ratio asymptotic of hermite pade orthogonal polynomials for nikishin systems ii we prove ratio asymptotic for sequences of multiple orthogonal polynomials with respect to a nikishin system of measures such that for each k sk has constant sign on its support consisting on an interval on which almost everywhere and a set without accumulation points in

computation of conformal representations of compact riemann surfaces we find a system of two polynomial equations in two unknowns whose solution allows us to give an explicit expression of the conformal representation of a simply connected three sheeted compact riemann surface onto the extended complex plane this function appears in the description of the ratio asymptotic of multiple orthogonal polynomials with respect to so called nikishin systems of two measures

holomorphic kernels and commuting operators necessary and sufficient conditions in terms of operator polynomials are obtained for an m tuple t tl t l of commuting bounded linear operators on a separable hilbert space x to extend to an m tuple s s1 sm of operators on some hilbert space x where each s is realized as a representation of the adjoint of a multiplication operator on the tensor product of a special type of functional hilbert spaces a1so necessary and sufficient conditions in terms of operator polynomials are obtained for t to have a commuting normal extension o introduction in this paper some results in 1 and 2 for a single bounded linear operator t on a separable hilbert space x are generalized to m commuting operators on x in 1 agler introduces a special class of functional hilbert spaces and describes conditions under which an operator t on x extends to m where m denotes the multiplication operator on x and m denotes the countable direct sum of m with itself special cases of spaces x of which the classical hardy space is the prototype are considered in 1 2 the relevant conditions for the kind of extension of t referred to above are expressed in terms of the positivity of certain operator polynomials involving t and t these conditions are closely related to the reproducing kernel associated with the space s in 2 necessary and sufficient conditions are also given for a contraction t to be subnormal these conditions are really the requirement that a certain sequence of polynomials in t and t be positive it is natural to seek generalizations of these results to m commuting operators t1 tm on x an appropriate model for this generalization is obtained by constructi ng a finite tensor product of spaces s and exploiting the well known fact that the reproducing kernel of such a tensor product is the product of the reproducing kernels of the illdividual spaces suitable modifications of the reproducing kernels which render them holomorphic on the unit polydisc can be used to describe analogous extension results for t t1 tm the question of t having a commuting normal extension n n1 nm turns out to have a direct link with the multi dimensional hausdorff moment problem from the theory of probability received by the editors march 21 1986 1980 mathematics subject classification primary 47a20 secondary 47b20 47b38

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

a vanishing viscosity approach to quasistatic evolution in plasticity with softening we deal with quasistatic evolution problems in plasticity with softening in the framework of small strain associative elastoplasticity the presence of a nonconvex term due to the softening phenomenon requires a nontrivial extension of the variational framework for rate independent problems to the case of a nonconvex energy functional we argue that in this case the use of global minimizers in the corresponding incremental problems is not justified from the mechanical point of view thus we analyse a different selection criterion for the solutions of the quasistatic evolution problem based on a viscous approximation this leads to a generalized formulation in terms of young measures developed in the first part of the paper in the second part we apply our approach to some concrete examples

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

real algebraic geometry for matrices over commutative rings abstract we define and study preorderings and orderings on rings of the form m n r where r is a commutative unital ring we extend the artin lang theorem and krivine stengle stellensatze both abstract and geometric from r to m n r this problem has been open since the seventies when hilbert s 17th problem was extended from usual to matrix polynomials while the orderings of m n r are in one to one correspondence with the orderings of r this is not true for preorderings therefore our theory is not morita equivalent to the classical real algebraic geometry

a variational formulation of rate independent phase transformations using an extremum principle we propose a rate independent mesoscopic model for the hysteretic evolution of phase transformations in shape memory alloys the model uses the deformation and phase indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws using the associated functionals admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality

an adaptive finite element approximation of a variational model of brittle fracture the energy of the francfort marigo model of brittle fracture can be approximated in the sense of gamma convergence by the ambrosio tortorelli functional in this work we formulate and analyze two adaptive finite element algorithms for the computation of its local minimizers for each algorithm we combine a newton type method with residual driven adaptive mesh refinement we present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the ambrosio tortorelli functional

oscillations and concentrations in weak solutions of the incompressible fluid equations the authors introduce a new concept of measure valued solution for the 3 d incompressible euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations one application of the concepts developed here is the following important result a sequence of leray hopf weak solutions of the navier stokes equations converges in the high reynolds number limit to a measure valued solution of 3 d euler defined for all positive times the authors present several explicit examples of solution sequences for 3 d incompressible euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations an extensions of the concept of young measure is developed to incorporate these complex phenomena in the measure valued solutions constructed here

quasistatic crack growth in nonlinear elasticity in this paper we prove a new existence result for a variational model of crack growth in brittle materials proposed in 19 we consider the case of n dimensional nonlinear elasticity for an arbitrary n 1 with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads both depending on time

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

a canonical family of multiple orthogonal polynomials for nikishin systems abstract for any pair of compact intervals of the real line 1 2 with 1 2 we obtain two probability measures 1 1 supported on 1 and 2 respectively such that the nikishin system n 1 1 has a sequence of monic multiple orthogonal polynomials which satisfy a four term recurrence relation with constant coefficients of period 2 the measures are obtained from the functions which give the ratio asymptotic of multiple orthogonal polynomials with respect to an arbitrary nikishin system n 1 2 on 1 2 such that i 0 a e on i i 1 2 the role of 1 1 is symmetric in the sense that the same construction is possible on 2 1 with n 1 1

localization study of a regularized variational damage model abstract the paper presents a detailed analysis and extended formulation of a rate independent regularized damage model proposed by mielke and roubicek 2006 localization properties are studied in the context of a simple one dimensional problem but the results reveal the fundamental features of the basic model and of its modified versions the initial bifurcation from a uniform solution is described analytically while the complete failure process is studied numerically modifications of the regularizing term and of the dissipation distance are introduced and their effect on the global response is investigated it is shown that with a proper combination of model parameters a realistic shape of the load displacement diagram can be achieved and pathological effects such as extremely brittle response or expansion of the damage zone accompanied by stress locking can be eliminated

a real nullstellensatz for free modules abstract let a be the algebra of all n n matrices with entries from r x 1 x d and let g 1 g m f a we will show that f a v 0 for every a r d and v r n such that g i a v 0 for all i if and only if f belongs to the smallest real left ideal of a which contains g 1 g m here a left ideal j of a is real if for every h 1 h k a such that h 1 t h 1 h k t h k j j t we have that h 1 h k j we call this result the one sided real nullstellensatz for matrix polynomials we first prove by induction on n that it holds when g 1 g m f have zeros everywhere except in the first row this auxiliary result can be formulated as a real nullstellensatz for the free module r x 1 x d n

positivstellensatze for algebras of matrices abstract noncommutative positivstellensatze express positive elements of algebras in terms of sums of squares here positive elements can be defined by means of representations point evaluations or abstract orderings squares are elements of the form a a we prove various types of noncommutative positivstellensatze for matrix algebras m n a where a is an algebra with involution and subalgebras of m n a such as path algebras crossed product algebras and cyclic algebras the notion of a noncommutative sum of squares is proposed and new versions of positivstellensatze are proved

moment problems for operator polynomials abstract haviland s theorem states that given a closed subset k in r n each functional l r x r positive on pos k p r x p k 0 admits an integral representation by a positive borel measure schmudgen proved that in the case of compact semialgebraic set k it suffices to check positivity of l on a preordering t having k as the non negativity set further he showed that the compactness of k is equivalent to the archimedianity of t the aim of this paper is to extend these results from functionals on the usual real polynomials to operators mapping from the real matrix or operator polynomials into r m n r or b k

maximally dissipative local solutions to rate independent systems and application to damage and delamination problems abstract the system of two inclusions u e t u t z t 0 and r z z e t u t z t 0 with the dissipation potential r degree 1 homogeneous and with the stored energy e t separately convex is considered the relation between conventional weak solutions and local solutions is shown and a suitably integrated maximal dissipation principle is devised to select force driven local solutions and eliminate solutions with too early jumps as it may occur in energy driven ones this is illustrated on scalar examples an approximation by a simple and efficient semi implicit time discretization of the fractional step type is shown to converge to local solutions on the scalar examples the approximate solutions are shown to satisfy the integrated maximal dissipation principle asymptotically while in general it is devised only to serve as an a posteriori tool to justify or possibly adaptively adjust thus obtained approximate solutions as in fact force driven applications of such solutions are illustrated on specific examples from continuum mechanics at small strains involving inelastic processes in a bulk or on a surface namely damage and delamination

equilibrium of vector potentials and uniformization of the algebraic curves of genus 0 we consider equilibrium problems for the logarithmic vector potential related to the asymptotics of the hermite pade approximants solutions of such problems can be expressed by means of algebraic functions the goal of this paper is to describe a procedure for determining the algebraic equation for this function in the case when the genus of this algebraic function is equal zero using the coefficients of the equation we compute the extremal cuts of the riemann surfaces these cuts are attractive sets for the poles of the hermite pade approximants we demonstrate the method by an example of the equilibrium problem related to a special system that is called the angelesco system

on the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials we study the evolution of systems described by internal variables after the introduction of thermodynamic forces and fluxes both the dissipation and dissipation potential are defined then the principle of maximum dissipation pmd and a minimum principle for the dissipation potential are developed in a variational formulation both principles are related to each other several cases are shown where both principles lead to the same evolution equations for the internal variables however also counterexamples are reported where such an equivalence is not valid in this case an extended pmd can be formulated

symmetry preserving return mapping algorithms and incrementally extremal paths a unification of concepts in this work we seek to characterize the conditions under which an elastic plastic stress update algorithm preserves the symmetries inherent to the material response from a numerical standpoint the aim is to determine under what conditions a stress update algorithm produces symmetric consistent tangents when applied to materials obeying normality for the ideally plastic solid we show that only the fully implicit or closest point return mapping algorithm is symmetry preserving for hardening plasticity symmetry cannot be preserved in general unless suitable restrictions are imposed on the constitutive equations we show that these restrictions amount to the existence of a pseudo internal energy function acting as a joint potential for both the direction of plastic flow and the hardening moduli in view of the fact that holonomic methods based on incrementally extremal paths also result in update rules possessing a potential structure and hence in symmetric tangents we address the question of whether any connections exist between the two approaches we show that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence

quasistatic evolution problems for linearly elastic perfectly plastic materials the problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate independent processes existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation this approach provides a new approximation result for the solutions of the quasistatic evolution problem which are shown to be absolutely continuous in time four equivalent formulations of the problem in rate form are derived a strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

finsler s lemma for matrix polynomials abstract finsler s lemma characterizes all pairs of symmetric n n real matrices a and b which satisfy the property that v t a v 0 for every nonzero v r n such that v t b v 0 we extend this characterization to all symmetric matrices of real multivariate polynomials but we need an additional assumption that b is negative semidefinite outside some ball we also give two applications of this result to noncommutative real algebraic geometry which for n 1 reduce to the usual characterizations of positive polynomials on varieties and on compact sets

archimedean operator theoretic positivstellensatze abstract we prove a general archimedean positivstellensatz for hermitian operator valued polynomials and show that it implies the multivariate fejer riesz theorem of dritschel rovnyak and positivstellensatze of ambrozie vasilescu and scherer hol we also obtain several generalizations of these and related results the proof of the main result depends on an extension of the abstract archimedean positivstellensatz for algebras that is interesting in its own right

non convex potentials and microstructures in finite strain plasticity a mathematical model for a finite strain elastoplastic evolution problem is proposed in which one time step of an implicit time discretization leads to generally non convex minimization problems the elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations the results for a single slip system and von mises plasticity illustrate that finite strain elastoplasticity generates reduced problems with non quasiconvex energy densities and so allows for non attainment of energy minimizers and microstructures

a model for the quasi static growth of brittle fractures existence and approximation results we give a precise mathematical formulation of a variational model for the irreversible quasi static evolution of brittle fractures proposed by g a francfort and j j marigo and based on griffith s theory of crack growth in the two dimensional case we prove an existence result for the quasi static evolution and show that the total energy is an absolutely continuous function of time although we cannot exclude the possibility that the bulk energy and the surface energy may present some jump discontinuities this existence result is proved by a time discretization process where at each step a global energy minimization is performed with the constraint that the new crack contains all cracks formed at the previous time steps this procedure provides an effective way to approximate the continuous time evolution

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy

on thermomechanical restrictions of continua the central idea proposed here is that in entropy producing processes a specific choice from among a competing class of constitutive functions can be made so that the state variables evolve in a way that maximizes the rate of entropy production when attention is restricted to quadratic forms for the rate of entropy production the assumption leads to results that are fully in keeping with linear phenomenological relations that satisfy the onsager relations in other words the usual linear evolution laws such as fourier s law of heat conduction fick s law darcy s law newton s law of viscosity etc all corroborate this assumption we clarify the difference between the maximum rate of entropy production criterion that characterizes choices among constitutive relations and the minimum entropy production theorem due to onsager 1931 that characterizes steady states for special choices of the rate of entropy production we then show that for other forms of entropy production that are not quadratic for which the onsager relations and related theorems cannot be applied we can use the procedure described here to obtain nonlinear laws we demonstrate by means of an example that even yield type phenomena can be accommodated within this framework while they cannot within the framework of onsager we also discuss issues concerning constraints especially in thermoelasticity within the context of our ideas

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