tightness of products of random matrices and stability of linear stochastic systems soit n la distribution d un produit de n matrices aleatoire i i d on etudie la compacite et la convergence de la suite n n on l applique aux equations differentielles stochastiques lineaires

circular free spectrahedra abstract this paper considers matrix convex sets invariant under several types of rotations it is known that matrix convex sets that are free semialgebraic are solution sets of linear matrix inequalities lmis they are called free spectrahedra we classify all free spectrahedra that are circular that is closed under multiplication by e i t up to unitary equivalence the coefficients of a minimal lmi defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal a matrix convex set is called free circular if it is closed under left multiplication by unitary matrices as a consequence of a hahn banach separation theorem for free circular matrix convex sets we show the coefficients of a minimal lmi defining a free circular free spectrahedron have up to unitary equivalence a block decomposition as above with only two blocks this paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials

matrix coefficient realization theory of noncommutative rational functions abstract noncommutative rational functions i e elements of the universal skew field of fractions of a free algebra can be defined through evaluations of noncommutative rational expressions on tuples of matrices this interpretation extends their traditionally important role in the theory of division rings and gives rise to their applications in other areas from free real algebraic geometry to systems and control theory if a noncommutative rational function is regular at the origin it can be described by a linear object called a realization in this article we present an extension of the realization theory that is applicable to arbitrary noncommutative rational functions and is well adapted for studying matrix evaluations of special interest are the minimal realizations which compensate the absence of a canonical form for noncommutative rational functions the non minimality of a realization is assessed by obstruction modules associated with it they enable us to devise an efficient method for obtaining minimal realizations with them we describe the stable extended domain of a noncommutative rational function and define a numerical invariant that measures its complexity using these results we determine concrete size bounds for rational identity testing construct minimal symmetric realizations and prove an effective local global principle for linear dependence of noncommutative rational functions

on the controllability of the linearized benjamin bona mahony equation we study the boundary controllability properties of the linearized benjamin bona mahony equation left begin array ll u t u xxt u x 0 x in 0 1 t 0 u t 0 0 u 1 t f t t 0 end array right we show that the equation is approximately controllable but not spectrally controllable no finite linear combination of eigenfunctions other than zero is controllable next we prove a finite controllability result and we estimate the norms of the controls needed in this case

an analogue of hincin s characterization of infinite divisibility for operator valued free probability abstract let b be a finite separable von neumann algebra we prove that a b valued distribution that is the weak limit of an infinitesimal array is infinitely divisible the proof of this theorem utilizes the steinitz lemma and may be adapted to provide a nonstandard proof of this type of theorem for various other probabilistic categories we also develop weak topologies for this theory and prove the corresponding compactness and convergence results

temporal aggregation of garch processes the authors derive low frequency say weekly models implied by high frequency say daily arma models with symmetric garch errors they show that low frequency models exhibit conditional heteroskedasticity of the garch form as well the parameters in the conditional variance equation of the low frequency model depend upon mean variance and kurtosis parameters of the corresponding high frequency model moreover strongly consistent estimators of the parameters in the high frequency model can be derived from low frequency data the common assumption in applications that rescaled innovations are independent is disputable since it depends upon the available data frequency copyright 1993 by the econometric society

financial asset returns direction of change forecasting and volatility dynamics we consider three sets of phenomena that feature prominently in the financial economics literature 1 conditional mean dependence or lack thereof in asset returns 2 dependence and hence forecastability in asset return signs and 3 dependence and hence forecastability in asset return volatilities we show that they are very much interrelated and explore the relationships in detail among other things we show that 1 volatility dependence produces sign dependence so long as expected returns are nonzero so that one should expect sign dependence given the overwhelming evidence of volatility dependence 2 it is statistically possible to have sign dependence without conditional mean dependence 3 sign dependence is not likely to be found via analysis of sign autocorrelations runs tests or traditional market timing tests because of the special nonlinear nature of sign dependence so that traditional market timing tests are best viewed as tests for sign dependence arising from variation in expected returns rather than from variation in volatility or higher moments 4 sign dependence is not likely to be found in very high frequency e g daily or very low frequency e g annual returns instead it is more likely to be found at intermediate return horizons and 5 the link between volatility dependence and sign dependence remains intact in conditionally non gaussian environments for example with time varying conditional skewness and or kurtosis

on the symmetric enveloping algebra of planar algebra subfactors we give a diagrammatic description of popa s symmetric enveloping algebras associated to planar algebra subfactors as an application we construct a natural family of derivations on these factors and compute a certain free entropy dimension type quantity 2013 american mathematical society

consistency and asymptotic normality of the quasi maximum likelihood estimator in igarch 1 1 and covariance stationary garch 1 1 models this paper provides a proof of the consistency and asymptotic normality of the quasi maximum likelihood estimator in garch 1 1 and igarch 1 1 models in contrast to the case of a unit root in the conditional mean the presence of a unit root in the conditional variance does not affect the limiting distribution of the estimators in both models estimators are normally distributed in addition a consistent estimator of the covariance matrix is available enabling the use of standard test statistics for inference

instances of the kaplansky lvov multilinear conjecture for polynomials of degree three abstract given a positive integer d the kaplansky lvov conjecture states that the set of values of a multilinear noncommutative polynomial f c x 1 x n on the matrix algebra m d c is a vector subspace in this article the technique of using one wiggle families of sylvester s clock and shift matrices is championed to establish the conjecture for polynomials f of degree three when d is even or d 17

limit laws for random matrices and free products in earlier articles we studied a kind of probability theory in the framework of operator algebras with the tensor product replaced by the free product we prove here that free random variables naturally arise as limits of random matrices and that wigner s semicircle law is a consequence of the central limit theorem for free random variables in this way we obtain a non commutative limit distribution of a general gaussian random matrix as an operator in a certain operator algebra wigner s law being given by the trace of the spectral measure of the selfadjoint component of this operator

a random matrix model from two dimensional yang mills theory we use fat graphs to give a unified treatment of various asymptotic freeness results in particular a random matrix model from two dimensional yang mills theory on the plane is presented

on fixed points of self maps of the free ball abstract in this paper we study the structure of the fixed point sets of noncommutative self maps of the free ball we show that for such a map that fixes the origin the fixed point set on every level is the intersection of the ball with a linear subspace we provide an application for the completely isometric isomorphism problem of multiplier algebras of noncommutative complete pick spaces

contemporaneous asymmetry in garch processes abstract the paper introduces a new concept of asymmetry contemporaneous asymmetry in conditional heteroskedasticity models we propose an original class of models aimed to capture the leverage effect contemporaneous asymmetry as well as time varying skewness and kurtosis not only past up and down moves have different impacts on the conditional variance but also positive and negative changes are governed by different conditional variances we give conditions for the existence of a second order and strictly stationary solution the paper also provides consistency results on the quasi maximum likelihood estimation finally an empirical analysis on the french cac 40 stock index is proposed

operator valued chordal loewner chains and non commutative probability abstract we adapt the theory of chordal loewner chains to the operator valued matricial upper half plane over a c algebra a we define an a valued chordal loewner chain as a subordination chain of analytic self maps of the a valued upper half plane such that each f t is the reciprocal cauchy transform of an a valued law t such that the mean and variance of t are continuous functions of t we relate a valued loewner chains to processes with a valued free or monotone independent increments just as was done in the scalar case by bauer 8 and schleisinger 48 we show that the loewner equation t f t z d f t z v t z when interpreted in a certain distributional sense defines a bijection between lipschitz mean zero loewner chains f t and vector fields v t z of the form v t z g t z where t is a generalized a valued law based on the loewner equation we derive a combinatorial expression for the moments of t in terms of t we also construct non commutative random variables on an operator valued monotone fock space which realize the laws t finally we prove a version of the monotone central limit theorem which describes the behavior of f t as t when t has uniformly bounded support

marginalization and contemporaneous aggregation in multivariate garch processes abstract we first of all show that contemporaneous aggregation of independent univariate garch processes yields a weak garch process as introduced by drost and nijman 1993 subsequently we analyze the dependence of the parameters in the aggregate on the parameters in the underlying models and show that the variance parameters after aggregation depend on the underlying variance and kurtosis parameters then we generalize the results by showing that a linear combination of variables generated by a multivariate garch process will also be weak garch finally we derive the marginal weak garch processes implied by several multivariate garch processes

the inverse function theorem and the jacobian conjecture for free analysis we establish an invertibility criterion for free polynomials and free functions evaluated on some tuples of matrices we show that if the derivative is nonsingular on some domain closed with respect to direct sums and similarity the function must be invertible thus as a corollary we establish the jacobian conjecture in this context furthermore our result holds for commutative polynomials evaluated on tuples of commuting matrices

pick interpolation for free holomorphic functions we give necessary and sufficient conditions to solve an interpolation problem for free holomorphic functions bounded in norm on a free polynomial polyhedron as an application we prove that every bounded holomorphic function on a polynomial polyhedron extends to a bounded free function

bootstrapping the box pierce q test a robust test of uncorrelatedness abstract this paper describes a test of the null hypothesis that the first k autocorrelations of a covariance stationary time series are zero in the presence of statistical dependence the test is based on the box pierce q statistic with bootstrap based p values the bootstrap is implemented using a double blocks of blocks procedure with prewhitening the finite sample performance of the bootstrap q test is investigated by simulation in our experiments the performance is satisfactory for samples of n 500 at this sample size the differences between the empirical and nominal rejection probabilities are essentially eliminated

compact sets in the free topology abstract subsets of the set of g tuples of matrices that are closed with respect to direct sums and compact in the free topology are characterized they are in a dilation theoretic sense the hull of a single point

multilinear function series and transforms in free probability theory abstract the algebra mul b of formal multilinear function series over an algebra b and its quotient symmul b are introduced as well as corresponding operations of formal composition in the setting of mul b the unsymmetrized r and t transforms of random variables in b valued noncommutative probability spaces are introduced these satisfy properties analogous to the usual r and t transforms the latter being just the reciprocal of the s transform but describe all moments of a random variable not only the symmetric moments the partially ordered set of noncrossing linked partitions is introduced and is used to prove properties of the unsymmetrized t transform

free biholomorphic functions and operator model theory ii let f f1 fn be an n tuple of formal power series in noncommutative indeterminates z1 zn such that f 0 0 and the jacobian detjf 0 0 and let g g1 gn be its inverse with respect to composition we assume that f and g have nonzero radius of convergence and g is a bounded free holomorphic function on the open unit ball b h n 1 where b h is the algebra of bounded linear operators an a hilbert space h in this paper several results concerning the noncommutative multivariable operator theory on the unit ball b h n 1 are extended to the noncommutative domain bf h x b h n g f x x and f x 1 for an appropriate evaluation x f x we develop an operator model theory and dilation theory for bf h where the associated universal model is an n tuple mz1 mzn of left multiplication operators acting on a hilbert space h2 f of formal power series all the results of this paper have commutative versions

toeplitz corona and the douglas property for free functions abstract the well known douglas lemma says that for operators a b on hilbert space that a a b b 0 implies b a c for some contraction operator c the result carries over directly to classical operator valued toeplitz operators simply by replacement of operator by toeplitz operator throughout free functions generalize the notion of free polynomials and formal power series and trace back to the work of j taylor in the 1970s they are of current interest in part because of their connections with free probability and engineering systems theory in this article for given free functions a and b with noncommutative domain k defined by free polynomial inequalities we obtain a sufficient condition in terms of the difference a a b b for the existence of a free function c taking contractive values on k such that b a c the connection to recent work of agler and mccarthy and their free toeplitz corona theorem is expounded

free pick functions representations asymptotic behavior and matrix monotonicity in several noncommuting variables r nevanlinna showed that elements of the pick class the set of complex analytic functions taking the upper half plane into itself have certain integral representations which reflect their asymptotic behavior at infinity later lowner connected the pick class to matrix monotone functions we generalize the nevanlinna representation theorems and lowner s theorem on matrix monotone functions to the free pick class the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive imaginary part which obey the free functional calculus

noncommutative polynomials nonnegative on a variety intersect a convex set abstract this paper gives a precise algebraic certificate for a noncommutative polynomial to be nonnegative on a free convex semialgebraic set intersect the variety of a free left ideal

bifurcation method and traveling wave solution to whitham broer kaup equation by using bifurcation method to whitham broer kaup shallow water equations bifurcation parameter sets are shown numbers of solitary waves kink waves and periodic waves are given under various parameter conditions all explicit formulas of solitary wave solutions kink wave solutions and periodic wave solutions are listed

algebras of noncommutative functions on subvarieties of the noncommutative ball the bounded and completely bounded isomorphism problem abstract given a noncommutative nc variety v in the nc unit ball b d we consider the algebra h v of bounded nc holomorphic functions on v we investigate the problem of when two algebras h v and h w are isomorphic we prove that these algebras are weak continuously isomorphic if and only if there is an nc biholomorphism g w v between the similarity envelopes that is bi lipschitz with respect to the free pseudo hyperbolic metric moreover such an isomorphism always has the form f f g where g is an nc biholomorphism these results also shed some new light on automorphisms of the noncommutative analytic toeplitz algebras h b d studied by davidson pitts and by popescu in particular we find that aut h b d is a proper subgroup of aut b d when d and the varieties are homogeneous we remove the weak continuity assumption showing that two such algebras are boundedly isomorphic if and only if there is a bi lipschitz nc biholomorphism between the similarity envelopes of the nc varieties we provide two proofs in the noncommutative setting our main tool is the noncommutative spectral radius about which we prove several new results in the free commutative case we use a new free commutative nullstellensatz that allows us to bootstrap techniques from the fully commutative case

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

implicit inverse function theorems for free noncommutative functions abstract we prove an implicit function theorem and an inverse function theorem for free noncommutative functions over operator spaces and on the set of nilpotent matrices we apply these results to study dependence of the solution of the initial value problem for odes in noncommutative spaces on the initial data and to extremal problems with noncommutative constraints

proper analytic free maps this paper concerns analytic free maps these maps are free analogs of classical analytic functions in several complex variables and are defined in terms of non commuting variables amongst which there are no relations they are free variables analytic free maps include vector valued polynomials in free non commuting variables and form a canonical class of mappings from one non commutative domain d in say g variables to another non commutative domain d in g variables as a natural extension of the usual notion an analytic free map is proper if it maps the boundary of d into the boundary of d assuming that both domains contain 0 we show that if f d d is a proper analytic free map and f 0 0 then f is one to one moreover if also g g then f is invertible and f 1 is also an analytic free map these conclusions on the map f are the strongest possible without additional assumptions on the domains d and d

arch models as diffusion approximations abstract this paper investigates the convergence of stochastic difference equations e g arch to stochastic differential equations as the length of the discrete time intervals between observations goes to zero these results are applied to the garch 1 1 model of bollerslev 1986 and to the ar 1 exponential arch model of nelson 1989 in their continuous time limits the conditional variance processes in these models have stationary distributions that are inverted gamma and lognormal respectively in addition a class of diffusion approximations based on the exponential arch model is developed

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

fixed point theorems for noncommutative functions abstract we establish a fixed point theorem for mappings of square matrices of all sizes which respect the matrix sizes and direct sums of matrices the conclusions are stronger if such a mapping also respects matrix similarities i e is a noncommutative function as a special case we prove the corresponding contractive mapping theorem which can be viewed as a new version of the banach fixed point theorem this result is then applied to prove the existence and uniqueness of a solution of the initial value problem for odes in noncommutative spaces as a by product of the ideas developed in this paper we establish a noncommutative version of the principle of nested closed sets

analytic mappings between noncommutative pencil balls abstract in this paper we analyze problems involving matrix variables for which we use a noncommutative algebra setting to be more specific we use a class of functions called nc analytic functions defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions we call such situations dimension free these types of functions have recently been used in the study of dimension free linear system engineering problems helton et al 2009 10 de oliviera et al 2009 8 in the earlier paper helton et al 2009 9 we characterized nc analytic maps that send dimension free matrix balls to dimension free matrix balls and carry the boundary to the boundary such maps we call nc ball maps in this paper we turn to a more general dimension free ball b l called a pencil ball associated with a homogeneous linear pencil l x a 1 x 1 a g x g a j c d d for x col x 1 x g c n n g define l x a j x j and let b l x c n n g l x 1 n n we study the generalization of nc ball maps to these pencil balls b l and call them pencil ball maps we show that every b l has a minimal dimensional in a certain sense defining pencil l up to normalization a pencil ball map is the direct sum of l with an nc analytic map of the pencil ball into the ball that is pencil ball maps are simple in contrast to the classical result of d angelo 1993 7 chapter 5 showing there is a great variety of such analytic maps from c g to c m when g m to prove our main theorem this paper uses the results of our previous paper helton et al 2009 9 plus entirely different techniques namely those of completely contractive maps what we do here is a small piece of the bigger puzzle of understanding how linear matrix inequalities lmis behave with respect to noncommutative change of variables

noncommutative reproducing kernel hilbert spaces abstract the theory of positive kernels and associated reproducing kernel hilbert spaces especially in the setting of holomorphic functions has been an important tool for the last several decades in a number of areas of complex analysis and operator theory an interesting generalization of holomorphic functions namely free noncommutative functions e g functions of square matrix arguments of arbitrary size satisfying additional natural compatibility conditions is now an active area of research with motivation and applications from a variety of areas e g noncommutative functional calculus free probability and optimization theory in linear systems engineering the purpose of this article is to develop a theory of positive kernels and associated reproducing kernel hilbert spaces for the setting of free noncommutative function theory

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples

asymptotic bias for quasi maximum likelihood estimators in conditional heteroskedasticity models virtually all applications of time varying conditional variance models use a quasimaximum likelihood estimator qmle consistency of a qmle requires an identification condition that the quasi log likelihood have a unique maximum at the true conditional mean and relative scale parameters we show that the identification condition holds for a non gaussian qmle if the conditional mean is identically zero or if a symmetry condition is satisfied without symmetry an additional parameter for the location of the innovation density must be added for identification we calculate the efficiency loss from adding such a parameter under symmetry when the parameter is not needed we also show that there is no efficiency loss for the conditional variance parameters of a garch process

analytic subordination for bi free convolution in this paper we study some analytic properties of bi free additive convolution both scalar and operator valued we show that using properties of voiculescu s subordination functions associated to free additive convolution of operator valued distributions simpler formulas for bi free convolutions can be derived we use these formulas in order to prove a result about atoms of bi free additive convolutions

bianalytic maps between free spectrahedra linear matrix inequalities lmis i d sum j 1 g a jx j sum j 1 g a j x j succeq 0 play a role in many areas of applications the set of solutions of an lmi is a spectrahedron lmis in dimension free matrix variables model most problems in linear systems engineering and their solution sets are called free spectrahedra free spectrahedra are exactly the free semialgebraic convex sets this paper studies free analytic maps between free spectrahedra and under certain generically valid irreducibility assumptions classifies all those that are bianalytic the foundation of such maps turns out to be a very small class of birational maps we call convexotonic the convexotonic maps in g variables sit in correspondence with g dimensional algebras if two bounded free spectrahedra mathcal d a and mathcal d b meeting our irreducibility assumptions are free bianalytic with map denoted p then p must after possibly an affine linear transform extend to a convexotonic map corresponding to a g dimensional algebra spanned by u i a 1 ldots u i a g for some unitary u furthermore b and ua are unitarily equivalent the article also establishes a positivstellensatz for free analytic functions whose real part is positive semidefinite on a free spectrahedron and proves a representation for a free analytic map from mathcal d a to mathcal d b not necessarily bianalytic another result shows that a function analytic on any radial expansion of a free spectrahedron is approximable by polynomials uniformly on the spectrahedron these theorems are needed for classifying free bianalytic maps

non commutative functions and the non commutative free levy hincin formula abstract the paper is discussing infinite divisibility in the setting of operator valued boolean free and more general c free independences particularly using hilbert bimodule and non commutative function techniques we obtain analogues of the levy hincin integral representation for infinitely divisible real measures

travelling wave solutions for a class of the generalized benjamin bona mahoney equations abstract in this paper a class of the generalized benjamin bona mahony gbbm equations with negative exponents are investigated by using the theory of bifurcations of dynamical systems as a result the dynamical behavior of different physical structure solitary patterns solitons perodic kink and anti kink wave solutions are obtained when parameters are varied under different parametric conditions various sufficient conditions to guarantee the existence of the above solutions are given and some exact solutions are shown

convergence in distribution of products of random matrices we consider a sequence a2 a2 of i i d nonnegative matrices of size d d and investigate convergence in distribution of the product mn a1 an when d 2 it is possible for mn to converge in distribution without normalization to a distribution not concentrated on the zero matrix several equivalent conditions for this to happen are given these lead to a fairly general family of examples these conditions can also be used to determine when the a s limit of 1 nlog mn equals the logarithm of the largest eigenvalue of e a1

bianalytic free maps between spectrahedra and spectraballs abstract linear matrix inequalities lmis are ubiquitous in real algebraic geometry semidefinite programming control theory and signal processing lmis with dimension free matrix unknowns are central to the theories of completely positive maps and operator algebras operator systems and spaces and serve as the paradigm for matrix convex sets the matricial feasibility set of an lmi is called a free spectrahedron in this article the bianalytic maps between a very general class of ball like free spectrahedra examples of which include row or column contractions and tuples of contractions and arbitrary free spectrahedra are characterized and seen to have an elegant algebraic form they are all highly structured rational maps in the case that both the domain and codomain are ball like these bianalytic maps are explicitly determined and the article gives necessary and sufficient conditions for the existence of such a map with a specified value and derivative at a point in particular this result leads to a classification of automorphism groups of ball like free spectrahedra the proofs depend on a novel free nullstellensatz established only after new tools in free analysis are developed and applied to obtain fine detail geometric in nature locally and algebraic in nature globally about the boundary of ball like free spectrahedra

strict stationarity of generalized autoregressive processes in this paper we consider the multivariate equation x n 1 a n 1 x n b n 1 with i i d coefficients which have only a logarithmic moment we give a necessary and sufficient condition for existence of a strictly stationary solution independent of the future as an application we characterize the multivariate arma equations with general noise which have such a solution

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given

null and positivstellensatze for rationally resolvable ideals abstract hilbert s nullstellensatz characterizes polynomials that vanish on the vanishing set of an ideal in c x in the free algebra c x the vanishing set of a two sided ideal i is defined in a dimension free way using images in finite dimensional representations of c x i in this article nullstellensatze for a simple but important class of ideals in the free algebra called tentatively rationally resolvable here are presented an ideal is rationally resolvable if its defining relations can be eliminated by expressing some of the x variables using noncommutative rational functions in the remaining variables whether such an ideal i satisfies the nullstellensatz is intimately related to embeddability of c x i into free skew fields these notions are also extended to free algebras with involution for instance it is proved that a polynomial vanishes on all tuples of spherical isometries iff it is a member of the two sided ideal i generated by 1 j x j x j this is then applied to free real algebraic geometry polynomials positive semidefinite on spherical isometries are sums of hermitian squares modulo i similar results are obtained for nc unitary groups

smooth and non smooth travelling wave solutions of generalized degasperis procesi equation abstract the dynamical behavior and special exact solutions of the generalized degasperis procesi equation u t c 0 u x u txx auu x 3 u x u xx uu xxx is analyzed by using bifurcation theory and the method of phase portraits analysis as a result the analytic expressions of peakon solutions compacton solutions and periodic cusp wave solutions are obtained and the condition under which peakon and compacton solutions appear are also given in the meantime the reason that solitary cusp wave and periodic cusp wave appear is given