculated by applying the gram charlier expansion to approximate probability density functions of the outputs finally we propose an efficient learning algorithm that incorporates with an adaptive method of estimating the unknown cumulants it is shown by computer simulation that the convergence of the stochastic descent algorithms is improved by using the natural gradient and the adaptively estimated cumulants

algebraic constructions in the category of lie algebroids the aim of this paper is to give the first general and abstract treatment of the algebraic properties of lie algebroids the concept of lie algebroid was introduced in 1967 by pradines ls as the basic infinitesimal invariant of a differentiable groupoid the construction of the lie algebroid of a differentiable groupoid is a direct generalization of the construction of the lie algebra of a lie group and 183 described a full lie theory for differentiable groupoids and lie algebroids encompassing many phenomena in the foundations of differential geometry a more detailed account and further references are given in 13 however at this stage the algebraic properties of lie algebroids were not pursued in 13 iii section 2 iv section 11 one of us gave a fairly detailed account of the abstract algebra of lie algebroids over a fixed base lie algebroids are vector bundles with several additional structures and a category of lie algebroids on a given base and morphisms which preserve that base has properties similar to those of the category of lie algebras using this algebra it was demonstrated in 13 that the lie theory of a category of locally trivial differentiable lie groupoids over a fixed base and the corresponding category of transitive lie algebroids over that base is coextensive with the standard theory of connections in principal bundles we now propose to show that the lie functor from the category of all differentiable groupoids over arbitrary bases and arbitrary smooth morphisms to the category of all lie algebroids preserves the basic algebraic constructions known to be possible in the category of differentiable groupoids the main work is to give abstract formulations of the corresponding lie algebroid concepts the groupoid results with which we

representations up to homotopy of lie algebroids we introduce and study the notion of representation up to homotopy of a lie algebroid paying special attention to examples we use representations up to homotopy to define the adjoint representation of a lie algebroid and show that the resulting cohomology controls the deformations of the structure the weil algebra of a lie algebroid is defined and shown to coincide with kalkman s brst model for equivariant cohomology in the case of group actions the relation of this algebra with the integration of poisson and dirac structures is explained in arias abad crainic ann inst fourier

necessary conditions for the existence of quasi polynomial invariants the quasi polynomial and lotka volterra systems abstract we show that any quasi polynomial invariant of a quasi polynomial dynamical system can be transformed into a quasi polynomial invariant of a homogeneous quadratic lotka volterra dynamical system we show how this quasi polynomial invariant can be decomposed in a simple manner this decomposition permits to conclude that the existence of polynomial semi invariants in lotka volterra systems is a necessary condition for the existence of quasi polynomial invariants we derive a method which allows to construct the necessary conditions for existence of semi invariants on lotka volterra dynamical systems applications are given

local classification of generalize complex structures we give a local classification of generalized complex structures about a point a generalized complex structure is equivalent to a product of a symplectic manifold with a holomorphic poisson manifold we use a nash moser type argument in the style of conn s linearization theorem

symmetries of differential equations from sophus lie to computer algebra the topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it the whole area naturally decomposes into two parts depending on whether ordinary or partial differential equations are considered we show how a symmetry may be applied to lower the order of an ordinary differential equation or to obtain similarity solutions of partial differential equations the computer algebra packages sods and spde respectively which have been developed to perform almost all algebraic manipulations necessary to determine the symmetry group of a given differential equation are presented furthermore it is argued that the application of computer algebra systems has qualitatively changed this area of applied mathematics

stability properties of a general class of nonlinear dynamical systems we establish sufficient conditions for the boundedness of the trajectories and the stability of the fixed points in a class of general nonlinear systems the so called quasi polynomial vector fields with the help of a natural embedding of such systems in a family of generalized lotka volterra lv equations a purely algebraic procedure is developed to determine such conditions we apply our method to obtain new results for lv systems by a reparametrization in time variable and to study general nonlinear vector fields originally far from the lv format

holomorphic jacobi manifolds and holomorphic contact groupoids this paper belongs to a series of works aiming at exploring generalized complex geometry in odd dimensions holomorphic jacobi manifolds were introduced and studied by the authors in a separate paper as special cases of generalized contact bundles in fact generalized contact bundles are nothing but odd dimensional analogues of generalized complex manifolds in the present paper we solve the integration problem for holomorphic jacobi manifolds by proving that they integrate to holomorphic contact groupoids a crucial tool in our proof is what we call the homogenization scheme which allows us to identify holomorphic jacobi manifolds with homogeneous holomorphic poisson manifolds and holomorphic contact groupoids with homogeneous complex symplectic groupoids

on the integrability of lie subalgebroids abstract let g be a lie groupoid with lie algebroid g it is known that unlike in the case of lie groups not every subalgebroid of g can be integrated by a subgroupoid of g in this paper we study conditions on the invariant foliation defined by a given subalgebroid under which such an integration is possible we also consider the problem of integrability by closed subgroupoids and we give conditions under which the closure of a subgroupoid is again a subgroupoid

contact reduction and groupoid actions we introduce a new method to perform reduction of contact manifolds that extends willett s and albert s results to carry out our reduction procedure all we need is a complete jacobi map j m 0 from a contact manifold to a jacobi manifold this naturally generates the action of the contact groupoid of 0 on m and we show that the quotients of fibers j 1 x by suitable lie subgroups x are either contact or locally conformal symplectic manifolds with structures induced by the one on m we show that willett s reduced spaces are prequantizations of our reduced spaces hence the former are completely determined by the latter since a symplectic manifold is prequantizable iff the symplectic form is integral this explains why willett s reduction can be performed only at distinguished points as an application we obtain kostant s prequantizations of coadjoint orbits finally we present several examples where we obtain classical contact manifolds as reduced spaces

integrals of quadratic ordinary differential equations in r3 the lotka volterra system abstract a method already introduced by the last two authors for finding the integrable cases of three dimensional autonomous ordinary differential equations based on the frobenius integrability theorem is described in detail using this method and computer algebra the so called three dimensional lotka volterra system is studied many cases of integrability are thus found the study of this system is completed by the application of painleve analysis and the jacobi last multiplier method the methods used are of general interest and can be applied to many other systems

blind separation using convex functions in this paper the problem of how to conveniently estimate independence from observations is addressed random variables rvs are transformed by their respective distribution functions and quantized then the uniformity of the joint probability of the obtained discrete rvs is evaluated using a strictly convex function an infinite class of new independence measures named quasientropy qe is thus proposed unbiased estimates of the values of the distribution functions at the observations are directly utilized in estimating qe the linear instantaneous blind source separation bss algorithm based on qe can separate signals with arbitrary continuous distributions

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

lie symmetries of the lorenz model abstract we study the generalized symmetries of the lorenz model to find the parameter values at which one or more time dependent integrals of motion exist in these cases the integrals are found trivially from the symmetries themselves a complete study of the one completely algebraically integrable case shows a the dynamics can be integrated exactly by reducing it first to a lower dimensional system b the symmetry vector field is hamiltonian these facts hold for other dissipative completely integrable dynamical systems as well the analytic study of a natural two form reveals that it is an entire function of time the foliation of phase space induced by the two form for the partially integrable cases has a simple description in terms of the coefficients occurring in the laurent series expansions of the dependent variables

vector bundles over lie groupoids and algebroids abstract we study vb groupoids and vb algebroids which are vector bundles in the realm of lie groupoids and lie algebroids through a suitable reformulation of their definitions we elucidate the lie theory relating these objects i e their relation via differentiation and integration we also show how to extend our techniques to describe the more general lie theory underlying double lie algebroids and la groupoids

poisson quasi nijenhuis manifolds we introduce the notion of poisson quasi nijenhuis manifolds generalizing poisson nijenhuis manifolds of magri morosi we also investigate the integration problem of poisson quasi nijenhuis manifolds in particular we prove that under some topological assumption poisson quasi nijenhuis manifolds are in one one correspondence with symplectic quasi nijenhuis groupoids as an application we study generalized complex structures in terms of poisson quasi nijenhuis manifolds we prove that a generalized complex manifold corresponds to a special class of poisson quasi nijenhuis structures as a consequence we show that a generalized complex structure integrates to a symplectic quasi nijenhuis groupoid recovering a theorem of crainic

multiplicative forms and spencer operators motivated by our attempt to recast cartan s work on lie pseudogroups in a more global and modern language we are brought back to the question of understanding the linearization of multiplicative forms on groupoids and the corresponding integrability problem from this point of view the novelty of this paper is that we study forms with coefficients however the main contribution of this paper is conceptual the discovery of the relationship between multiplicative forms and cartan s work is explored here to provide a completely new approach to integrability theorems for multiplicative forms the multiplicative point of view shows that modulo lie s functor the cartan pfaffian system itself a multiplicative form with coefficients is the same thing as the classical spencer operator

algebraic structures and invariant manifolds of differential systems algebraic tools are applied to find integrability properties of odes bilinear nonassociative algebras are associated to a large class of polynomial and nonpolynomial systems of differential equations since all equations in this class are related to a canonical quadratic differential system the lotka volterra system these algebras are classified up to dimension 3 and examples for dimension 4 and 5 are given their subalgebras are associated to nonlinear invariant manifolds in the phase space these manifolds are calculated explicitly more general algebraic invariant surfaces are also obtained by combining a theorem of walcher and the lotka volterra canonical form applications are given for lorenz model lotka may leonard and rikitake systems

nonexistence and existence of various order integrals for two and three dimensional polynomial potentials the nonexistence of integrals of the motion which are sixth and fourth degree polynomials in the velocities was established for a range of polynomial potentials the first known systematic search for sixth order integrals was performed for cubic and quartic polynomial potentials it revealed that there exist no nondegenerate cases with such integrals for either potential similarly there exists no nondegenerate fifth or sixth degree polynomial potentials possessing quartic invariants a complete list of three dimensional cubic potentials with quadratic integrals is given all these integrable three dimensional potentials can be interpreted as orthogonal superpositions of known integrable two dimensional potentials possessing quadratic integrals they correspond to 3 of 11 possible coordinate systems in which three dimensional potentials separate