embedding of the operator space oh and the logarithmic little grothendieck inequality we use voiculescu s concept of free probability to construct a completely isomorphic embedding of the operator space oh in the predual of a von neumann algebra we analyze the properties of this embedding and determine the operator space projection constant of oh n frac 1 108 sqrt frac n 1 ln n le inf p mathcal b ell 2 to oh n p 2 p left p right cb le 288 pi sqrt frac 2n 1 ln n the lower estimate is a recent result of pisier and shlyakhtenko that improves an estimate of order 1 1 lnn of the author the additional factor 1 sqrt 1 ln n indicates that the operator space oh n behaves differently than its classical counterpart ell 2 n we give an application of this formula to positive sesquilinear forms on mathcal b ell 2 this leads to logarithmic characterization of c algebras with the weak expectation property introduced by lance

a transference method in quantum probability abstract working with a rather general notion of independence we provide a transference method which allows to compare the p norm of sums of independent copies with the p norm of sums of free copies our main technique is to construct explicit operator space l p embeddings preserving independence to reduce the problem to l 1 where some recent results by the first named author can be used we find applications on noncommutative khintchine rosenthal type inequalities and on noncommutative l p embedding theory

pseudo localization of singular integrals and noncommutative calderon zygmund theory abstract the weak type 1 1 boundedness of singular integrals acting on matrix valued functions has remained open since the 1980s mainly because the methods provided by the vector valued theory are not strong enough in fact we can also consider the action of generalized calderon zygmund operators on functions taking values in any other von neumann algebra our main tools for its solution are two first the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases l 2 sense outside of the support of the function on which it acts this gives rise to a pseudo localization principle which is of independent interest even in the classical theory second we construct a noncommutative form of calderon zygmund decomposition by means of the recent theory of noncommutative martingales this is a corner stone in the theory as application we obtain the sharp asymptotic behavior of the constants for the strong l p inequalities also unknown up to now our methods settle some basics for a systematic study of a noncommutative calderon zygmund theory

gundy s decomposition for non commutative martingales and applications we provide an analogue of gundy s decomposition for l 1 bounded non commutative martingales an important difference from the classical case is that for any l 1 bounded non commutative martingale the decomposition consists of four martingales this is strongly related with the row column nature of non commutative hardy spaces of martingales as applications we obtain simpler proofs of the weak type 1 1 boundedness for non commutative martingale transforms and the non commutative analogue of burkholder s weak type inequality for square functions a sequence x n n ge 1 in a normed space mathrm x is called 2 co lacunary if there exists a bounded linear map from the closed linear span of x n n ge 1 to l 2 taking each x n to the n th vector basis of l 2 we prove using our decomposition that any relatively weakly compact martingale difference sequence in l 1 mathcal m tau whose sequence of norms is bounded away from zero is 2 co lacunary generalizing a result of aldous and fremlin to non commutative l 1 spaces

convergence of the strongly damped nonlinear klein gordon equation in ℝn with radial symmetry on considere l equation de klein gordon fortement amortie dans r 3 dans le cas ou les donnees initiales possedent une symetrie radiale on construit une solution faible globale v de l equation non amortie pour des puissances elevees p qui peuvent etre approchees par les solutions globales fortes de l equation amortie

φ moment inequalities for independent and freely independent random variables abstract this paper is devoted to the study of moments of sums of independent freely independent random variables more precisely let f k k 1 n be a sequence of positive symmetrically distributed independent random variables and let be an orlicz function with 2 condition we provide an equivalent expression for the quantity e k 1 n f k in term of the sum of disjoint copies of the sequence f k k 1 n we also prove an analogous result in the setting of free probability furthermore we provide an equivalent characterization of sup 1 k n x k for positive freely independent random variables and also present some new results on free johnson schechtman inequalities in the quasi banach symmetric operator space

q gaussian processes non commutative and classical aspects we examine for 1 q 1 q gaussian processes i e families of operators non commutative random variables where the a t fulfill the q commutation relations for some covariance function equipped with the vacuum expectation state we show that there is a q analogue of the gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on q gaussian processes into corresponding and much simpler questions in the underlying hilbert space in particular we use this idea to show that a large class of q gaussian processes possesses a non commutative kind of markov property which ensures that there exist classical versions of these non commutative processes this answers an old question of frisch and bourret fb

overlapping block balanced canonical forms and parametrizations the stable siso case the balanced canonical form and parametrization of ober for the case of siso stable systems are extended to block balanced canonical forms and related input normal forms and parametrizations they form an overlapping atlas of parametrizations of the manifold of stable siso systems of given order this extends the usefulness of these parametrizations e g in gradient algorithms for system identification as an implication of our construction it follows that each of the subsets of the parametrization of r ober internat j control 46 1987 pp 643 670 corresponding to a choice for the structural indices is in fact an imbedded submanifold of the manifold of stable siso systems of fixed order

polynomial optimization and a jacobi davidson type method for commuting matrices in this paper we introduce a new jacobi davidson type eigenvalue solver for a set of commuting matrices called jdcomm used for the global optimization of so called minkowski norm dominated polynomials in several variables the stetter moller matrix method yields such a set of real non symmetric commuting matrices since it reformulates the optimization problem as an eigenvalue problem a drawback of this approach is that the matrix most relevant for computing the global optimum of the polynomial under investigation is usually large and only moderately sparse however the other matrices are generally much sparser and have the same eigenvectors because of the commutativity this fact is used to design the jdcomm method for this problem the most relevant matrix is used only in the outer loop and the sparser matrices are exploited in the solution of the correction equation in the inner loop to greatly improve the efficiency of the method some numerical examples demonstrate that the method proposed in this paper is more efficient than approaches that work on the main matrix standard jacobi davidson and implicitly restarted arnoldi as well as conventional solvers for computing the global optimum i e sostools gloptipoly and phcpack

grothendieck s theorem for operator spaces we prove several versions of grothendieck s theorem for completely bounded linear maps t e f when e and f are operator spaces we prove that if e f are c algebras of which at least one is exact then every completely bounded t e f can be factorized through the direct sum of the row and column hilbert operator spaces equivalently t can be decomposed as t t r t c where t r resp t c factors completely boundedly through a row resp column hilbert operator space this settles positively at least partially some earlier conjectures of effros ruan and blecher on the factorization of completely bounded bilinear forms on c algebras moreover our result holds more generally for any pair e f of exact operator spaces this yields a characterization of the completely bounded maps from a c algebra or from an exact operator space to the operator hilbert space oh as a corollary we prove that up to a complete isomorphism the row and column hilbert operator spaces and their direct sums are the only operator spaces e such that both e and its dual e are exact we also characterize the schur multipliers which are completely bounded from the space of compact operators to the trace class

weighted h 2 rational approximation and consistency we investigate consistency properties of rational approximation of prescribed type in the weighted hardy space h 2 mu for the exterior of the unit disk where mu is a positive symmetric measure on the unit circle mathbb t the question of consistency which is especially significant for gradient algorithms that compute local minima concerns the uniqueness of critical points in the approximation criterion for the case when the approximated function is itself rational in addition to describing some basic properties of the approximation problem we prove for measures mu having a rational function distribution weight with respect to arclength on mathbb t that consistency holds only under rather restricted conditions

rosenthal type inequalities for free chaos let a denote the reduced amalgamated free product of a family a 1 a 2 an of von neumann algebras over a von neumann subalgebra b with respect to normal faithful conditional expectations e k a k b we investigate the norm in lp a of homogeneous polynomials of a given degree d we first generalize voiculescu s inequality to arbitrary degree d 1 and indices 1 p this can be regarded as a free analogue of the classical rosenthal inequality our second result is a length reduction formula from which we generalize recent results of pisier ricard and the authors all constants in our estimates are independent of n so that we may consider infinitely many free factors as applications we study square functions of free martingales more precisely we show that in contrast with the khintchine and rosenthal inequalities the free analogue of the burkholder gundy inequalities does not hold in l a at the end of the paper we also consider khintchine type inequalities for shlyakhtenko s generalized circular systems

johnson schechtman inequalities for noncommutative martingales abstract in this paper we study johnson schechtman inequalities for noncommutative martingales more precisely disjointification inequalities of noncommutative martingale difference sequences are proved in an arbitrary symmetric operator space e m of a finite von neumann algebra m without making any assumption on the boyd indices of e we show that we can obtain johnson schechtman inequalities for arbitrary martingale difference sequences and that in contrast with the classical case of independent random variables or the noncommutative case of freely independent random variables the inequalities are one sided except when e l 2 0 1 as an application we partly resolve a problem stated by randrianantoanina and wu in 46 we also show that we can obtain sharp moment analogues for orlicz functions satisfying p convexity and q concavity for 1 p 2 q 2 and p 2 2 q this is new even for the classical case we also extend and strengthen the noncommutative burkholder gundy inequalities in symmetric spaces and in the moment case

johnson schechtman inequalities in the free probability theory abstract we introduce and study a free kruglov operator as an application of this study we prove an analogue of the johnson schechtman inequalities in the setting of free probability theory and an arbitrary symmetric operator space this extends and complements earlier l p results of junge et al 2007 12

cylindrical algebraic decomposition i the basic algorithm given a set of r variate integral polynomials a cylindrical algebraic decomposition cad of euclidean r space e r partitions e r into connected subsets compatible with the zeros of the polynomials by compatible with the zeros of the polynomials we mean that on each subset of e r each of the polynomials either vanishes everywhere or nowhere for example consider the bivariate polynomial y 4 2 y 3 y 2 3 x 2 y 2 x 4

operator khintchine inequality in non commutative probability abstract for the operator t sum n k 1 a k otimes r k where a k belongs to the schatten class s 2n and where r k are non commutative random variables with mixed moments satisfying a specific condition we prove the following khintchine inequality vert t vert 2n leq d 2n left left vert left sum n k001 a ka k right frac 1 2 right vert s 2n quad quad left vert left sum n k 1 a k a k right frac 1 2 right vert s 2n right we find the optimal constants d 2n in the case when r k are the q gaussian and circular random variables moreover we show that the moments of any probability symmetric measure appear as the optimal constants for some random variables

roots of bivariate polynomial systems via determinantal representations we give two determinantal representations for a bivariate polynomial they may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two parameter eigenvalue problem the first determinantal representation is suitable for polynomials with scalar or matrix coefficients and consists of matrices with asymptotic order n 2 4 where n is the degree of the polynomial the second representation is useful for scalar polynomials and has asymptotic order n 2 6 the resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10 as well as for polynomials with a small number of terms

optimal angle reduction a behavioral approach to linear system approximation we investigate the problem of optimal state reduction under minimization of the angle between system behaviors the angle is defined in a worst case sense as the largest angle that can occur between a system trajectory and its optimal approximation in the reduced order model this problem is analyzed for linear time invariant finite dimensional systems in a behavioral l2 setting without reference to input output decompositions and stability considerations the notion of a weakest past future link is introduced and it is shown how this concept is applied for the purpose of model reduction a method that reduces the state dimension by one is presented and shown to be optimal specific algorithms are provided for the numerical implementation of the approximation method the concepts and results are explicitly translated to an input output setting and related to balancing hankel norm reduction and normalized doubly coprime factorizations

identification of microwave filters by analytic and rational h2 approximation in this paper an original approach to frequency identification is explained and demonstrated through an application in the domain of microwave filters this approach splits into two stages a stable and causal model of high degree is first computed from the data completion stage then model reduction is performed to get a rational low order model in the first stage the most is made of the data taking into account the expected behavior of the filter a reduced order model is then computed by rational h 2 approximation a new and efficient method has been developed improved over the years and implemented to solve this problem it heavily relies on the underlying hilbert space structure and on a nice parameterization of the optimization set this approach guarantees the stability of the mimo approximant of prescribed mcmillan degree

noncommutative burkholder rosenthal inequalities we investigate martingale inequalities in noncommutative lp spaces associated with a von neumann algebra equipped with a faithful normal state we prove the noncommutative analogue of the classical burkholder inequality on the conditioned or little square function and extend the noncommutative burkholder gundy inequalities from comm math phys 189 1997 667 698 to this nontracial setting we include several related results

constructive algebra methods for the l 2 problem for stable linear systems abstract we investigate the feasibility of using computer algebra for solving l 2 system approximation problems the first order optimality conditions yield a set of polynomial equations which can in principle be solved using grobner basis methods a general solution along these lines would be tremendously useful although it does not appear to be feasible at present except for rather low mcmillan degrees we demonstrate that it can be feasible for specific examples in such cases global optima can be found reliably we show that the use of a schwarz like canonical form simplifies the structure of the problem

balanced realizations of discrete time stable all pass systems and the tangential schur algorithm in this paper the connections are investigated between two different approaches towards the parametrization of multivariable stable all pass systems in discrete time the first approach involves the tangential schur algorithm which employs linear fractional transformations it stems from the theory of reproducing kernel hilbert spaces and enables the direct construction of overlapping local parametrizations using schur parameters and interpolation points the second approach proceeds in terms of state space realizations in the scalar case a balanced canonical form exists that can also be parametrized by schur parameters this canonical form can be constructed recursively using unitary matrix operations here this procedure is generalized to the multivariable case by establishing the connections with the first approach it gives rise to balanced realizations and overlapping canonical forms directly in terms of the parameters used in the tangential schur algorithm

cylindrical algebraic decomposition ii an adjacency algorithm for the plane in part i of the present paper we defined cylindrical algebraic decompositions cad s and described an algorithm for cad construction in part ii we give an algorithm that provides information about the topological structure of a cad of the plane informally two disjoint cells in e r r 1 are adjacent if they touch each other formally they are adjacent if their union is connected in a picture of a cad eg fig 2 of part i it is obvious to the eye which pairs of cells are adjacent however the cad algorithm of part i does not actually produce this information nor did the original version of that algorithm in collins 1975

overlapping block balanced canonical forms for various classes of linear systems abstract through the use of balanced realizations it has been possible to derive parametrizations and canonical forms for various classes of minimal linear systems of given dimension a possible problem of these parametrizations is that they are not overlapping this could be a drawback for the application of balanced parametrizations in such areas as system identification model reduction and optimization it is the topic of this paper to derive overlapping parametrizations which are closely related to the existing balanced parametrizations we first introduce input normal canonical forms which are defined through a novel way of choosing nice selections of columns of the reachability matrix these canonical forms provide overlapping parametrizations in the sense that they form a real analytic atlas of the manifold of systems which are considered then we introduce so called block balanced input normal forms which use the previously constructed input normal forms as building blocks the classes of systems for which such parametrizations are given are the stable minimal systems positive real minimal systems bounded real minimal systems and the class of all minimal systems of given mcmillan degree the results include both the single input single output and the multivariable case in the single input single output case however the issue of choosing nice selections of columns does not occur therefore in this case the derivation and presentation of the results is considerably simplified

convergence properties of the strongly damped nonlinear klein gordon equation abstract we consider the nonlinear klein gordon equation u tt u i m 2 u u p 1 u 0 over a domain in r 3 in 2 aviles and sandefur established global existence of strong solutions when 0 for p 3 for each 0 let u be such a solution in this paper we show that if is a bounded domain with smooth boundary then there exists a sequence k and a global weak solution v of the undamped equation where 0 such that lim k 0 u k v in l 2 uniformly on any finite interval 0 t with t 0 we also show that if u is a strong local solution with 0 then for smooth enough initial data there exists an interval 0 t such that lim 0 u u uniformly in a much stronger norm we conclude by noting a connection between these two results and by discussing the difficulty of extending the first result to the case r 3 we also note in section 2 a natural extension of theorem 1 1 to bounded domains in r n n 3

non injectivity of the q deformed von neumann algebra in this paper we prove that the von neumann algebra generated by q gaussians is not injective as soon as the dimension of the underlying hilbert space is greater than 1 our approach is based on a suitable vector valued khintchine type inequality for wick products the same proof also works for the more general setting of a yang baxter deformation our techniques can also be extended to the so called q araki woods von neumann algebras recently introduced by hiai in this latter case we obtain the non injectivity under some asssumption on the spectral set of the positive operator associated with the deformation

non commutative martingale inequalities we prove the analogue of the classical burkholder gundy inequalites for non commutative martingales as applications we give a characterization for an ito clifford integral to be an lp martingale via its integrand and then extend the ito clifford integral theory in l2 developed by barnett streater and wilde to lp for all 1 p we include an appendix on the non commutative analogue of the classical fefferman duality between h1 and bmo

a new algorithm for l 2 optimal model reduction in this paper the quadratically optimal model reduction problem for single input single output systems is considered the reduced order model is determined by minimizing the integral of the magnitude squared of the transfer function error it is shown that the numerator coefficients of the optimal approximant satisfy a weighted least squares problem and on this basis a two step iterative algorithm is developed combining a least squares solver with a gradient minimizer convergence of the proposed algorithm to stationary values of the quadratic cost function is proved the formulation is extended to handle the frequency weighted optimal model reduction problem three examples demonstrate the optimization algorithm

a weak type inequality for non commutative martingales and applications we prove a weak type 1 1 inequality for square functions of non commutative martingales that are simultaneously bounded in l 2 and l 1 more precisely the following non commutative analogue of a classical result of burkholder holds there exists an absolute constant k 0 such that if m is a semi finite von neumann algebra and mn 1 1 is an increasing filtration of von neumann subalgebras of m then for any given martingale x xn 1 1 that is bounded in l 2 m l 1 m adapted to mn 1 1 there exist two martingale difference sequences a an 1 1 and b bn 1 1 with dxn an bn for every n 1 1 x n 1 aan 1 2