irrational behavior of algebraic discrete valuations abstract we study algebraic discrete valuations dominating normal local domains of dimension two we construct a family of examples to show that the hilbert samuel function of the associated graded ring of the valuation can fail to be asymptotically of the form quasi polynomial plus a bounded function we also show that the associated multiplicity can be irrational or even transcendental

computer evaluation of kauffman polynomials by using gauss codes with a skein template algorithm homfly polynomials of knots and links are evaluated on computers by using an algebraic version with gauss codes of the skein template algorithm the computer codes are written with the language c whose structure is particularly suitable to such evaluations jones conway and alexander polynomials may then be readily obtained as special cases

an invariant of regular isotopy this paper studies a two variable laurent polynomial invariant of regular isotopy for classical unoriented knots and links this invariant is denoted lk for a link k and it satisfies the axioms 1 regularly isotopic links receive the same polynomial 2 lo0 1 3 l al l a l 4 l small diagrams indicate otherwise identical parts of larger diagrams regular isotopy is the equivalence relation generated by the reidemeister moves of type ii and type iii invariants of ambient isotopy are obtained from l by writhe normalization

ramification of valuations abstract in this paper we develop a very explicit theory of ramification of general valuations in algebraic function fields in characteristic zero and arbitrary dimension we obtain the strongest possible generalization of the classical ramification theory of local dedekind domains we further develop a ramification theory of algebraic functions fields of dimension two in positive characteristic we prove that local monomialization and simultaneous resolution hold under very mild assumptions and give pathological examples

counterexamples to local monomialization in positive characteristic a local monomialization of an algebraic map between varieties along a valuation is a sequence of blowups of nonsingular subvarieties along the valuation in the domain and target of the map such that the resulting map can be expressed by monomial inclusion in a suitable sense in this paper we give examples showing that a local monomialization does not always exist over fields of positive characteristic it was shown in earlier work of the author that a local monomialization always exists over fields of characteristic zero

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

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

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

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

semigroups of valuations on local rings ii given a noetherian local domain r and a valuation nu of its field of fractions which is nonnegative on r we derive some very general bounds on the growth of the number of distinct valuation ideals of r corresponding to values lying in certain parts of the value group gamma of nu we show that this growth condition imposes restrictions on the semigroups nu r setminus 0 for noetherian r which are stronger than those resulting from the previous paper of the first author given an ordered embedding gamma subset bf r h hbox rm lex where h is the rank of nu we also study the shape in bf r h of the parts of gamma which appear naturally in this study we give examples which show that this shape can be quite wild in a way which does not depend on the embedding and suggest that it is a good indicator of the complexity of the semigroup nu r setminus 0

analysis of slip planes in three dimensional solids abstract three dimensional solids in which displacement jumps develop across planes are analysed critical to such analysis is the identification of the correct orientation of the discontinuity plane to do this strain softening continuum relationships are examined for the limit case of a strong displacement discontinuity which then yields a localisation condition for the introduction of a displacement jump a numerical procedure is proposed to correctly orientate the discontinuity plane in space for the von mises and drucker prager yield criteria traction displacement relationships are developed and implemented by embedding the slip planes in conventional solid finite elements this makes the analysis ideal for large three dimensional calculations the performance of the developed discrete constitutive models and that of the embedded discontinuity elements in three dimensions are scrutinised with special attention paid to mesh objectivity and volumetric locking

valuation semigroups of two dimensional local rings we consider the question of when a semigroup is the semigroup of a valuation dominating a two dimensional noetherian domain giving some surprising examples we give a necessary and sufficient condition for the pair of a semigroup s and a field extension l k to be the semigroup and residue field of a valuation dominating a regular local ring r of dimension two with residue field k generalizing the theorem of spivakovsky for the case when there is no residue field extension

combinatorial expression for universal vassiliev link invariant a general model similar to r matrix type models for link invariants is constructed it contains all r matrix invariants and is a generating function for universal vassiliev link invariants this expression is simpler than kontsevich s expression for the same quantity because it is defined combinatorially and does not contain any integrals except for an expression for the universal drinfeld s associator

extension d une valuation we want to determine all the extensions of a valuation v of a field k to a cyclic extension l of k i e l k x is the field of rational functions of x or l k is the finite separable extension generated by a root 0 of an irreducible polynomial g x in two articles from 1936 saunders maclane has introduced the notions of key polynomial and of augmented valuation for a given valuation of k x and has shown how we can recover any extension to l of a discrete rank one valuation v of k by a countable sequence of augmented valuations with i n the valuation i is defined by induction from the valuation i 1 from a key polynomial o i and from the value oi in this article we study some properties of the augmented valuations and we generalize the results of maclane to the case of any valuation v of k for this we need to introduce simple admissible families of augmented valuations a a where a is not necessarily a countable set and to define a limit key polynomial and limit augmented valuation for such families then any extension to l of a valuation on k is again a limit of a family of augmented valuations we also get a factorization theorem which gives a description of the values f for any polynomial f in k x

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

an oriented state model for the jones polynomial and its applications alternating links in this paper we define a polynomial invariant of regular isotopy g l for oriented knot and link diagrams l from g l by multiplying it by a normalizing factor we obtain an ambient isotopy invariant n l for oriented knots and links we compare the polynomial n l with the original jones polynomial and with the normalized bracket polynomial we show that the polynomial n l yields the jones polynomial and the normalized bracket polynomial as examples we give the polynomial g l of some knot and link diagrams and compute the polynomial g l for torus links of type 2 n and applying computer algebra maple techniques we calculate the polynomial g l of torus links of type 2 n furthermore we give its applications to alternating links

valuations in algebraic field extensions let k l be an algebraic field extension and a valuation of k the purpose of this paper is to describe the totality of extensions of to l using a refined version of maclane s key polynomials in the basic case when l is a finite separable extension and rk 1 we give an explicit description of the limit key polynomials which can be viewed as a generalization of the artin schreier polynomials we also give a realistic upper bound on the order type of the set of key polynomials namely we show that if chark 0 then the set of key polynomials has order type at most n while in the case chark p 0 this order type is bounded above by logpn 1 where n l k our results provide a new point of view of the well known formula j 1sejfjdj n and the notion of defect

strain localization in frictional materials exhibiting displacement jumps abstract this paper presents a mathematical model for analyzing strain localization in frictional solids exhibiting displacement jumps precise conditions for the appearance of slip lines including their initiation and evolution are outlined for a rate independent strain softening drucker prager model and explicit analytical expressions are used to describe the orientation of the slip line a stress displacement relation obtained through the consistency condition is also formulated to describe the quasi static response in the post localization regime the mathematical model which is cast within the framework of finite element analysis employing the assumed enhanced strain method circumvents mesh dependence issues often associated with rate independent plasticity models it is shown that the enhancement equation is nothing else but the consistency condition imposed on the band numerical examples involving plane strain compression are described to demonstrate objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions

knot invariants and cellular automata abstract there is a deep connection between the theory of invariants of knots in three dimensions and certain classes of solvable models in statistical mechanics we study the consequences of this for a class of cellular automaton systems knot invariants furnish both a non perturbative framework for the classification and analysis of such models and a primary tool with which to study the emergence of thermodynamic behavior in extended systems

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

a rigorous bound on error in backward difference elastoplastic time integration abstract the finite element analysis of elastoplastic structures requires in general a time stepping procedure and in most cases the integration of the constitutive law within each time step has to be carried out by numerical integration the error associated to this numerical integration depends on the degree of non linearity of the structural response and can be used as an indicator for the adaptive definition of the time step size based on martin s and ortiz theorem on minimum total work a simple estimate of the integration error associated to a backward difference scheme for elastoplastic models is derived it is shown that the proposed estimate is a rigorous upper bound on the error in the case of assigned constant strain rate finally a simple strategy for the automatic definition of the time step size is proposed the estimator and the adaptive strategy are validated by application to problems with a perfectly plastic material model

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

a new exact integration method for the drucker prager elastoplastic model with linear isotropic hardening abstract this paper presents the exact stress solution of the non associative drucker prager elastoplastic model governed by linear isotropic hardening rule the stress integration is performed under constant strain rate assumption and the derived formulas are valid in the setting of small strain elastoplasticity theory based on the time continuous stress solution a complete discretized stress updating algorithm is also presented providing the solutions for the special cases when the initial stress state is located in the apex and when the increment produces a stress path through the apex explicit expression for the algorithmically consistent tangent tensor is also derived in addition a fully analytical strain solution is also derived for the stress driven case using constant stress rate assumption in order to get a deeper understanding of the features of these solutions two numerical test examples are also presented

a finite element model for strain localization analysis of strongly discontinuous fields based on standard galerkin approximation abstract this paper presents a finite element model for strain localization analysis of elastoplastic solids subjected to discontinuous displacement fields based on standard galerkin approximation strain enhancements via jumps in the displacement field are captured and condensed on the material level leading to a formulation that does not require static condensation to be performed on the element level the mathematical formulation revolves around the dual response of a macroscopic point cut by a shear band which requires the satisfaction of the yield condition on the band as the same stress point unloads elastically just outside the band precise conditions for the appearance of slip lines including their initiation and evolution are outlined for a rate independent strain softening drucker prager model and explicit analytical expressions are used to describe the orientation of the slip line in a plane strain setting at post localization the stress point integration algorithm along the band is exact and amenable to consistent linearization numerical examples involving simple shearing of elastoplastic solids with deviatoric plastic flow as well as plane strain compression of dilatant cohesive frictional materials are presented to demonstrate absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions

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

the yang baxter equation and invariants of links on considere le fait que l on peut construire des invariants d isotopie p et f pour des liens en utilisant certaines solutions des equations de yang baxter

extending a valuation centred in a local domain to the formal completion let r m k be a local noetherian domain with eld of fractions k and r a valuation ring dominating r not necessarily birationally let jk k be the restriction of to k by denition jk is centered at r let r denote the m adic completion of r in the applications of valuation theory to commutative algebra and the study of singularities one is often induced to replace r by its m adic completion r and by a suitable extension to r p for a suitably chosen prime ideal p such that p r 0 the purpose of this paper is to give assuming that r is excellent a systematic description of all such extensions and to identify certain classes of extensions which are of particular interest for applications

on the integration schemes for drucker prager s elastoplastic models based on exponential maps rate plasticity equations for the case of drucker prager s model in small strain regime are considered by defining an augmented stress vector the formulations convert the original problem into a quasi linear differential equation system two new exponential mapping schemes for integrating model equations are proposed in addition two traditional schemes for solving the dynamical system in an explicit manner are discussed the two semi implicit schemes developed pose higher accuracy and better convergency error contours are provided for all four methods to display the accuracy of each scheme in order to compare the results these contours for the classical one step backward euler integration method are also displayed accuracy and efficiency along with the rate of convergency of the existing and the proposed methods are examined by numerical examples copyright 2007 john wiley sons ltd

knot polynomials and vassiliev s invariants a fundamental relationship is established between jones knot invariants and vassiliev s knot invariants since vassiliev s knot invariants have a firm grounding in classical topology one obtains as a result a first step in understanding the jones polynomial by topological methods