controlling chaotic dynamical systems we review the major ideas involved in the control of chaos we present the ott grebogi yorke ogy method of controlling chaos which is a particular case of the pole placement technique but which is the one leading to the shortest time to achieve the control of chaotic systems implementation using only measured time series in experimental settings is also described

cyclic plasticity using prager s translation rule and both nonlinear kinematic and isotropic hardening theory validation and algorithmic implementation abstract finite element analysis of structures under elasto plastic nonproportional cyclic loadings is useful in seismic engineering fatigue analysis and ductile fracture usual models with nonlinear stress strain curves in cyclic behavior are based on mroz multisurface plasticity bounding surface models or models derived from the armstrong frederick rule these models depart from the associative prager s rule with the main purpose of modeling aspects of cyclic nonlinear hardening in this paper we develop a model for cyclic plasticity within the framework of the associative classical plasticity theory using prager s rule accounting for anisotropic nonlinear kinematic hardening coupled with nonlinear isotropic hardening we include the validation of the theory against several uniaxial and multiaxial cyclic experiments and an efficient fully implicit radial return algorithm the parameters of the model are obtained directly by a discretization of the uniaxial stress strain behavior remarkably both the presented theory and the computational algorithm automatically recover classical bi linear plasticity and the krieg and key algorithm if the user prescribed stress strain curve is bilinear

multi dimensioned intertwined basin boundaries basin structure of the kicked double rotor abstract using numerical computations on a map which describes the time evolution of a particular mechanical system in a four dimensional phase space the kicked double rotor we have found that the boundaries separating basins of attraction can have different properties in different regions and that these different regions can be intertwined on arbitrarily fine scale in particular for the double rotor map if one chooses a restricted region of the phase space and examines the basin boundary in that region then either one observes that the boundary is a smooth three dimensional surface or one observes that the boundary is fractal with dimension d 3 9 and which of these two possibilities applies depends on the particular phase space region chosen for examination furthermore for any region no matter how small for which d 3 9 one can choose subregions within it for which d 3 hence d 3 9 region and d 3 region are intertwined on arbitrarily fine scale other examples will also be presented and analyzed to show how this situation can arise these include one dimensional map cases a map of the plane and the lorenz equations in one of our one dimensional map cases the boundary will be fractal everywhere but the dimension can take on either of two different values both of which lie between 0 and 1 these examples lead us to conjecture that basin boundaries typically can have at most a finite number of possible dimension values more specifically let these values be denoted d 1 d 2 d n choose a volume region of phase space whose interior contains some part of the basin boundary and evaluate the dimension of the boundary in that region then our conjecture is that for all typical volume choices the evaluated dimension within the chosen volume will be one of the values d 1 d 2 d n for example in our double rotor map it appears that n 2 and d 1 3 0 and d 2 3 9

the henselization of a valued division algebra the main result of this paper is the following if f v is a valued field fh its henselization d a division algebra with center f and d f then v extends to d iff d ffh is a division algebra it is also shown that if p char f p 1if char f 0 then d f d f d f pb for some nonnegative integer b

adaptive recognition and control of chaos abstract exploiting the information provided by the local variation rates of a dynamical system we show how an adaptive algorithm is able to recognize and stabilize the unstable periodic orbits embedded in a chaotic attractor through additive corrections to the dynamics the relative weight of each correction is adjusted upon the size of the local expansion rate application of this method to chaotic lorenz and four dimensional roessler systems is discussed

non canonical minkowski and pseudo riemann frames of plasticity models with anisotropic quadratic yield criteria abstract including the von mises the drucker prager and the hill there are many material plastic models with the yield criteria based on the quadratic yield functions in this paper we study the associated plastic flow model with an anisotropic yield criterion tc 1 2vt 2sy where is a suitable stress vector and v reflects the asymmetric yield strength in tension and compression and with an anisotropic elastic law k e in the elastic phase we first consider the term tc 1 to be a strain energy which means that the anisotropic tensor c is positive definite and coincides with the elastic tensor k under this additional assumption we can prove that the governing equations of plasticity can be linearized in a non canonical minkowski space the internal symmetry of the plastic equation is a special orthochronous pseudo linear group sl n 1 r a further variable transformation reveals a canonical minkowski structure of which the plastic equation exhibits the proper orthochronous lorentz group symmetry soo n 1 then we release the constraint by allowing the above c 1 to be free which may be positive non positive or even singular for this case the plastic equation becomes highly non linear we approach this problem by converting the non linear constitutive equations to a lie system x a x t x on a pseudo riemann manifold where a sl n 1 r is a real lie algebra of the special orthochronous pseudo linear group sl n 1 r the underlying space in the plastic phase is a cone with the metric tensor indefinite having a signature n 1 and also dependent on the temporal component based on these symmetry groups the exact solutions or the cayley transform are developed which can update the stress point exactly on the yield surface at every time increment without any iteration as numerical examples we calculate the stress responses for the drucker prager and hill models

spectre premier de oq mn k image canonique et separation normale abstract given any commutative field k denote r oq mn k the coordinate ring of quantum n n matrices over k and assume q is a nonzero element in k which is not a root of unity recall that r is generated by n2 variables xi i 1 n 2 subject only to the following relations if x y z t is any 2 2 sub matrix of x x i then a yx q 1xy zx q 1xz tz q 1zt ty q 1yt zy yz b tx xt q q 1 yz denote r the k algebra generated by the same variables xi subject to the same relations except relations b which are replaced by c tx xt so that r is just the algebra of regular functions on some quantum affine space of dimension n2 over k the theory of derivative elimination defines a natural embedding spec r spec r and asserts that the canonical image spec r is a union of strata spec w r in the sense of goodearl letzter in cms conf proc vol 22 1998 39 58 where w describes some subset w of p 1 n 2 the sets spec w r 1 spec w r w w define the goodearl letzter h stratification of spec r in the sense of goodearl letzter trans amer math soc 352 2000 1381 1403 in this paper we give the precise description of the set w and we compute its cardinality using that description and the derivative elimination algorithm we can verify theorems 6 3 1 6 3 2 that h spec r has an h normal separation in the sense of goodearl in lecture notes in pure and appl math 210 2000 205 237 so that spec r has normal separation in the sense of brown goodearl trans amer math soc 348 1996 2465 2502 this property was conjectured by k brown and k goodearl since r is auslander regular and cohen macaulay this implies by goodearl lenagan j pure appl algebra 111 1996 123 142 that r is catenary and satisfies the tauvel s height formula

a semi analytical integration method for j2 flow theory of plasticity with linear isotropic hardening abstract in this paper an exact time integration scheme is presented for von mises elastoplasticity with linear isotropic hardening at small deformations the method is based on the constant strain rate assumption which is widely accepted in displacement based finite element applications the deviatoric form of the rate equation using the method proposed by krieg and krieg in r d krieg d b krieg accuracies of numerical solution methods for the elastic perfectly plastic model j press vess technol trans asme 99 1977 510 515 is rewritten to a system of nonlinear odes with two scalar functions the radius of the mises circle and an angle obtained by using the scalar product of the stress deviator and the strain rate the time integration of these functions gives an exact solution in implicit form using the incomplete beta function in addition by the exact linearization of this new semi analytical solution a stress updating algorithm and the consistent tangent operator are derived the numerical performance and accuracy of the proposed method are illustrated on numerical examples moreover a comparison of the proposed method with the well known radial return method is included

reduced forms and reduced witt rings of higher level the present paper and its successor s deal with an extension of the socalled reduced theory of quadratic forms over a formally real field k 5 7 lo this theory deals with the reduced witt ring wred k w k nil w k and its natural embedding into c x z the ring of integer valued continuous functions on the space x x k of orderings of k recently it has been found that there is a far reaching generalization of the notion of ordering namely the orderings of higher level 11 the orderings of level s where s divides a given natural number n are related to sums of 2nth powers in the same way as the usual orderings are to sums of squares their intersection is the set of all sums of 2nth powers in k 11 in 6 orderings of levels were replaced by signatures of this level these are those characters x i s of order 2s whose kernels are additively closed where k k o it is exactly this change in point of view which leads to a natural definition of reduced witt rings of higher level let x x k denote the set of all signatures of k whose order divides 2n so that x k x k x being a closed subspace of the compact character group of x the set x k is also a compact topological space each element a e i induces a continuous function a x c in c x c defined by x x a where c is given the discrete topology by analogy with the natural representation of the ordinary reduced witt ring as a subring of 477 0021 8693 85 3 00

prandtl reuss elastoplasticity on off switch and superposition formulae abstract constitutive postulates for prandtl reuss elastoplasticity are selected based on them sufficient and necessary conditions for plastic irreversibility are found to be the yield condition and the straining condition this is then analyzed and it is pointed out that prandtl reuss elastoplasticity is nothing but a two phase linear system with an on off switch which is operated in the pace of an intrinsic measure of plastic irreversibility then the temporally global concept of the switch on time t on and the switch off time t off and their determination and bound estimation is developed owing to the implicit linearity superposition formulae for the stress response are discovered as an example the exact constitutive solutions for circular paths based on the superposition formulae are obtained and t on and t off are determined

an algorithm to control chaotic behavior in one dimensional maps abstract an algorithm is proposed whereby any chaotic transformation i e one merely possessing a dense orbit can be modified slightly to in order to attain a desired target probability density function f the algorithm can be used in practical examples such as controlling the long term population distribution of a species whose dynamics is described by a point transformation

non smooth multisurface plasticity and viscoplasticity loading unloading conditions and numerical algorithms rate independent plasticity and viscoplasticity in which the boundary of the elastic domain is defined by an arbitrary number of yield surfaces intersecting in a non smooth fashion are considered in detail it is shown that the standard kuhn tucker optimality conditions lead to the only computationally useful characterization of plastic loading on the computational side an unconditionally convergent return mapping algorithm is developed which places no restrictions aside from convexity on the functional forms of the yield condition flow rule and hardening law the proposed general purpose procedure is amenable to exact linearization leading to a closed form expression of the so called consistent algorithmic tangent moduli for viscoplasticity a closed form algorithm is developed based on the rate independent solution the methodology is applied to structural elements in which the elastic domain possesses a non smooth boundary numerical simulations are presented that illustrate the excellent performance of the algorithm

a lie poisson bracket formulation of plasticity and the computations based on the lie group so n abstract in this paper we develop a generalized hamiltonian formulation of a perfectly elastoplastic model which is a typical dissipative system on the cotangent bundle of the yield manifold a lie poisson bracket is used to construct the differential equations system the stress trajectory is a coadjoint orbit on the poisson manifold under a coadjoint action by the lie group so n the plastic differential equation is an affine non linear system of which a finite dimensional lie algebra can be constructed and the superposition principle is available for this system accordingly we can construct numerical schemes to automatically preserve the yield surface for perfect plasticity for isotropic hardening material as well as for an anisotropic elastic plastic model then we describe an anisotropic elastic plastic material model without entering the work hardening range and deforming under a specified dissipation rate which can be achieved through a stress dependent feedback control law of strain rate

utilization of soma and differential evolution for robust stabilization of chaotic logistic equation this paper deals with the utilization of two evolutionary algorithms self organizing migrating algorithm soma and differential evolution de for the optimization of the control of chaos this paper is aimed at an explanation on how to use evolutionary algorithms eas and how to properly define the advanced targeting cost function cf securing fast precise and mainly robust stabilization of selected chaotic system on a desired state for any initial conditions the role of ea here is as a powerful tool for an optimal tuning of control technique input parameters as a model of deterministic chaotic system the one dimensional discrete logistic equation was used the four canonical strategies of soma and six canonical strategies of de were utilized for each ea strategy repeated simulations were conducted to outline the effectiveness and robustness of used method and targeting cf securing robust solution satisfactory results obtained by both heuristic and the two proposed cost functions are compared with previous research given by different cost function designs

definition of a general implicit orthotropic yield criterion abstract the definition of an orthotropic yield criterion presents a serious challenge in the formulation of constitutive models based on such theories as elastoplasticity viscoplasticity damage etc the need to model the behavior of a real orthotropic material requires the formulation of orthotropic yield criteria and these may be difficult to obtain for metals orthotropic yield functions have been formulated by hill proc roy soc lon ser a 193 1948 281 j mech phys solids 38 3 1990 200 barlat int j plasticity 5 1989 51 7 1991 693 chu numisheet 93 1993 199 and dutko et al comput methods appl mech engrg 109 1993 73 but in many cases these functions do not describe the true behavior of the metal the situation is worse when one attempts to represent a nonmetal such as a polymer ceramic or composite in this paper we present a general definition of an explicit orthotropic yield criterion together with a general method for defining implicit orthotropic yield functions the latter formulation is based on the transformed tensor method whose principal advantage lies in the possibility of adjusting an arbitrary isotropic yield criterion to the behavior of an anisotropic material as example we choose the adjustment to the hill hoffman j comp materials 1 1967 200 and tsai wu j comp materials 5 1971 58 criteria but these particular cases serve to establish the methodology for achieving the desired function adjustment for any other well known criterion or experimental set of data obtained from laboratory

synchronization of chaotic trajectories using control we demonstrate that two identical chaotic systems can be made to synchronize by applying small judiciously chosen temporal parameter perturbations to one of them this idea is illustrated with a numerical example other issues related to synchronization are also discussed

real spectra of quantum groups abstract the only noncommutative ring for which the real spectrum has been described so far is the quantum affine ring r q x y j algebra 212 1999 190 the aim of this paper is to describe the real spectra of quantum affine rings k q x 1 x n where k is a formally real affine r algebra and q m n r as a by product we describe the real spectra of quantized enveloping algebra u q s l 2 r and quantum special linear group o q sl 2 r formal reality and semireality is characterized for the following classes of quantum groups quantum affine rings quantized enveloping algebras quantized function algebras quantized weyl algebras

systems of ordinary differential equations with nonlinear superposition principles abstract this work is concerned with the derivation of superposition rules which express the general solution of ordinary differential equations x x t x r n t r in terms of a finite number of particular solutions the point of departure is lie s criterion according to which such a rule exists if and only if the vector fields x t generate a finite dimensional lie algebra we provide three different constructive methods for deriving superposition rules and apply them to systems of coupled riccati equations of the projective and conformal types based respectively on the lie algebra sl n 1 r and o p 1 n p 1

effacement des derivations et spectres premiers des algebres quantiques abstract given any commutative field k and any iterated ore extension r k x1 x2 2 2 xn n n satisfying some suitable assumptions we construct the so called derivative elimination algorithm it consists of a sequence of changes of variables inside the division ring f fract r starting with the indeterminates x1 xn and terminating with new variables t1 tn these new variables generate some quantum affine space r such that f fract r this algorithm induces a natural embedding spec r spec r which satisfies the following property if p is in spec r then fract r p fract r p we study both the derivative elimination algorithm and natural embedding and use them to produce for the general case a common proof of the quantum gelfand kirillov property for the prime homomorphic images of the following quantum algebras u q w g b q w g w w r q g where g denotes any complex semi simple connected simply connected lie group with associated lie algebra g and weyl group w quantum matrices algebras quantum weyl algebras and quantum euclidean respectively symplectic spaces another application will be given in g cauchon j algebra to appear in the general case the prime spectrum of any quantum matrices algebra satisfies the normal separation property

renormalization approach to optimal limiter control in 1 d chaotic systems optimal limiter control of chaos in 1 d systems is described by flat topped maps when we study the properties of this control by a bifurcation analysis of the latter we find partial universal behavior the optimality of the control method is expressed by an exponentially fast control onto selected periodic orbits making targeting algorithms idle

witt rings and orderings of skew fields while ordered skew fields have been known for a long time they have received very little attention one purpose of this paper is to show that studying orderings and associated structures is one way of obtaining information about skew fields which are infinite dimensional over their centers all of our work is restricted to skew fields of characteristic not equal to 2 of course ordered skew fields have characteristic zero since they induce orderings on their centers the first example of a noncommutative ordered field appears to be due to d hilbert in grundlagen der geometrie 115 331 the example follows a proof of the fact that if a skew lield d has an archimedean ordering every positive element of d is less than some rational integer then d is commutative 15 321 another early result of a general nature was proved by a a albert i 1 based on earlier work of l dickson albert showed that an ordered skew field which is algebraic over its center must be commutative we shall generalize this result in the next section in 1952 t szele 28 extended to skew fields the basic results of artin and schreier on sums of squares 2 3 sums of products of squares for skew fields and a criterion for extension of orderings due to serre 1271 the second and third sections of our paper are devoted to extending these results and others to orderings of higher level in skew fields orderings of higher level were recently defined by e becker in 4 for commutative fields and appear to be of great importance in extending results for quadratic forms to forms of higher degree in particular 2 th powers take the place of squares in the artin schreier theory in fact in 5 1 he has extended this to arbitrary even powers but the valuation theory needed exceeds that which we have developed in section 3 for skew fields section 2 covers the extension to skew

reconcile the perfectly elastoplastic model to simulate the cyclic behavior and ratcheting perfectly elastoplastic constitutive model is modified through a smoothing factor introduced by liu liu c s 2003 smoothing elastoplastic stress strain curves obtained by a critical modification of conventional models int j solids struct 40 2121 2145 the new model allows plasticity to happen in a non zero measure yield volume in stress space rather than that of conventional zero measure yield surface and within the yield volume the plastic modulus is varying continuously it endows a specific strain hardening rule of flow stress and is able to describe the phenomena of strain hardening cyclic hardening the bauschinger effect mean stress relaxation strain ratcheting out of phase hardening as well as erasure of memory in order to suppress the over prediction of ratcheting we consider a scalar function of smoothing factor which can simulate the saturation behavior of uniaxial multiaxial strain ratcheting these effects are demonstrated through numerical examples the existence of stress equilibrium point and limiting surface is a natural result without requiring an extra design moreover the non linear constitutive equations can be converted into a linear system for augmented stress in the minkowski space of which the symmetry group is a proper orthochronous lorentz group soo 5 1 the augmented stress is a time like vector moving on hyperboloids inside the cone when taking the prager kinematic hardening rule into account we can simulate some cyclic behaviors of sae 4340 and grade 60 steels within a certain accuracy through the use of only three material constants and a fixed smoothing factor to simulate the ratcheting behaviors of ss304 stainless steel we allow the smoothing factor to be an exponential decaying function of k 2005 elsevier ltd all rights reserved

optimization of local control of chaos by an evolutionary algorithm an evolutionary algorithm for optimizing local control of chaos is presented based on a lyapunov approach a linear control law and the state space region in which this control law is activated are determined in addition we study a relation between certain adjustable design parameters and a particular measure of the uncontrolled chaotic attractor in the state space region of control ssrc from this relation the objective function to be optimized is derived in that context we assume a linear control law to be given and optimize size and shape of the ssrc using an evolutionary algorithm it is shown by examples how the algorithm can also be applied to higher dimensional systems with possibly more than one positive lyapunov exponent 2000 elsevier science b v all rights reserved

n real valuations and the higher level version of the krull baer theorem abstract in comm algebra 29 2001 193 201 cimpric gave examples of division rings containing an ordering of level 2m but not of level m for m n his examples were quite complicated we give substantially simplified examples in section 2 in sections 3 and 4 we investigate this phenomenon using valuation theory we define almost real and n real valuations and study liftings of orderings from the residue division ring to the original division ring such liftings are not always possible as is the case in the commutative setting but we give a necessary and sufficient condition for a lifting to exist we also prove a suitable generalization of the baer krull theorem finally in the last section we use our examples and the theory developed to answer a question given by marshall and zhang j algebra 212 1 1999 190 207

on the consistency of nested surfaces models and their kinematic hardening rules abstract in this paper we briefly address the consistency of formulations for nested surfaces plasticity and their kinematic hardening translation rules some requirements for these formulations are then given it is shown and discussed that multilayer plasticity based on nested yield surfaces present some inconveniences when modelling multiaxial cyclic loading on the other hand the use of hardening surfaces instead of yield surfaces solves the problem partially it is also shown that multilayer plasticity based on the mroz kinematic rule yields clearly different multiaxial predictions when using the same uniaxial bilinear curve and different number of surfaces which is a priori inconsistent since the same monotonic stress strain curve should not result into a clearly different multiaxial behavior simply because of the discretization employed by the user it is demonstrated that in contrast with the mroz translation rule multilayer plasticity using the prager translation rule results in the same prediction regardless of the number of surfaces in which the hardening part of the bilinear curve is discretized this rule is also compliant with the principle of maximum dissipation it also allows for a vanishing elastic region without falling into theoretical or numerical difficulties hence it should not be discarded when developing advanced models

comments on the mroz multiple surface type plasticity models abstract several questions are addressed regarding the role of number of surfaces in the mroz type multiple surface models firstly increasing the number of surfaces improves the accuracy of the plastic modulus function but at the same time alters the translation behavior of surfaces two 90 out of phase axial torsion loading experiments are chosen to illustrate the number of surface influence on the model behavior the stresses exceed the experimental levels with increasing numbers of surfaces from 5 to 100 secondly for proportional loading the multiple surface models do not predict ratchetting progressive plastic strain accumulation in one direction however these models predict ratchetting for general nonproportional loading an ellipse shaped axial torsional loading path has been considered where predicted ratchetting rates far exceeded their experimental counter parts an explanation is forwarded to address these properties of the models it is further demonstrated that the multiple surface model of mroz and its modification by garud produce identical stress strain predictions when the number of surfaces exceeds a certain value for infinitesimal loading increment intersection of surfaces does not occur when using either the mroz or garud model however when finite loading increment is selected in numerical calculations the intersection problem arises in the mroz model

chain signatures and higher level witt rings in many respects becker s theory of orderings of higher level is very much reminiscent of the familiar artin schreier theory of ordered fields a few examples of similarities are fields with higher level orderings have real closures 1 3 orderings of higher level can be used to study sums of powers in fields 3 41 there is a theory of witt rings for fields with higher level preorderings 7 81 however among all these similarities one very obvious difference stands out a field with a higher level ordering has nonisomorphic real closures becker showed that in order to describe the isomorphism classes of real closures a whole sequence of higher level orderings is required 1 p 162 theorem 121 subsequently this has been relined and reformulated through the introduction of chains of higher level orderings 121 and chain signatures 27 see alo 10 11 i in view of this situation one should expect that if results from the usual theory of ordered fields level 1 carry over to the case of orderings of higher level then quite different results will also be brought in by the chains of higher level orderings the purpose of this paper is to look for such results in the theory of higher level witt rings developed recently by becker and rosenberg 7 81 to keep things from getting too complicated this will be done only for the case where all levels are powers of 2 thus throughout the term higher level will be synonymous with level 2 for some n iv w k denotes the witt ring of the field k with respect to the higher level preordering k c k2 nonzero sums of 2 th powers of 9 just as the use of signatures s leads to homomorphisms w 7 chain signatures lead to homomorphisms

quantification de la localisation de dixmier de u sln 1 c resume in this article we extend to the quantum case the classical result of dixmier according to which up to localization the enveloping algebra of sl n 1 c is a central polynomial extension of the enveloping algebra of some parabolic subalgebra in fact the latter has a semiinvariant element d by which it is sufficient to localize to obtain the result the methods we use here are quite different from those used by dixmier indeed in the quantum case the powers of the indeterminate q would make calculations similar to those of dixmier much too complicated on the other hand when we use the rosso form which allows us to work in the hopf dual of q sl n 1 c the calculations become much easier than they would be in q sl n 1 c

holomorphy rings and higher level orders on skew fields the intimate connections which exist between higher level orders sums of 2n th powers valuation theory and real valued places on a field have been explored extensively craven initiated the study of similar notions for skew fields in 4 in 12 we pursued these ideas but the best results required a 2 primary assumption this paper extends the work of 12 in particular we prove an artin schreier theorem for preordered skew fields namely that a preorder t in a skew field is the intersection of the kernels of all the signatures that contain t this allows us to obtain the results of 12 without the 2 primary assumption in particular we have that preordered skew fields give rise to spaces of signatures mulcahy s abstract setting for the theory of higher level forms and reduced witt rings our main tools are valuation theory and the kadison dubois representation theorem for partially ordered rings as in the field case we make use of the ring a associated to a preorder t which is the intersection of certain valuation rings along the way to our main theorem we prove some facts about noncommutative priifer rings and we show that a is a priifer ring while our study of preordered skew fields follows along lines which are familiar from the field context there are significant differences between the two theories for example there are skew fields which have no orders but admit valuation rings with ordered residue fields

structure theorems for ap rings abstract in new proofs of the structure theorems for witt rings the first author shows how the standard ring theoretic results on the witt ring can be deduced in a quick and elementary way from the fact that the witt ring of a field is integral and from the specific nature of the explicit annihilating polynomials he provides we will show in the present article that the same structure results hold for larger classes of commutative rings and not only for witt rings we will construct annihilating polynomials for these rings

lie symmetries of finite strain elastic perfectly plastic models and exactly consistent schemes for numerical integrations abstract for finite strain elastic perfectly plastic models the use of non spinning objective stress rates leads to cubic degree non linear differential equations as well as the coupling of deviatoric and volumetric constitutive equations the governing equations reduce to quadratic non linearity and the coupling disappears when spinning objective stress rates are used in order to effectively deal with these non linearities we first convert the non linear constitutive equations into a lie type system x ax a so 5 1 for x in the minkowski spacetime m 5 1 which has an indefinite metric g of signature 5 1 however for spinning objective stress rates a merely depends on t through deformation history and the above equation reduces to a time varying linear system the new representation admits a lie symmetry of the proper orthochronous lorentz group soo 5 1 in the plastic phase in terms of x the yield condition is transformed to a null cone condition x t gx 0 in m 5 1 due to this nullity we thus assign a quaternionic hermitian matrix to represent x and convert the six dimensional system to a more economic lower dimensional quaternionic two component spinor system q q sl 2 h in the two dimensional spinor space the lie symmetry is found to be sl 2 h the mapping of which onto soo 5 1 is derived in addition to the mapping between so 5 1 and sl 2 h an isomorphic mapping between sl 2 h and su 4 is also established the latter of which generates the group su 4 finally according to these lie symmetries the numerical schemes preserving the group properties are developed which satisfy the consistency condition 1 exactly for every time step without iterations at all

on behavior of perfect elastoplasticity under rectilinear paths abstract a perfect elastoplastic model for the relation of generalized stresses and their conjugate generalized strain rates is investigated in the paper the exact solutions are solved for the model subjected to rectilinear generalized strain paths 1 and the concepts of response subspace and visualization are introduced such that the n dimensional problem is reduced to a two dimensional problem in the response subspace the response path is found to be geometrically speaking a geodesic in the n dimensional closed ball b n of generalized stresses the phase portrait of dissipation is studied and it is a remarkable fact that in the response subspace the coordinate x is exactly the dissipation power per unit generalized strain rate when x x on see eqn 63 the generalized stress strain curves are classified by geometrical shapes into ten types in contrast to the superficial impression of only one type of shape an inclined straight line followed by a horizontal line the existence of a limit strength vector and a limit power of dissipation is demonstrated by the long term behavior based on the exact solutions and the existence together with the response s stability is further verified by lyapunov s direct method the closeness to the limit strength point is also estimated and the exact formula accounted for prestress is derived the dependence of the responses on the product space of initial values input paths property constants and closeness to the limit is thoroughly investigated with control parameters identified

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

the steady state responses of s d o f viscous elasto plastic oscillator under sinusoidal loadings abstract depending on the values of the parameters identified the long term behaviour of s d o f viscous elasto plastic oscillator under sinusoidal loadings may be periodic with or without intermediate unloading or elastic shakedown including purely elastic a new phase plane estimation method for the steady state elasto plastic solutions is presented we identify three dimensionless ratios namely damping ratio force ratio rf and frequency ratio rw as well as an elastic phase duration variable y the new estimate offers closed form formulae for the force ratio rf and for the ductility ratio in terms of rw and y applying numerical method to the function rf rf rw y we can obtain the inverse function y y rw rf such that the variation of in terms of rw and rf can be evaluated alternatively for a given ductility ratio we can solve rw y numerically for y and then obtain the curves of rf versus rw to meet specified ductility ratio the proposed method can estimate the steady state responses for any applied load and forcing frequency the results calculated are in very good agreement with the exact time marching solutions a simple criterion of parameters values for elastic shakedown is derived by which we can calculate the maximum driving force amplitude to avoid structures oscillating in the plastic range it is found that there exists a best driving force amplitude for maximizing dissipation efficiency the distribution of periodic points with and without intermediate unloading in the parametric plane rw 1 rf for given damping ratio is clarified the intermediate unloading motion region locates within the range of r w 1 2 3 and its size decreases when damping ratio increases

quotients premiers deoq mn k resume consider a fieldk some nonzero elementqofkwhich is not a root of unity and some nonnegative integern k r goodearl and e s letzter have proved 1991 prime factor algebras of the coordinate ring of quantum matrices preprint that any prime factor algebra of the coordinate ringoq m n k of quantumn nmatrices overkis a domain in that paper it is proved that the division ring of fractions of such a domain is always isomorphic to the division ring of fractions of the coordinate ring of some quantum space of dimension at mostn2over some fieldkwhich is an extension ofk a similar result is also proved for quantum weyl algebras

optimal control of nonlinear systems to given orbits abstract using optimal control techniques we derive and demonstrate the use of an explicit single step control method for directing a nonlinear system to a target orbit and keeping it there we require that control values remain near the uncontrolled settings the full nonlinearity of the problem in state space variables is retained the one step of the control is typically a composition of known or learned maps over a the time required to learn the state b the time to compute the control and c the time to apply the control no special targeting is required yet the time to control is quite rapid working with the dynamics of a well studied nonlinear electrical circuit we show how this method works efficiently and accurately in two situations when the known circuit equations are used and when control is performed only on a poincare section of the reconstructed phase space in each case because the control rule is known analytically the control strategy is computationally efficient while retaining high accuracy the target locations on the selected target trajectory at each control stage are determined dynamically by the initial conditions and the system dynamics and the target trajectory is an approximation to an unstable periodic orbit of the uncontrolled system a linear stability analysis shows that dissipation in the dynamical system is essential for reaching a controllable state

higher dimensional targeting this paper describes a procedure to steer rapidly successive iterates of an initial condition on a chaotic attractor to a small target region about any prespecified point on the attractor using only small controlling perturbations such a procedure is called targeting previous work on targeting for chaotic attractors has been in the context of one and two dimensional maps here it is shown that targeting can also be done in higher dimensional cases the method is demonstrated with a mechanical system described by a four dimensional mapping whose attractor has two positive lyapunov exponents and a lyapunov dimension of 2 8 the target is reached by making very small successive changes in a single control parameter in one typical case 35 iterates on average are required to reach a target region of diameter 10 sup minus 4 as compared to roughly 10 sup 11 iterates without the use of the targeting procedure

the real spectrum of a noncommutative ring spaces of orderings were introduced by the second author in a series of w x papers 15 19 and various structure results were obtained generalizing results proved earlier for formally real fields by various people e becker w x l brocker r brown t craven a prestel and n schwartz in 6 26 w x also see 20 23 25 craven and tschimmel prove that formally real skew w x w x fields give rise to spaces of orderings in 8 also see 9 kalhoff extends w x this to ternary fields also called planar ternary rings in 10 11 also see w x 28 kleinstein and rosenberg and knebusch respectively show how spaces of orderings arise in the study of real commutative semi local rings orderings on commutative rings have been studied extensively since the introduction of the real spectrum by coste and roy in the early 1980s

a new approach to controlling chaotic systems abstract given a piecewise monotonic transformation which is strongly transitive and preserves an invariant probability density function f a construction is described which modifies very slightly but which changes f significantly in order to possess desired properties this construction permits control of the global long term behavior of chaotic dynamical systems

optimal multi dimensional ogy controller abstract a technique which generalizes the standard ogy control scheme and may be applied to higher dimensional systems to stabilize both saddle nodes and fully unstable fixed points is presented the proposed method is designed for nonlinear flows rather than maps and also provides a powerful technique to assess the controllability of nonlinear systems with ogy type controllers an optimal ogy control scheme designed for larger dimensional systems having improved convergence properties compared to the standard ogy controller is also presented numerical examples for the well known van der pol duffing oscillator are presented to illustrate the proposed control schemes

comments on loss of strong ellipticity in elastoplasticity the loss of strong ellipticity is analyzed for a rate independent infinitesimal elastoplastic model this local stability condition corresponds to the loss of positive definiteness of the symmetric part of the acoustic tensor first in the case of multisurface plasticity some expressions for the plastic hardening moduli are obtained for various bifurcation criteria next explicit expressions for the critical plastic hardening modulus and the critical orientation are obtained in the case of single surface plasticity hill type comparison solid the analysis is based on a geometric method linear isotropic elasticity and a general nonassociative flow rule are assumed however the principal axes of the second order tensors of the plastic potential and yield surfaces gradients are coaxial it is shown that similar to the loss of ellipticity the direction of the critical orientation is identical to one of the principal directions except in the particular case where the gradient of the plastic potential and yield surfaces each have a double eigenvalue in particular explicit expression for the plastic hardening modulus using the same geometric method is also presented for the raniecki type comparison solid as an illustrative example the critical orientation for the loss of strong ellipticity and the classical shear band localization loss of ellipticity are compared for axially symmetric compression and tension

using comparison theorem to compare corotational stress rates in the model of perfect elastoplasticity abstract for the simple shear problem of a perfectly elastoplastic body we convert the non linear governing equations into a third order linear differential system then into a second order linear differential system and further into a sturm liouville equation thus sturm s comparison theorem can be employed and extended to compare the simple shear responses based on different objective corotational stress rates it is proved that the rates of jaumann green naghdi sowerby chu xiao bruhns meyers and lee mallett wertheimer render non oscillatory stress responses with the jaumann equation as a sturm majorant for the other four equations for an objective corotational stress rate with the general plane spin a sufficient non oscillation criterion is found to be that the plane spin must not exceed the shear strain rate

valuations and higher level orders in commutative rings valuation theory is one of the main tools for studying higher level orders and the reduced theory of forms over fields see for example br in mw the theory of higher level orders and reduced forms was generalized to rings with many units and many of the results for fields carried over to this setting while it seems desirable to extend these results further the techniques used for rings with many units will not work for general commutative rings at the same time there is a general theory of valuations in commutative rings see lm m and g which in ma was used to study orders and the reduced theory of quadratic forms over general commutative rings thus it seems natural to ask if the connections between valuations and higher level orders in fields exist in commutative rings in this paper we use valuation theory to study the space of orders and the reduced witt ring relative to a higher level preorder in a commutative ring as in ma we first localize our ring at a multiplicative set without changing the space of orders in order to make the valuation theory work better this is a standard idea from real algebraic geometry remarkably many of the notions methods and results for fields carry over to this new setting we define compatiblity between valuations and orders and preorders and the ring a t associated to a preorder t which turns out to be prufer ring as in the field case we define the relation of dependency on the set of valuations associated to a preorder and we use this to prove a decomposition theorem for the space of orders we can then apply this to show that under a certain finiteness condition the space of orders is equivalent to the space of orders of a preordered field

internal symmetry groups for the drucker prager material model of plasticity and numerical integrating methods abstract internal symmetries for the drucker prager material model and numerical methods based on them are investigated here we first convert the non linear constitutive equations to a lie type system x a x t x where a sl 5 2 r is a lie algebra of the special orthochronous pseudo linear group sl 5 2 r the underlying space of drucker prager model in the plastic phase is a pseudo sphere in the pseudo riemann manifold which is locally a pseudo euclidean space e 7 5 2 and whose metric tensor is indefinite having signature 5 2 and also depends on the temporal component then we split the drucker prager yield condition into two sub yield conditions which together with two integrating factors idea led us to derive two lie type systems in the product space m 5 1 m 1 1 the lie algebra is the direct sum so 5 1 so 1 1 and the symmetry group is thus soo 5 1 soo 1 1 these results are essential from computational aspect based on these symmetry groups exponential mappings are developed which update stress points exactly on the yield surface at every time increment without any iteration as an example the results calculated by using the two different group preserving schemes are compared to that calculated by the runge kutta scheme together with the newton raphson iterative solution of a non linear algebraic equation which enforcing the consistency condition

implicit j2 bounding surface plasticity using prager s translation rule a bounding surface j2 plasticity model that uses prager s translation rule is presented the model preserves masing s rules and is developed from the same ideas as classical infinitesimal j2 plasticity resulting in the same formulation with the exception of the algorithm for the computation of the hardening function instead of utilizing a loading surface as in a previous formulation hardening surfaces are introduced the formulation is similar to that of multilayer plasticity using prager s rule presented in previous work an implicit algorithm based on the radial return concept is used and the consistent elastoplastic tangent is also developed in closed form examples illustrating anisotropic behaviour are presented and compared to that predicted by a multilayer j2 plasticity model the model is also applied to a soil dynamics problem to show the robustness of the algorithm and its applicability to complex loading copyright 2002 john wiley sons ltd

corners in plasticity koiter s theory revisited a general theory for plastic loading at corners is presented that includes koiter s theory as a special case this theory is derived within a thermodynamic framework and includes non associated as well as associated theory the non associated theory even allows the number of potential functions to differ from the number of yield functions the properties of the matrix of plastic moduli as well as of another important matrix are discussed in detail and hardening perfect and softening plasticity are concisely defined the existence of limit points is also discussed the strain driven format turns out to be the most general moreover consistent loading and unloading criteria are established for general non associated plasticity an explicit criterion for uniqueness is derived and finally some of the general findings are illustrated by means of specific plasticity formulations often encountered in practice less

determinants of r fold symmetric multilinear forms in hi harrison has constructed and investigated higher degree forms and the witt grothendieck ring of forms we speculate that the signature theory of ordered fields may be extended to the signature theory of partially ordered fields which will be needed for higher degree forms cf br k it seems that higher degree forms and the witt grothendieck ring have important meaning for it in this paper we introduce a notion of determinant as an extension of ordinary determinant and an invariant of forms we shall use the terminologies and notations in h let r be a commutative ring with identity 1 for an r module e let 8 e x e x x e r be an r fold symmetric multilinear form that is 0 x x2 x 0 x x x for every permutation p of 1 2 r and 8 ax byl x2 x ao x

controlling chaotic transport through recurrence abstract transport times for a chaotic system are highly sensitive to initial conditions and parameter values we present a technique to find rough orbits epsilon chains that achieve a desired transport rapidly and which can be stabilized with small parameter perturbations the strategy is to build the epsilon chain from segments of a long orbit the point is that long orbits have recurrences in neighborhoods where faster orbits must also pass the recurrences are used as the switching points between segments the resulting epsilon chain can be refined by gluing orbit segments over the switching points providing that a local hyperbolicity condition is satisfied as an example we show that transport times for the standard map can be reduced by factors of 10 4 the techniques presented here can be easily generalized to higher dimensions and to systems where the dynamics is known only as a time series

controlling hamiltonian chaos the method for stabilizing an unstable periodic orbit in chaotic dynamical systems originally formulated by ott grebogi and yorke ogy is not directly applicable to chaotic hamiltonian systems the reason is that an unstable periodic orbit in such systems often exhibits complex conjugate eigenvalues at one or more of its orbit points in this paper we extend the ogy stabilization method to control hamiltonian chaos by incorporating the notion of stable and unstable directions at each periodic point we also present an algorithm to calculate the stable and unstable directions other issues specific to the control of hamiltonian chaos are also discussed

nonlinear action of lie groups and superposition principles for nonlinear differential equations abstract recently obtained results on superposition principles for systems of ordinary nonlinear equations are reviewed in particular the general solution of the matrix riccati equation is expressed in terms of five particular solutions the study of superposition laws is related to the problem of backlund transformations for nonlinear partial differential equations to the problem of transforming nonlinear equations into linear ones and to the classification of primitive filtered lie algebras

a representation theorem for higher level reduced witt rings in 6 becker and rosenberg defined the reduced witt ring wak of a formally real field k with respect to a higher level preorder tin k wak is a subring of the ring c x of continuous c valued functions on the space x of signatures of k which are trivial on t and is built up from higher level t forms over k this higher level reduced theory is a natural generalization of the reduced theory of quadratic forms over fields spaces of signatures were introduced in 18 to provide an abstract framework for the higher level reduced theory over fields along the lines of marshall s spaces of orderings 13 h treatment of the classical reduced theory of quadratic forms there are many advantages to this axiomatic approach it allows for a unified study of reduced witt rings over fields and skew fields powers has shown that becker and rosenberg s study of higher level reduced witt rings can also be carried out over skew fields 22 the absence of technical considerations such as valuation theory not only yields simpler proofs of known results but also new results about the theory over fields 183 the space of orderings version 151 of the result which we prove here has found application in real algebraic geometry and we hope that our abstract higher level result might also prove useful in this context the question which concerns us here is the so called representation problem given fe c x how can we tell if f f for some f w x this has been solved for the higher level reduced witt ring of a field by becker and rosenberg 6 see s also and in the case of spaces of orderings by marshall 15 both solutions being generalizations of the original representation theorems of becker and brlicker 4 5 31 and brown and marshall 9 4 21 in the context of the reduced quadratic form theory over fields under a 2 power assumption we simultaneously extend all of these representation theorems to our abstract higher level setting this entails studying fans and more importantly a generalization called quasifans we not only obtain a common proof of the representation 105 0021 8693 88 3 00