weak amenability of banach algebras on locally compact groups abstract recent work by various authors has considered the implications of banach algebra amenability for various algebras defined over locally compact groups one of the basic tools being the fact that a continuous homomorphic image of an amenable algebra is again amenable in the present paper we look at the consequences of weak amenability here the homomorphism property fails in general however it remains true for suitable direct summands it is this technique that we make much use of here

darboux method and search of invariants for the lotka volterra and complex quadratic systems the darboux method introduces algebraic solutions quite useful to obtain invariant and first integrals of polynomial differential systems here we study the 2d lotka volterra lvs and the complex quadratic system qs using straight lines for both and conics for the lvs the conditions needed to obtain these invariants are given and a study of the phase space portrait is done

estimating multivariate latent structure models a constructive proof of identification of multilinear decompositions of multiway arrays is presented it can be applied to show identification in a variety of multivariate latent structures examples are finite mixture models and hidden markov models the key step to show identification is the joint diagonalization of a set of matrices in the same non orthogonal basis an estimator of the latent structure model may then be based on a sample version of this simultaneous diagonalization problem simple algorithms are available for computation asymptotic theory is derived for this joint approximate diagonalization estimator

hypergeometric first integrals of the duffing and van der pol oscillators abstract the autonomous duffing oscillator and its van der pol modification are known to admit time dependent first integrals for specific values of parameters this corresponds to the existence of darboux polynomials and in fact more can be shown that there exist liouvillian first integrals which do not depend on time they can be expressed in terms of the gauss and kummer hypergeometric functions and are neither analytic algebraic nor meromorphic a criterion for this to happen in a general dynamical system is formulated as well

on the dynamics of lotka volterra equations having an invariant hyperplane the dynamics of n dimensional lotka volterra equations having an invariant hyperplane is investigated herein there exists a fundamental difference between the even and the odd dimensional case for even dimensions the dynamics are generated by a hamiltonian system with respect to an appropriately chosen poisson structure for odd dimensions the dynamics are hamiltonian if there is a continuum of interior fixed points and gradient like if the interior fixed point is unique in this case the invariant hyperplane is part of the omega limit set and the dynamics restricted to this set are those of a hamiltonian system furthermore an example of a completely integrable four dimensional hamiltonian lotka volterra system is presented

homological properties of modules over semigroup algebras abstract let s be a semigroup in this paper we investigate the injectivity of l 1 s as a banach right module over l 1 s for weakly cancellative s this is the same as studying the flatness of the predual left module c 0 s for such semigroups s we also investigate the projectivity of c 0 s we prove that for many semigroups s for which the banach algebra l 1 s is non amenable the l 1 s module l 1 s is not injective the main result about the projectivity of c 0 s states that for a weakly cancellative inverse semigroup s c 0 s is projective if and only if s is finite

investigation of painleve property under time singularities transformations in this paper the relation between the painleve property for ordinary differential equations and some coordinate transformations in the complex plane is investigated two sets of transformations are introduced for which the transformations of singularities structure in the complex plane are explicitly computed the first set acts only on the dependent variables while the second one acts also on the independent variables it allows for defining an extended painleve test which is coordinates invariant our results are applied on various problems arising in the singularity analysis such as the problem of negative resonances and the weak painleve conjecture the method for finding new first integrals for dynamical systems is also shown

tensor decompositions for learning latent variable models this work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models including gaussian mixture models hidden markov models and latent dirichlet allocation which exploits a certain tensor structure in their low order observable moments typically of second and third order specifically parameter estimation is reduced to the problem of extracting a certain orthogonal decomposition of a symmetric tensor derived from the moments this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices although tensor decompositions are generally intractable to compute the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches including power iterations and maximization approaches similar to the case of matrices a detailed analysis of a robust tensor power method is provided establishing an analogue of wedin s perturbation theorem for the singular vectors of matrices this implies a robust and computationally tractable estimation approach for several popular latent variable models

on the connection between hyperelliptic separability and painleve integrability we consider systems of odes which are associated with some physically significant examples shallow water equilibrium solutions travelling waves of the harry dym equation a lotka volterra system of competing species and the geodesic flow on the triaxial ellipsoid the first three are shown to share the following properties i they are hyperelliptically separable systems hss and after a suitable nonlinear time transformation become algebraically completely integrable aci and ii they are of the weak painleve type and become full painleve after the application of this transformation the geodesic flow on the other hand although it passes the usual painleve test does not possess a full set of free constants and thus one may not conclude whether it has the painleve property or not this system is also hss and becomes aci after the application of a suitable nonlinear time transformation we also combine our geometric analytical investigation with a numerical analysis of the system in the complex plane and show that there is perfect correspondence between the results of the two approaches this correspondence strengthens the reliability of such numerical studies and helps us better understand their implication in cases where such nonlinear transformations to complete integrability are not available

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

on wielandt s inequality and its application to the asymptotic distribution of the eigenvalues of a random symmetric matrix a relatively obscure eigenvalue inequality due to wielandt is used to give a simple derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix the asymptotic distributions are obtained under a fairly general setting an application of the general theory to the bootstrap distribution of the eigenvalues of the sample covariance matrix is given 1 introduction and summary the derivation of the asymptotic distribution of the eigenvalues of a random symmetric matrix arises in many papers in multivariate analysis although the main idea behind most of the derivations is quite basic i e the expansion of the sample roots about the population roots the derivations themselves are often quite involved these complications are primarily due to the mathematical rather than statistical nature of the eigenvalue problem one of the main objectives of this paper is to introduce a simple method for obtaining the asymptotic distribution of the eigenvalue of random symmetric matrices the method is based upon a relatively obscure eigenvalue inequality

on the general solution for the modified emden type equation in this paper we demonstrate that the modified equation of emden type mee is integrable either explicitly or by quadrature for any value of and we also prove that the mee possesses appropriate time independent hamiltonian functions for the full range of parameters and in addition we show that the mee is intimately connected with two well known nonlinear models namely the equation of force free duffing type oscillator and the two dimensional lotka volterra equation and thus the complete integrability of the latter two models can also be understood in terms of the mee

improving point and interval estimators of monotone functions by rearrangement suppose that a target function is monotonic and an available original estimate of this target function is not monotonic rearrangements univariate and multivariate transform the original estimate to a monotonic estimate that always lies closer in common metrics to the target function furthermore suppose an original confidence interval which covers the target function with probability at least 1 is defined by an upper and lower endpoint functions that are not monotonic then the rearranged confidence interval defined by the rearranged upper and lower endpoint functions is monotonic shorter in length in common norms than the original interval and covers the target function with probability at least 1 we illustrate the results with a growth chart example

lagrangian structures integrability and chaos for 3d dynamical equations in this paper we consider the general setting for constructing action principles for three dimensional first order autonomous equations we present the results for some integrable and non integrable cases of the lotka volterra equation and show lagrangian descriptions which are valid for systems satisfying shil nikov criteria on the existence of strange attractors though chaotic behaviour has not been verified up to now the euler lagrange equations we get for these systems usually present time reparametrization invariance though other kinds of invariance may be found according to the kernel of the associated symplectic 2 form the formulation of a hamiltonian structure poisson brackets and hamiltonians for these systems from the lagrangian viewpoint leads to a method of finding new constants of the motion starting from known ones which is applied to some systems found in the literature known to possess a constant of the motion to find the other and thus showing their integrability in particular we show that the so called abc system is completely integrable if it possesses one constant of the motion

darboux theory of integrability in cn taking into account the multiplicity abstract darboux in 1878 provided a theory on the existence of first integrals of polynomial systems based on the existence of sufficient invariant algebraic hypersurfaces called now the darboux theory of integrability in 1979 jouanolou successfully improved the darboux theory of integrability characterizing the existence of rational first integrals for this he used sophisticated tools of algebraic geometry the aim of this paper is to improve the classical result of darboux and the new one of jouanolou taking into account the multiplicity of the invariant algebraic hypersurfaces additionally our proof of the improved result of jouanolou is much simpler and elementary than the original one some examples show that the improved darboux theory of integrability with multiplicity is much useful than the classical one

the integration of three dimensional lotka volterra systems the general solutions of many three dimensional lotka volterra systems previously known to be at least partially integrable are constructed with the aid of special functions examples include certain abc and may leonard systems the special functions used are elliptic and incomplete beta functions in some cases the solution is parametric with the independent and dependent variables expressed as functions of a new time variable this auxiliary variable satisfies a nonlinear third order differential equation of a generalized schwarzian type and results of carton lebrun on the equations of this type that have the painleve property are exploited so as to produce solutions in closed form for several especially difficult lotka volterra systems the solutions are expressed in terms of painleve transcendents an appendix on incomplete beta functions and closed form expressions for their inverses is included

instrumental variable treatment of nonclassical measurement error models while the literature on nonclassical measurement error traditionally relies on the availability of an auxiliary data set containing correctly measured observations we establish that the availability of instruments enables the identification of a large class of nonclassical nonlinear errors in variables models with continuously distributed variables our main identifying assumption is that conditional on the value of the true regressors some measure of location of the distribution of the measurement error e g its mean mode or median is equal to zero the proposed approach relies on the eigenvalue eigenfunction decomposition of an integral operator associated with specific joint probability densities the main identifying assumption is used to index the eigenfunctions so that the decomposition is unique we propose a convenient sieve based estimator derive its asymptotic properties and investigate its finite sample behavior through monte carlo simulations

on improving density estimators which are not bona fide functions in order to improve the rate of decrease of the imse for nonparametric kernel density estimators with nonrandom bandwidth beyond o n 4 5 all current methods must relax the constraint that the density estimate be a bona fide function that is be nonnegative and integrate to one in this paper we show how to achieve similar improvement without relaxing any of these constraints the method can also be applied for orthogonal series adaptive orthogonal series spline jackknife and other density estimators and assures an improvement of the imse for each sample size

nonparametric estimation of non exchangeable latent variable models we propose a two step method to nonparametrically estimate multivariate models in which the observed outcomes are independent conditional on a discrete latent variable applications include microeconometric models with unobserved types of agents regime switching models and models with misclassification error in the first step we estimate weights that transform moments of the marginal distribution of the data into moments of the conditional distribution of the data for given values of the latent variable in the second step these conditional moments are estimated as weighted sample averages we illustrate the method by estimating a model of wages with unobserved heterogeneity on psid data

bell polynomial approach and n soliton solution for the extended lotka volterra equation in plasmas symbolically investigated in this paper is the extended lotka volterra elv equation which can govern the kinetics of the discrete peaks of the weak langmuir turbulence in plasmas without the linear damping and random noise binary bell polynomials are applied to the bilinearization of the discrete system bilinear backlund transformation of the elv equation is constructed n soliton solution in terms of the extended casorati determinant is also presented and verified propagation and interaction behaviors of the langmuir turbulence are analyzed it is demonstrated that the number of the interacting langmuir waves can influence the soliton velocity and amplitude as well as the collision phase shift graphic illustrations of the solitonic collisions show that the repulsion effects and nonlinear interactions are also associated with the number of the interacting langmuir waves

non parametric estimation of finite mixtures from repeated measurements this paper provides methods to estimate finite mixtures from data with repeated measurements non parametrically we present a constructive identification argument and use it to develop simple two step estimators of the component distributions and all their functionals we discuss a computationally efficient method for estimation and derive asymptotic theory simulation experiments suggest that our theory provides confidence intervals with good coverage in small samples

banach algebra structure and amenability of a class of matrix algebras with applications abstract we introduce a new category of banach algebras l 1 munn algebras which we use as a tool in the study of semigroup algebras then we characterize amenable l 1 munn algebras and also semisimple ones in this category applying these results to the semigroup algebras provides some characterizations of amenable semigroup algebras we also provide a counter example to a conjecture of duncan and paterson

non parametric identification and estimation of the number of components in multivariate mixtures this article analyzes the identiability of the number of components in k variate mcomponent nite mixture models in which each component distribution has independent marginals including models in latent class analysis without making parametric assumptions on the component distributions we investigate how one can identify the number of components from the distribution function of the observed data when k 2 a lower bound on the number of components m is nonparametrically identiable from the rank of a matrix constructed from the distribution function of the observed variables building on this identication condition we develop a procedure to consistently estimate a lower bound on the number of components

a new jacobi like method for joint diagonalization of arbitrary non defective matrices this paper addresses the problem of joint diagonalization of a set of matrices a new jacobi like method that has the advantages of computational efficiency and of generality is presented the proposed algorithm brings the general matrices into normal ones and performs a joint diagonalization by a combination of unitary and shears non unitary transformations it is based on the iterative minimization of an appropriate cost function using generalized jacobi rotation matrices

topological centers of certain dual algebras let a be a banach algebra with a bounded approximate identity let z1 and z2 be respectively the topological centers of the algebras a and a a in this paper for weakly sequentially complete banach algebras in particular for the group and fourier algebras l1 g and a g we study the sets z1 z2 the relations between them and with several other subspaces of a or a

connection between the existence of first integrals and the painleve property in two dimensional lotka volterra and quadratic systems taking advantage of the considerable amount of work done in the search for first integrals invariants for the two dimensional lotka volterra system and the quadratic system lvs and qs we compare the relations needed to exhibit invariants one for the lvs at least three for the qs to the two conditions of the painleve test index and compatibility we find that eventually restricting the invariants to those which are analytic all exponents integers and thereby adding new constraints these constraints always coalesce with the two painleve conditions we conclude that straightforward application of the painleve test picks up only these simple analytic invariants and that possession of the painleve property is too strong a condition for the existence of the invariants

maximum smoothed likelihood for multivariate mixtures summary we introduce an algorithm for estimating the parameters in a finite mixture of completely unspecified multivariate components in at least three dimensions under the assumption of conditionally independent coordinate dimensions we prove that this algorithm based on a majorization minimization idea possesses a desirable descent property just as any em algorithm does we discuss the similarities between our algorithm and a related one the so called nonlinearly smoothed em algorithm for the non mixture setting we also demonstrate via simulation studies that the new algorithm gives very similar results to another algorithm that has been shown empirically to be effective but that does not satisfy any descent property we provide code for implementing the new algorithm in a publicly available r package

identifiability of parameters in latent structure models with many observed variables while hidden class models of various types arise in many statistical applications it is often difficult to establish the identifiability of their parameters focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones we demonstrate a general approach for establishing identifiability utilizing algebraic arguments a theorem of j kruskal for a simple latent class model with finite state space lies at the core of our results though we apply it to a diverse set of models these include mixtures of both finite and nonparametric product distributions hidden markov models and random graph mixture models and lead to a number of new results and improvements to old ones in the parametric setting this approach indicates that for such models the classical definition of identifiability is typically too strong instead generic identifiability holds which implies that the set of nonidentifiable parameters has measure zero so that parameter inference is still meaningful in particular this sheds light on the properties of finite mixtures of bernoulli products which have been used for decades despite being known to have nonidentifiable parameters in the nonparametric setting we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed but we explicitly describe the conditions

an integrating factor matrix method to find first integrals in this paper we develop an integrating factor matrix method to derive conditions for the existence of first integrals we use this novel method to obtain first integrals along with the conditions for their existence for two and three dimensional lotka volterra systems with constant terms the results are compared to previous results obtained by other methods

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

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

nonparametric estimation of component distributions in a multivariate mixture suppose k variate data are drawn from a mixture of two distributions each having independent components it is desired to estimate the univariate marginal distributions in each of the products as well as the mixing proportion this is the setting of two class fully parametrized latent models that has been proposed for estimating the distributions of medical test results when disease status is unavailable the problem is one of inference in a mixture of distributions without training data and until now it has been tackled only in a fully parametric setting we investigate the possibility of using nonparametric methods of course when k 1 the problem is not identifiable from a nonparametric viewpoint we show that the problem is almost identifiable when k 2 there the set of all possible representations can be expressed in terms of any one of those representations as a two parameter family furthermore it is proved that when k geq3 the problem is nonparametrically identifiable under particularly mild regularity conditions in this case we introduce root n consistent nonparametric estimators of the 2k univariate marginal distributions and the mixing proportion finite sample and asymptotic properties of the estimators are described

nonparametric identification of finite mixture models of dynamic discrete choices in dynamic discrete choice analysis controlling for unobserved heterogeneity is an important issue and finite mixture models provide flexible ways to account for it this paper studies nonparametric identifiability of type probabilities and type specific component distributions in finite mixture models of dynamic discrete choices we derive sucient conditions for nonparametric identification for various finite mixture models of dynamic discrete choices used in applied work under dierent assumptions on the markov property stationarity and type invariance in the transition process three elements emerge as the important determinants of identification the time dimension of panel data the number of values the covariates can take and the heterogeneity of the response of dierent types to changes in the co

homological properties of modules over group algebras let g be a locally compact group and let l 1 g be the banach algebra which is the group algebra of g we consider a variety of banach left l 1 g modules over l 1 g and seek to determine conditions on g that determine when these modules are either projective or injective or flat in the category the answers typically involve g being compact or discrete or amenable for example in the case where g is discrete and 1 p infty we find that the module ell p g is injective whenever g is amenable and that if it is amenable then g is pseudo amenable a property very close to that of amenability

darboux integrability for 3d lotka volterra systems we describe the improved darboux theory of integrability for polynomial ordinary differential equations in three dimensions using this theory and computer algebra we study the existence of first integrals for the three dimensional lotka volterra systems only working up to degree two with the invariant algebraic surfaces and the exponential factors we find the major part of the known first integrals for such systems and in addition we find three new classes of integrability the method used is of general interest and can be applied to any polynomial ordinary differential equations in arbitrary dimension

identification and estimation of nonlinear models with misclassification error using instrumental variables a general solution abstract this paper provides a general solution to the problem of identification and estimation of nonlinear models with misclassification error in a general discrete explanatory variable using instrumental variables the misclassification error is allowed to be correlated with all the explanatory variables in the model it is not enough to identify the model by simply generalizing the identification in the binary case with a claim that the number of restrictions is no less than that of the unknowns such a claim requires solving a complicated nonlinear system of equations this paper introduces a matrix diagonalization technique which allows one to easily find the unique solution of the system the solution shows that the latent model can be expressed as an explicit function of directly observed distribution functions therefore the latent model is nonparametrically identifiable and directly estimable using instrumental variables the results show that certain monotonicity restrictions on the latent model may lead to its identification with virtually no restrictions on the misclassification probabilities an alternative identification condition suggests that the nonparametric identification may rely on the belief that people always have a higher probability of telling the truth than of misreporting the nonparametric identification in this paper directly leads to a n consistent semiparametric estimator the monte carlo simulation and empirical illustration show that the estimator performs well with a finite sample and real data

isometric isomorphisms between banach algebras related to locally compact groups let g1 g2 be locally compact groups we prove in this paper that if t is an isometric isomorphism from the banach algebra luc g1 the continuous dual of the banach space of left uniformly continuous functions on g1 equipped with arens multiplication onto luc g2 i then t maps m g1 onto m g2 and li g onto l g2 we also prove that any isometric isomorphism from li g second conjugate algebra of l gi onto ll g2 maps l g1 onto l g2 0 introduction and preliminaries let g1 g2 be locally compact groups let m gi i 1 2 be the banach algebra of regular borel measures on gi a well known result of b e johnson 10 asserts that if t is an isometric isomorphism from m g1 onto m g2 then t maps ll g onto l g2 and hence g1 and g2 must be isomorphic by wendel s theorem 21 in this paper we prove theorem 3 1 c among other things that if t is an isometric isomorphism from ll g1 onto l1 g2 then t maps l1 g1 onto ll g2 this answers affirmatively a question raised in 4 theorem 3 1 c was proved for abelian locally compact groups by lau and losert in 13 and for compact and discrete groups by ghahramani and lau in 4 let g be a locally compact group let c g denote the space of bounded continuous complex valued functions on g with the sup norm topology and luc g denote the closed subspace of bounded left uniformly continuous functions on g i e all f e c g such that the map x lxf from g into c g is continuous where lxf y f xy x y e g then luc g is a banach algebra with the arens multiplication defined by nm f n m1f n m e luc g f e luc g where mlf x m lxf x e g furthermore m g may be identified with a closed subspace of luc g by the natural embedding u f f f x du x f e luc g u e m g it was received by the editors november 1 1988 1980 mathematics subject classification 1985 revision primary 43a20