topological noetherianity of polynomial functors we prove that any finite degree polynomial functor over an infinite field is topologically noetherian this theorem is motivated by the recent resolution by ananyan hochster of stillman s conjecture and a recent noetherianity proof by derksen eggermont snowden for the space of cubics via work by erman sam snowden our theorem implies stillman s conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees

neural network adaptive control for a class of nonlinear uncertain dynamical systems with asymptotic stability guarantees in this paper a neuroadaptive control framework for continuous and discrete time nonlinear uncertain dynamical systems with input to state stable internal dynamics is developed the proposed framework is lyapunov based and unlike standard neural network nn controllers guaranteeing ultimate boundedness the framework guarantees partial asymptotic stability of the closed loop system that is asymptotic stability with respect to part of the closed loop system states associated with the system plant states the neuroadaptive controllers are constructed without requiring explicit knowledge of the system dynamics other than the assumption that the plant dynamics are continuously differentiable and that the approximation error of uncertain system nonlinearities lie in a small gain type norm bounded conic sector this allows us to merge robust control synthesis tools with nn adaptive control tools to guarantee system stability finally two illustrative numerical examples are provided to demonstrate the efficacy of the proposed approach

gl equivariant modules over polynomial rings in infinitely many variables consider the polynomial ring in countably infinitely many variables over a field of characteristic zero together with its natural action of the infinite general linear group g we study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible g action we define and prove finiteness properties for analogues of hilbert series systems of parameters depth local cohomology koszul duality and regularity we also show that this category is built out of a simpler more combinatorial quiver category which we describe explicitly our work is motivated by recent papers in the literature which study finiteness properties of infinite polynomial rings equipped with group actions for example the paper by church ellen berg and farb on the category of fi modules which is equivalent to our category along the way we see several connections with the character polynomials from the representation theory of the symmetric groups several examples are given to illustrate that the invariants we introduce are explicit and computable

kp hierarchy for hodge integrals abstract starting from the elsv formula we derive a number of new equations on the generating functions for hodge integrals over the moduli space of complex curves this gives a new simple and uniform treatment of certain known results on hodge integrals like witten s conjecture virasoro constrains faber s g conjecture etc among other results we show that a properly arranged generating function for hodge integrals satisfies the equations of the kp hierarchy

pfaffian formulas and schur q function identities abstract we establish pfaffian analogues of the cauchy binet formula and the ishikawa wakayama minor summation formula each of these pfaffian analogues expresses a sum of products of subpfaffians of two skew symmetric matrices in terms of a single pfaffian by using these pfaffian formulas we give transparent proofs to several identities for schur q functions

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

decentralized control design for large scale systems with strong interconnections using neural networks we propose a decentralized neural network nn controller for a class of large scale nonlinear systems with the strong interconnections the nns are used to approximate the unknown subsystems and interconnections due to the functional approximation capabilities of nns the additional precautions are not required to be made for avoiding the possible control singularity problems semiglobal asymptotic stability results are obtained and the tracking error converges to zero furthermore the issue of transient performance of the subsystems is also addressed under an analytical framework

finite generation of symmetric ideals let be a commutative noetherian ring and let be the polynomial ring in an infinite collection of indeterminates over let be the group of permutations of the group acts on in a natural way and this in turn gives the structure of a left module over the group ring we prove that all ideals of invariant under the action of are finitely generated as modules the proof involves introducing a certain well quasi ordering on monomials and developing a theory of grobner bases and reduction in this setting we also consider the concept of an invariant chain of ideals for finite dimensional polynomial rings and relate it to the finite generation result mentioned above finally a motivating question from chemistry is presented with the above framework providing a suitable context in which to study it

finiteness properties of congruence classes of infinite by infinite matrices abstract we look at affine spaces of n n matrices over a field k and consider varieties that are stable under simultaneous row and column operations we prove that up to symmetry any of these varieties is defined by finitely many equations

the isserlis matrix and its application to non decomposable graphical gaussian models the operations of matrix completion and isserlis matrix construction arise in the statistical analysis of graphical gaussian models in both the bayesian and frequentist approaches in this paper we study the properties of the isserlis matrix of the completion of a positive definite matrix and propose an edge set indexing notation which highlights the symmetry existing between and its isserlis matrix in this way well known properties of can be exploited to give an easy proof of certain asymptotic results for decomposable graphical gaussian models as well as to extend such results to the non decomposable case in particular we consider the asymptotic variance of maximum likelihood estimators the non informative jeffreys prior the laplace approximation to the bayes factor the asymptotic distribution of the maximum likelihood estimators and the asymptotic posterior distribution of the parameters in a bayesian conjugate analysis in dealing with distributions over non decomposable models an extension of the hyper markov property to models with non chordal graphs is required our proposed extension is justified by an analysis of the conditional independence structure of the sufficient statistic

betti numbers of symmetric shifted ideals abstract we introduce a new class of monomial ideals which we call symmetric shifted ideals symmetric shifted ideals are fixed by the natural action of the symmetric group and within the class of monomial ideals fixed by this action they can be considered as an analogue of stable monomial ideals within the class of monomial ideals we show that a symmetric shifted ideal has linear quotients and compute its equivariant graded betti numbers as an application of this result we obtain several consequences for graded betti numbers of symbolic powers of defining ideals of star configurations

the character of the infinite wedge representation abstract we study the character of the infinite wedge projective representation of the algebra of differential operators on the circle we prove quasi modularity of this character and also compute certain generating functions for traces of differential operators which we call correlation functions these correlation functions are sums of determinants built from genus 1 theta functions and their derivatives

the possible dimensions of the higher secant varieties let x c p be an irreducible closed subvariety of projective space the kth higher secant variety of x denoted x is the closure of the union of all linear spaces spanned by k points of x let a ix dim x dim x l it is known that the sequence ty x is nonincreasing and that there may be at most one 1 in such a sequence we prove that these two conditions are also sufficient conditions for an arbitrary sequence a of nonnegative integers to be the fl x s for an irreducible variety x to this end we exhibit the ideals of the higher secant varities of rational normal scrolls as determinantal ideals this allows us to compute their dimensions showing that there is in fact always a rational normal scroll x such that ak x a for any sequence a satisfying the two conditions above adlandsvik al a2 and flenner and vogel f v much of this work has centered on the behavior of the differences ak x dimx dimx 1 of the dimensions of consecutive higher secant varieties it is well known that this sequence is nonincreasing and that there is at most one 1 in the sequence palatini claimed moreover that once ak x 1 were counterexamples a2 we answer here the question of what sequences actually appear as the ai x s of some variety x some of the guiding examples of varieties with degenerate higher secant varieties come from rational normal scrolls explicit equations for the scrolls themselves have been known for some time cf acgh we give a determi nantal presentation of the ideals of their higher secant varieties generalizing a result for rational normal scrolls in w this presentation enables us to compute the dimensions of all the higher secant varieties of a rational normal scroll and gives examples of varieties x with a prescribed nonincreasing ak x with at most one 1

on some models for multivariate binary variables parallel in complexity with the multivariate gaussian distribution it is shown that both the simple form of the rasch model for binary data and a generalisation are essentially equivalent to special dichotomised gaussian models in these the underlying gaussian structure is of single factor form that is the correlations between the binary variables arise via a single underlying variable called in psychometrics a latent trait the implications for scoring of the binary variables are discussed in particular regarding the scoring system as in effect estimating the latent trait in particular the role of the simple sum score in effect the total number of successes is examined relations with the principal component analysis of binary data are outlined and some connections with the quadratic exponential binary model are sketched

noetherian quotients of the algebra of partial difference polynomials and grobner bases of symmetric ideals abstract in this paper we develop a grobner bases theory for ideals of partial difference polynomials with constant or non constant coefficients in particular we introduce a criterion providing the finiteness of such bases when a difference ideal contains elements with suitable linear leading monomials this can be explained in terms of noetherianity of the corresponding quotient algebra among these noetherian quotients we find finitely generated polynomial algebras where the action of suitable finite dimensional commutative algebras and in particular finite abelian groups is defined we obtain therefore a consistent grobner bases theory for ideals that possess such symmetries

a bayesian information criterion for singular models summary we consider approximate bayesian model choice for model selection problems that involve models whose fisher information matrices may fail to be invertible along other competing submodels such singular models do not obey the regularity conditions underlying the derivation of schwarz s bayesian information criterion bic and the penalty structure in bic generally does not reflect the frequentist large sample behaviour of the marginal likelihood although large sample theory for the marginal likelihood of singular models has been developed recently the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning guided by examples such as determining the number of components in mixture models the number of factors in latent factor models or the rank in reduced rank regression we propose a resolution to this paradox and give a practical extension of bic for singular model selection problems

a continuous asymptotic tracking control strategy for uncertain nonlinear systems in this note we present a new continuous control mechanism that compensates for uncertainty in a class of high order multiple input multiple output nonlinear systems the control strategy is based on limited assumptions on the structure of the system nonlinearities a new lyapunov based stability argument is employed to prove semiglobal asymptotic tracking

discussion latent variable graphical model selection via convex optimization we want to congratulate the authors for a thought provoking and very interesting paper sparse modeling of the concentration matrix has enjoyed popularity in recent years it has been framed as a computationally convenient convex 1constrained estimation problem in yuan and lin 2007 and can be applied readily to higher dimensional problems the authors argue we think correctly that the sparsity of the concentration matrix is for many applications more plausible after the effects of a few latent variables have been removed the most attractive point about their method is surely that it is formulated as a convex optimization problem latent variable fitting and sparse graphical modeling of the conditional distribution of the observed variables can then be obtained through a single fitting procedure practical aspects the method deserves wide adoption but this will only be realistic if software is made available for example as an r package not many users will go to the trouble of implementing the method on their own so we will strongly urge the authors to do so an imputation method in the absence of readily available software it is worth thinking whether the proposed fitting procedure can be approximated by methods involving known and well tested computational techniques the concentration matrix of observed and hidden variables is k ko koh k ho kh where we have deviated from the notation in the paper by omitting the asterisk the proposed estimator sn ko of ko was defined as

stochastic choice of basis functions in adaptive function approximation and the functional link net a theoretical justification for the random vector version of the functional link rvfl net is presented in this paper based on a general approach to adaptive function approximation the approach consists of formulating a limit integral representation of the function to be approximated and subsequently evaluating that integral with the monte carlo method two main results are 1 the rvfl is a universal approximator for continuous functions on bounded finite dimensional sets and 2 the rvfl is an efficient universal approximator with the rate of approximation error convergence to zero of order o c spl radic n where n is number of basis functions and with c independent of n similar results are also obtained for neural nets with hidden nodes implemented as products of univariate functions or radial basis functions some possible ways of enhancing the accuracy of multivariate function approximations are discussed

rise based adaptive control of a control affine uncertain nonlinear system with unknown state delays a continuous robust adaptive control method is designed for a class of uncertain nonlinear systems with unknown constant time delays in the states specifically a robust adaptive control method and a delay free gradient based desired compensation adaptation law dcal are utilized to compensate for unknown time delays linearly parameterizable uncertainties and additive bounded disturbances for a general nonlinear system despite these disturbances a lyapunov krasovskii based analysis is used to conclude that the system output asymptotically tracks a desired time varying bounded trajectory

stratified exponential families graphical models and model selection we describe a hierarchy of exponential families which is useful for distinguishing types of graphical models undirected graphical models with no hidden variables are linear exponential families lefs directed acyclic graphical dag models and chain graphs with no hidden variables including dag models with several families of local distributions are curved exponential families cefs graphical models with hidden variables are what we term stratified exponential families sefs a sef is a finite union of cefs of various dimensions satisfying some regularity conditions we also show that this hierarchy of exponential families is noncollapsing with respect to graphical models by providing a graphical model which is a cef but not a lef and a graphical model that is a sef but not a cef finally we show how to compute the dimension of a stratified exponential family these results are discussed in the context of model selection of graphical models

the maximum likelihood degree maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials we study the algebraic degree of the critical equations of this optimization problem this degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces and to the euler characteristic of the complexified complement under suitable hypotheses the maximum likelihood degree equals the top chern class of a sheaf of logarithmic differential forms exact formulae in terms of degrees and newton polytopes are given for polynomials with generic coefficients

semidefinite tests for latent causal structures testing whether a probability distribution is compatible with a given bayesian network is a fundamental task in the field of causal inference where bayesian networks model causal relations here we consider the class of causal structures where all correlations between observed quantities are solely due to the influence from latent variables we show that each model of this type imposes a certain signature on the observable covariance matrix in terms of a particular decomposition into positive semidefinite components this signature and thus the underlying hypothetical latent structure can be tested in a computationally efficient manner via semidefinite programming this stands in stark contrast with the algebraic geometric tools required if the full observable probability distribution is taken into account the semidefinite test is compared with tests based on entropic inequalities

finiteness for the k factor model and chirality varieties this paper deals with two families of algebraic varieties arising from applications first the k factor model in statistics consisting of n times n covariance matrices of n observed gaussian variables that are pairwise independent given k hidden gaussian variables second chirality varieties inspired by applications in chemistry a point in such a chirality variety records chirality measurements of all k subsets among an n set of ligands both classes of varieties are given by a parameterisation while for applications having polynomial equations would be desirable for instance such equations could be used to test whether a given point lies in the variety we prove that in a precise sense which is different for the two classes of varieties these equations are finitely characterisable when k is fixed and n grows

algebraic analysis for nonidentifiable learning machines this article clarifies the relation between the learning curve and the algebraic geometrical structure of a nonidentifiable learning machine such as a multilayer neural network whose true parameter set is an analytic set with singular points by using a concept in algebraic analysis we rigorously prove that the bayesian stochastic complexity or the free energy is asymptotically equal to 1 log n m1 1 log log n constant where n is the number of training samples and 1 and m1 are the rational number and the natural number which are determined as the birational invariant values of the singularities in the parameter space also we show an algorithm to calculate 1 and m1 based on the resolution of singularities in algebraic geometry in regular statistical models 2 1 is equal to the number of parameters and m1 1 whereas in nonregular models such as multilayer networks 2 1 is not larger than the number of parameters and m1 1 since the increase of the stochastic complexity is equal to the learning curve or the generalization error the nonidentifiable learning machines are better models than the regular ones if bayesian ensemble learning is applied

adaptive control of a class of nonlinear systems with fuzzy logic an adaptive tracking control architecture is proposed for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture employs fuzzy systems which are expressed as a series expansion of basis functions to adaptively compensate for the plant nonlinearities global asymptotic stability of the algorithm is established in the lyapunov sense with tracking errors converging to a neighborhood of zero simulation results for an unstable nonlinear plant are included to demonstrate that incorporating the linguistic fuzzy information from human experts results in superior tracking performance

discrete time neural net controller for a class of nonlinear dynamical systems a family of two layer discrete time neural net nn controllers is presented for the control of a class of mnth order mimo dynamical system no initial learning phase is needed so that the control action is immediate in other words the neural network nn controller exhibits a learning while functioning feature instead of a learning then control feature a two layer nn is used which is linear in the tunable weights the structure of the neural net controller is derived using a filtered error approach it is indicated that delta rule based tuning when employed for closed loop control can yield unbounded nn weights if 1 the net cannot exactly reconstruct a certain required function or 2 there are bounded unknown disturbances acting on the dynamical system certainty equivalence is not used overcoming a major problem in discrete time adaptive control in this paper new online tuning algorithms for discrete time systems are derived which are similar to spl epsiv modification for the case of continuous time systems that include a modification to the learning rate parameter and a correction term to the standard delta rule

asymptotic tracking of uncertain systems with continuous control using adaptive bounding this paper presents a robust adaptive control design method for a class of multiple input multiple output uncertain nonlinear systems in the presence of parametric and nonparametric uncertainties and bounded disturbances using the approximation properties of the unknown continuous nonlinearities and the adaptive bounding technique the developed controller achieves asymptotic convergence of the tracking error to zero while ensuring boundedness of parameter estimation errors the algorithm does not assume the knowledge of any bound on the unknown quantities in designing the controller it is based on an integral technique involving the filtered tracking error and produces a continuous control theoretical developments are illustrated via simulation results

on identification of multi factor models with correlated residuals summary we specify some conditions for the identification of a multi factor model with correlated residuals uncorrelated factors and zero restrictions in the factor loadings these conditions are derived from the results of stanghellini 1997 and vicard 2000 which deal with single factor models with zero restrictions in the concentration matrix like these authors we make use of the complementary graph of residuals and the conditions build on the role of odd cycles in this graph however in contrast to these authors we consider the case where the conditional dependencies of the residuals are expressed in terms of a covariance matrix rather than its inverse the concentration matrix we first derive the corresponding condition for identification of single factor models with structural zeros in the covariance matrix of the residuals this is extended to the case where some factor loadings are constrained to be zero we use these conditions to obtain a sufficient and a necessary condition for identification of multi factor models

the correlation space of gaussian latent tree models and model selection without fitting we provide a complete description of possible distributions consistent with any gaussian latent tree model this description consists of polynomial equations and inequalities involving covariances between the observed variables testing inequality constraints can be done using the inverse wishart distribution and this leads to simple preliminary assessment of tree compatibility to test equality constraints we employ general techniques of tetrad analyses this approach is effective even for small sample sizes and can be easily adjusted to test either entire models or just particular macrostructures of a tree our methods are simple to implement and do not require fitting of the model the versatility of the techniques is illustrated by performing exploratory and confirmatory tetrad analyses in linguistic and biological settings respectively

algebraic factor analysis tetrads pentads and beyond factor analysis refers to a statistical model in which observed variables are conditionally independent given fewer hidden variables known as factors and all the random variables follow a multivariate normal distribution the parameter space of a factor analysis model is a subset of the cone of positive definite matrices this parameter space is studied from the perspective of computational algebraic geometry grobner bases and resultants are applied to compute the ideal of all polynomial functions that vanish on the parameter space these polynomials known as model invariants arise from rank conditions on a symmetric matrix under elimination of the diagonal entries of the matrix besides revealing the geometry of the factor analysis model the model invariants also furnish useful statistics for testing goodness of fit

nonintersecting paths pfaffians and plane partitions gessel and viennot have developed a powerful technique for enumerating various classes of plane partitions the purpose of this article is to show by similar means that one may use pfaffiants to enumerate configurations of nonintersecting paths in which the initial and or terminal vertices of the paths are allowed to vary over specified regions of the digraph this leads to the possibility of enumerating classes of plane partitions in which the shape is allowed to vary whereas the previous applications of gessel and viennot were largely confined to plane partitions of a given shape

feedback linearization using neural networks abstract for a class of single input single output continuous time nonlinear systems a multilayer neural network based controller that feedback linearizes the system is presented control action is used to achieve tracking performance for a state feedback linearizable but unknown nonlinear system the control structure consists of a feedback linearization portion provided by two neural networks plus a robustifying portion that keeps the control magniture bounded a stability proof is given in the sense of lyapunov it is shown that all the signals in the closed loop system are uniformly ultimately bounded no off line learning phase is needed initialization of the network weights is straightforward

noetherianity and specht problem for varieties of bicommutative algebras abstract nonassociative algebras satisfying the polynomial identities x 1 x 2 x 3 x 2 x 1 x 3 and x 1 x 2 x 3 x 1 x 3 x 2 are called bicommutative we prove the following results i finitely generated bicommutative algebras are weakly noetherian i e satisfy the ascending chain condition for two sided ideals ii we give the positive solution to the specht problem or the finite basis problem for varieties of bicommutative algebras over an arbitrary field of any characteristic

em algorithms for ml factor analysis the details of em algorithms for maximum likelihood factor analysis are presented for both the exploratory and confirmatory models the algorithm is essentially the same for both cases and involves only simple least squares regression operations the largest matrix inversion required is for aq q symmetric matrix whereq is the matrix of factors the example that is used demonstrates that the likelihood for the factor analysis model may have multiple modes that are not simply rotations of each other such behavior should concern users of maximum likelihood factor analysis and certainly should cast doubt on the general utility of second derivatives of the log likelihood as measures of precision of estimation

hall littlewood symmetric functions and the bkp equation the connection between solutions of the bkp equation and hall littlewood symmetric functions is utilised in a unified approach to soliton and polynomial solutions this is analogous to the wronskian formulation of the solution of the kp equation as a by product two novel expressions for certain hall littlewood functions in terms of pfaffians are derived

equivariant grobner bases and the gaussian two factor model exploiting symmetry in grobner basis computations is difficult when the symmetry takes the form of a group acting by automorphisms on monomials in finitely many variables this is largely due to the fact that the group elements being invertible cannot preserve a term order by contrast inspired by work of aschenbrenner and hillar we introduce the concept of equivariant grobner basis in a setting where a monoid acts by homomorphisms on monomials in potentially infinitely many variables we require that the action be compatible with a term order and under some further assumptions derive a buchberger type algorithm for computing equivariant grobner bases using this algorithm and the monoid of strictly increasing functions we prove that the kernel of the ring homomorphism y ij i j i j s i t i i y ij s i s j t i t j is generated by two types of polynomials off diagonal 3 x 3 minors and pentads this confirms a conjecture by drton sturmfels and sullivant on the gaussian two factor model from algebraic statistics

gaussian networks for direct adaptive control a direct adaptive tracking control architecture is proposed and evaluated for a class of continuous time nonlinear dynamic systems for which an explicit linear parameterization of the uncertainty in the dynamics is either unknown or impossible the architecture uses a network of gaussian radial basis functions to adaptively compensate for the plant nonlinearities under mild assumptions about the degree of smoothness exhibit by the nonlinear functions the algorithm is proven to be globally stable with tracking errors converging to a neighborhood of zero a constructive procedure is detailed which directly translates the assumed smoothness properties of the nonlinearities involved into a specification of the network required to represent the plant to a chosen degree of accuracy a stable weight adjustment mechanism is determined using lyapunov theory the network construction and performance of the resulting controller are illustrated through simulations with example systems

the equations for the moduli space of n points on the line a central question in invariant theory is that of determining the relations among invariants geometric invariant theory git quotients come with a natural ample line bundle and hence often a natural projective embedding this question translates to determining the equations of the moduli space under this embedding this article deals with one of the most classical quotients the space of ordered points on the projective line we show that under any weighting of the points this quotient is cut out scheme theoretically by a particularly simple set of quadric relations with the single exception of the segre cubic threefold the space of six points with equal weight we also show that the ideal of relations is generated in degree at most four and we give an explicit description of the generators if all the weights are even e g in the case of equal weight for odd n then we show that the ideal of relations is generated by quadrics

asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and rise feedback control structure the use of a neural network nn as a feedforward control element to compensate for nonlinear system uncertainties has been investigated for over a decade typical nn based controllers yield uniformly ultimately bounded uub stability results due to residual functional reconstruction inaccuracies and an inability to compensate for some system disturbances several researchers have proposed discontinuous feedback controllers e g variable structure or sliding mode controllers to reject the residual errors and yield asymptotic results the research in this paper describes how a recently developed continuous robust integral of the sign of the error rise feedback term can be incorporated with a nn based feedforward term to achieve semi global asymptotic tracking to achieve this result the typical stability analysis for the rise method is modified to enable the incorporation of the nn based feedforward terms and a projection algorithm is developed to guarantee bounded nn weight estimates

plucker varieties and higher secants of sato s grassmannian every grassmannian in its plucker embedding is defined by quadratic polynomials we prove a vast qualitative generalisation of this fact to what we call plucker varieties a plucker variety is in fact a family of varieties in exterior powers of vector spaces that like the grassmannian is functorial in the vector space and behaves well under duals a special case of our result says that for each fixed natural number k the k th secant variety of any plucker embedded grassmannian is defined in bounded degree independent of the grassmannian our approach is to take the limit of a plucker variety in the dual of a highly symmetric space known as the infinite wedge and to prove that up to symmetry the limit is defined by finitely many polynomial equations for this we prove the auxilliary result that for every natural number p the space of p tuples of infinite by infinite matrices is noetherian modulo row and column operations our results have algorithmic counterparts every bounded plucker variety has a polynomial time membership test and the same holds for zariski closed basis independent properties of p tuples of matrices

chattering analysis a formal mathematical definition of chattering is proposed chattering phenomena are classified into three types in particular the first type is harmless and cannot be avoided chattering properties of various control approaches are considered the dangerous second and third types of chattering phenomena are proved to be removable by proper use of high order sliding modes hosm fast stable actuators and sensors only generate the first type of chattering in hosm systems and practically never affect the sliding motion computer simulation confirms the theoretical results

robust integral of neural network and error sign control of mimo nonlinear systems this paper presents a novel state feedback control scheme for the tracking control of a class of multi input multioutput continuous time nonlinear systems with unknown dynamics and bounded disturbances first the control law consisting of the robust integral of a neural network nn output plus sign of the tracking error feedback multiplied with an adaptive gain is introduced the nn in the control law learns the system dynamics in an online manner while the nn residual reconstruction errors and the bounded disturbances are overcome by the error sign signal since both of the nn output and the error sign signal are included in the integral the continuity of the control input is ensured the controller structure and the nn weight update law are novel in contrast with the previous effort and the semiglobal asymptotic tracking performance is still guaranteed by using the lyapunov analysis in addition the nn weights and all other signals are proved to be bounded simultaneously the proposed approach also relaxes the need for the upper bounds of certain terms which are usually required in the previous designs finally the theoretical results are substantiated with simulations

adaptive neural control of high order uncertain nonaffine systems a transformation to affine systems approach abstract this brief investigates the adaptive neural network nn control of a class of high order nonaffine nonlinear systems with completely unknown dynamics since the control terms appear within the unknown nonlinearity traditional control schemes and stability analysis are usually rendered extremely complicated our main contribution includes a novel system transformation that converts the nonaffine system into an affine system through a combination of a low pass filter and state transformation as a result the state feedback control of the nonaffine system can be viewed as the output feedback control of an affine system in normal form the transformed system becomes linear with respect to the new input while the traditional backstepping approach is not needed thus allowing the synthesis to be extremely simplified it is theoretically proven that all the signals in the closed loop system are uniformly ultimately bounded uub simulation results are provided to demonstrate the performance of the developed controller

separability problem for multipartite states of rank at most 4 one of the most important problems in quantum information is the separability problem which asks whether a given quantum state is separable we investigate multipartite states of rank at most 4 which are ppt i e all their partial transposes are positive semidefinite we show that any ppt state of rank 2 or 3 is separable and has length at most 4 for separable states of rank 4 we show that they have length at most 6 it is six only for some qubit qutrit or multiqubit states it turns out that any ppt entangled state of rank 4 is necessarily supported on a 3 3 or a 2 2 2 subsystem we obtain a very simple criterion for the separability problem of the ppt states of rank at most 4 such a state is entangled if and only if its range contains no product vectors this criterion can be easily applied since a four dimensional subspace in the 3 3 or 2 2 2 system contains a product vector if and only if its plucker coordinates satisfy a homogeneous polynomial equation the chow form of the corresponding segre variety we have computed an explicit determinantal expression for the chow form in the former case while such an expression was already known in the latter case

finite grobner bases in infinite dimensional polynomial rings and applications abstract we introduce the theory of monoidal grobner bases a concept which generalizes the familiar notion in a polynomial ring and allows for a description of grobner bases of ideals that are stable under the action of a monoid the main motivation for developing this theory is to prove finiteness results in commutative algebra and applications a basic theorem of this type is that ideals in infinitely many indeterminates stable under the action of the symmetric group are finitely generated up to symmetry using this machinery we give new streamlined proofs of some classical finiteness theorems in algebraic statistics as well as a proof of the independent set conjecture of hosten and the second author

the rank of reduced dispersion matrices psychometricians working in factor analysis and econometricians working in regression with measurement error in all variables are both interested in the rank of dispersion matrices under variation of the diagonal elements psychometricians concentrate on cases in which low rank can be attained preferably rank one the spearman case econometricians cocentrate on cases in which the rank cannot be reduced below the number of variables minus one the frisch case in this paper we give an extensive historial discussion of both fields we prove the two key results in a more satisfactory and uniform way we point out various small errors and misunderstandings and we present a methodological comparison of factor analysis and regression on the basis of our results

actions of cocommutative hopf algebras abstract let h be a cocommutative hopf algebra acting on an algebra a assuming the base field to be algebraically closed and the h action on a to be integral that is it is given by a coaction of some hopf subalgebra of the finite dual h that is an integral domain we stratify the prime spectrum spec a in terms of the prime spectra of certain commutative algebras for arbitrary h actions in characteristic 0 we show that the largest h stable ideal of a that is contained in a given semiprime ideal of a is semiprime as well

hilbert series of symmetric ideals in infinite polynomial rings via formal languages abstract let r be the polynomial ring k x i j where 1 i r and j n and let i be an ideal of r stable under the natural action of the infinite symmetric group s nagel romer recently defined a hilbert series h i s t of i and proved that it is rational we give a much shorter proof of this theorem using tools from the theory of formal languages and a simple algorithm that computes the series

bounded rank tensors are defined in bounded degree matrices of rank at most k are defined by the vanishing of polynomials of degree k 1 in their entries namely their k 1 k 1 subdeterminants regardless of the size of the matrix we prove a qualitative analogue of this statement for tensors of arbitrary dimension where matrices correspond to two dimensional tensors more specifically we prove that for each k there exists an upper bound d d k such that tensors of border rank at most k are defined by the vanishing of polynomials of degree at most d regardless of the dimension of the tensor and regardless of its size in each dimension our proof involves passing to an infinite dimensional limit of tensor powers of a vector space whose elements we dub infinite dimensional tensors and exploiting the symmetries of this limit in crucial ways