the whitham hierarchies reductions and hodograph solutions a general scheme for analysing reductions of whitham hierarchies is presented it is based on a method for determining the s function by means of a system of first order partial differential equations compatibility systems of differential equations characterizing both reductions and hodograph solutions of whitham hierarchies are obtained the method is illustrated by exhibiting solutions of integrable models such as the dispersionless toda equation heavenly equation and the generalized benney system

consistent tangent operator for plasticity models based on the plastic strain rate potential a stress updating integration algorithm is derived for use with plastic strain rate potentials and with the backward implicit euler method an explicit expression is also deduced for the consistent tangent modulus this analysis extends an earlier treatment developed for elastic perfectly plastic materials in that it applies to materials which harden isotropically the method is intended for use in finite element calculations that employ plasticity models based on the plastic strain rate potential it is particularly suitable for cases where the form of the plastic dual potential has been obtained from texture measurements or from crystallographic models that describe texture evolution during metal forming

constrained kp hierarchy and bi hamiltonian structures the kakomtsev petviashvili kp hierarchy is considered together with the evolutions of eigenfunctions and adjoint eigenfunctions constraining the kp flows in terms of squared eigenfunctions one obtains 1 1 dimensional integrable equations with scattering problems given by pseudo differential lax operators the bi hamiltonian nature of these systems is shown by a systematic construction of two general poisson brackets on the algebra of associated lax operators gauge transformations provide miura links to modified equations these systems are constrained flows of the modified kp hierarchy for which again a general description of their bi hamiltonian nature is given the gauge transformations are shown to be poisson maps relating the bi hamiltonian structures of the constrained kp hierarchy and the modified kp hierarchy the simplest realization of this scheme yields the akns hierarchy and its miura link to the kaup broer hierarchy

the r matrix theory and the reduction of poisson manifolds the relation between the r matrix approach to the construction of bi hamiltonian structures and the reduction of poisson manifolds is discussed the crucial role of the reduction process is emphasized by showing on the basis of specific examples that the knowledge of a r matrix may not be sufficient the reduction of the poisson structures related with r from the entire algebra to the lax submanifold may become an obstruction in constructing the bi hamiltonian structures of a specific hierarchy of integrable equations

generalized drinfeld sokolov reductions and kdv type hierarchies generalized drinfeld sokolov ds hierarchies are constructed through local reductions of hamiltonian flows generated by monodromy invariants on the dual of a loop algebra following earlier work of de groot et al reductions based upon graded regular elements of arbitrary heisenberg subalgebras are considered we show that in the case of the nontwisted loop algebra l gl n graded regular elements exist only in those heisenberg subalgebras which correspond either to the partitions ofn into the sum of equal numbersn pr or to equal numbers plus onen pr 1 we prove that the reduction belonging to the grade 1 regular elements in the casen pr yields thep p matrix version of the gelfand dickeyr kdv hierarchy generalizing the scalar casep 1 considered by ds the methods of ds are utilized throughout the analysis but formulating the reduction entirely within the hamiltonian framework provided by the classical r matrix approach leads to some simplifications even forp 1

classical r matrix theory of dispersionless systems ii 2 1 dimension theory a systematic way of constructing 2 1 dimensional dispersionless integrable hamiltonian systems is presented the method is based on the so called central extension procedure and classical r matrix applied to the poisson algebras of formal laurent series results are illustrated with the known and new 2 1 dimensional dispersionless systems

classical r matrix theory for bi hamiltonian field systems this is a survey of the application of the classical r matrix formalism to the construction of infinite dimensional integrable hamiltonian field systems the main point is the study of bi hamiltonian structures appropriate constructions on poisson noncommutative and loop algebras as well as the central extension procedure are presented the theory is developed for 1 1 and 2 1 dimensional field and lattice soliton systems as well as hydrodynamic systems the formalism presented contains sufficiently many proofs and important details to make it self contained and complete the general theory is applied to several infinite dimensional lie algebras in order to construct both dispersionless and dispersive soliton integrable field systems

analysis of the anisotropic viscoplastic damage response of composite laminates continuum basis and computational algorithms the mathematical structure underlying the rate equations of a recently developed constitutive model for the coupled viscoplastic damage response of anisotropic composites is critically examined in this regard a number of tensor projection operators have been identified and their properties were exploited to enable the development of a general computational framework for their numerical implementation using the euler fully implicit integration method in particular this facilitated i the derivation of explicit expressions of the consistent material tangent stiffnesses that are valid for both three dimensional as well as subspace e g plane stress formulations ii the implications of the symmetry or unsymmetry properties of these tangent operators from a thermodynamic standpoint and iii the development of an effective time step control strategy to ensure accuracy and convergence of the solution in addition the special limiting case of inviscid elastoplasticity is treated the results of several numerical simulations are given to demonstrate the effectiveness of the schemes developed

a plasticity model and algorithm for mode i cracking in concrete summary a class of plasticity models which utilize rankine s principal stress yield locus is formulated to simulate cracking in concrete and rock under monotonic loading conditions the formulation encompasses isotropic and kinematic hardenindsoftening rules and incremental flow theory as well as total deformation theory formats are considered an euler backward algorithm is used to integrate the stresses and internal variables over a finite loading step and an explicit expression is derived for a consistently linearized tangent stiffness matrix associated with the euler backward scheme particular attention is paid to the corner regime that is when the two major principal stresses become equal a detailed comparison has been made of the proposed plasticity based crack formulations and the traditional fixed and rotating smeared crack models for a homogeneously stressed sample under a non proportional loading path a comparison between the flow theory based plasticity crack models and experimental data has been made for a single edge notched plain concrete specimen under mixed mode loading conditions

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

on the integrability of 2 1 dimensional quasilinear systems a 2 1 dimensional quasilinear system is said to be integrable if it can be decoupled in infinitely many ways into a pair of compatible n component one dimensional systems in riemann invariants exact solutions described by these reductions known as nonlinear interactions of planar simple waves can be viewed as natural dispersionless analogues of n gap solutions it is demonstrated that the requirement of the existence of sufficiently many n component reductions provides the effective classification criterion as an example of this approach we classify integrable 2 1 dimensional systems of conservation laws possessing a convex quadratic entropy

universal anisotropic yield criterion based on superquadric functional representation part 1 algorithmic issues and accuracy analysis abstract a robust and efficient stress update algorithm for the anisotropic yield criterion represented in the form of superquadric function introduced by barlat et al is presented it relies on a discrete variational formulation of the lagrangian functional which results from the principle of maximum plastic dissipation numerical solution of the discretized equations of evolution is based on the operator split methodology and the newton raphson method a line search algorithm is implemented to overcome numerical instabilities and to expand the convergence region of the basic newton raphson solution procedure the consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discrete kuhn tucker condition numerical testing of the integration algorithm based on iso error maps is provided for several variants of the yield function

integrable equations on time scales integrable systems are usually given in terms of functions of continuous variables on r in terms of functions of discrete variables on z and recently in terms of functions of q variables on kq we formulate the gel fand dikii gd formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations in particular they yield integrable equations over integers difference equations and over q numbers q difference equations we formulate the gd formalism also in terms of shift operators for all regular discrete time scales we give a method allowing to construct the recursion operators for integrable systems on time scales finally we give a trace formula on time scales and then construct infinitely many conserved quantities casimirs of the integrable systems on time scales

waterbag reductions of the dispersionless discrete kp hierarchy we consider the dispersionless limit of the discrete kp hierarchy we describe a family of n parameter waterbag reductions of the hierarchy and give sufficient conditions for the reduced equations of motion to be hyperbolic using geogdzhaev s method we solve the initial value problem for the reduced system the solution is implicit in generalized hodograph form tsarev s p 1985 sov math dokl 31 488 the relationship of this form of the solution to the properties of the hierarchy the reduction and the initial data is discussed briefly indicating how this construction might be generalized

a return mapping algorithm for a class of cyclic plasticity models a return mapping algorithm for a class of cyclic plasticity models is presented the constitutive model includes a quadratic yield criterion and multi component formulations of non linear isotropic and kinematic hardening thus both initially isotropic and anisotropic materials can be described and a proper description of the hysteresis loops and the cyclic hardening response of materials under cyclic loading is obtained the return mapping algorithm employs the closest point projection scheme in combination with a decomposition method to obtain an efficient and robust integration of the constitutive model the consistent tangent operator is derived in closed form and is found to be unsymmetric due to the non linear evolution of the kinematic hardening terms the accuracy and robustness of the algorithm are assessed through numerical examples

on deformations of standard r matrices for integrable infinite dimensional systems simple deformations with a parameter of classical r matrices which follow from decomposition of appropriate lie algebras are considered as a result nonstandard lax representations for some well known integrable systems are presented and integrable evolution equations are constructed

classical r matrices and compatible poisson structures for lax equations on poisson algebras given a classical r matrix on a poisson algebra we show how to construct a natural family of compatible poisson structures for the hamiltonian formulation of lax equations examples for which our formalism applies include the benny hierarchy the dispersionless toda lattice hierarchy the dispersionless kp and modified kp hierarchies the dispersionless dym hierarchy etc

the bi hamiltonian structure of fully supersymmetric korteweg de vries systems the bi hamiltonian structure of integrable supersymmetric extensions of the korteweg de vries kdv equation related to then 1 and then 2 superconformal algebras is found it turns out that some of these extensions admit inverse hamiltonian formulations in terms of presymplectic operators rather than in terms of poisson tensors for one extension related to then 2 case additional symmtries are found with bosonic parts that cannot be reduced to symmetries of the classical kdv they can be explained by a factorization of the corresponding lax operator all the bi hamiltonian formulations are derived in a systematic way from the lax operators

weak leopoldt s conjecture for hecke characters of imaginary quadratic fields we give a proof of weak leopoldt s conjecture a la perrin riou under some technical condition for the p adic realizations of the motive associated to hecke characters over an imaginary quadratic fleld k of class number 1 where p is a prime 3 and where the cm elliptic curve associated to the hecke character has good reduction at the primes above p in k this proof makes use of the 2 variable iwasawa main conjecture proved by rubin thus we prove the jannsen conjecture for the above p adic realizations for almost all tate twists 1 the formulation of weak leopoldt s conjecture and weak jannsen s conjecture for hecke characters let us flrst introduce the p adic integer realizations associated to hecke characters the aim of this paper is verify for these realizations the weak leopoldt conjecture a la perrin riou and jannsen s conjecture for almost all tate twists let e be a flxed elliptic curve with cm which is deflned over an imaginary quadratic fleld k with cm by ok the ring of integers of k note that this implies cl k 1 associated to this elliptic curve there is a cm character of the imaginary quadratic fleld k with conductor f an ideal of ok that coincides with the conductor of the elliptic curve e let us consider once for all p be an odd prime where e has good reduction at the primes above p in k ordinary or supersingular we impose p 6 3 in order that any primitive p root of unity is in k thus in particular p dk the discriminant of k let us introduce the motives and the p adic realizations on which we study the conjectures consider the category of chow motives mq k over k with morphisms induced by graded correspondences in chow theory tensored with q then the motive of the elliptic curve e has a canonical decomposition h e q h 0 e q h 1 e q h 2 e q the motive h 1 e q has a multiplication by k 5 let us consider the motive wh 1 e q for w a strictly positive integer which has multiplication by tw wk observe that tw has a decomposition q

the moyal bracket and the dispersionless limit of the kp hierarchy a new lax equation is introduced for the kp hierarchy which avoids the use of pseudo differential operators as used in the sate approach this lax equation is closer to that used in the study of the dispersionless kp hierarchy and is obtained by replacing the poisson bracket with the moyal bracket the dispersionless limit under which the moyal bracket collapses to the poisson bracket is particularly simple

lie algebraic approach to the construction of 2 1 dimensional lattice field and field integrable hamiltonian equations two different methods for the construction of 2 1 dimensional integrable lattice field and field hamiltonian dynamical systems are presented the first method is based on the so called central extension procedure applied to the lie algebra of shift operators and the lie algebra of pseudodifferential operators the second method is the so called operand formalism both methods allow a construction of some new integrable nonlinear hamiltonian lattice field and field equations in 2 1 dimensional space

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

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

nonstandard drinfeld sokolov reduction subject to some conditions the input data for the drinfeld sokolov construction of kdv type hierarchies is a quadruplet where the are gradations of a loop algebra and is a semisimple element of the nonzero grade a new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented the proposal relies on splitting the grade zero part of into a vector space direct sum of two subalgebras this permits one to interpret certain gelfand dickey type systems associated with a nonstandard splitting of the algebra of pseudodifferential operators in the drinfeld sokolov framework

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

on the formulation and numerical solution of problems in anisotropic finite plasticity abstract this article discusses an extension of the green naghdi finite plasticity theory to explicitly include anisotropic effects in the stress response yield condition flow rule and hardening rule constitutive models are developed within the context of the extended theory and used in formulating implicit integration algorithms which inherit the characteristics of the classical return mapping scheme of isotropic plasticity representative numerical simulations demonstrate the applicability and predictive capacity of the proposed model in the presence of large plastic deformations

geometrically non linear and elastoplastic three dimensional shear flexible beam element of von mises type hardening material a three dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures the continuum type equations of plastic model of j2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses an effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so called multistep method which is a method of successive control of residuals on yield surfaces it avoids severe divergence when the displacement increments become large which is usual for the continuation methods material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix different from other elements the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion a great number of numerical examples are analysed and compared with the literature the proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre buckling and large deflection collapse of spatial framed structures

studies in material parameter sensitivity of softening solids a methodology is presented for the evaluation of the material parameter sensitivity of softening solids the algorithm is based on the differentiation of the implicit euler backward return mapping algorithm used for integration of the constitutive equations and the system of nonlinear algebraic equations obtained from the discretization in finite elements together with an arc length constraint the necessity of a stable equilibrium path and the meaning of left hand and right hand derivatives are discussed examples illustrate the theoretical developments as well as the capabilities of the algorithm to predict possible localization patterns

from dispersionless to soliton systems via weyl moyal like deformations the formalism of quantization deformation is reviewed and the weyl moyal like deformation is applied to systematic construction of the field and lattice integrable soliton systems from poisson algebras of dispersionless systems

numerical technique in plasticity including solution advancement control numerical techniques applied to the consistent formulation of plasticity which is based on convex analysis are investigated for each time step the stress is found as the projection in complementary energy of the elastic stress onto the set of plastically admissible stresses while the velocity field is the extremal of a non quadratic functional explicit formulas for von mises yield criterion with mixed hardening are developed and the nonlinear equations arising from finite element discretization are solved for comparison by a number of newton type iteration procedures with line search and are length control a few numerical examples with proportional and non proportional loading are analyzed

a return mapping algorithm for plane stress elastoplasticity an unconditionally stable algorithm for plane stress elastoplasticity is developed based upon the notion of elastic predictor return mapping plastic corrector enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of newton methods the accuracy of the algorithm is assessed by means of iso error maps the excellent performance of the algorithm for large time steps is illustrated in numerical experiments

topological landau ginzburg theory with a rational potential and the dispersionless kp hierarchy based on the dispersionless kp dkp theory we study a topological landau ginzburg lg theory characterized by a rational potential writing the dkp hierarchy in a general form treating all the primaries in an equal basis we find that the hierarchy naturally includes the dispersionless continuous limit of toda hierarchy and its generalizations having a finite number of primaries several flat solutions of the topological lg theory are obtained in this formulation and are identified with those discussed by dubrovin we explicitly construct gravitational descendants for all the primary fields giving a residue formulae for the 3 point functions of the fields we show that these 3 point functions satisfy the topological recursion relation the string equation is obtained as the generalized hodograph solutions of the dkp hierarchy which show that all the gravitational effects to the constitutive equations 2 point functions can be renormalized into the coupling constants in the small phase space

the dispersionless lax equations and topological minimal models it is shown that perturbed rings of the primary chiral fields of the topological minimal models coincide with some particular solutions of the dispersionless lax equations the exact formulae for the tree level partition functions of an topological minimal models are found the virasoro constraints for the analogue of the function of the dispersionless lax equation correspond ing to these models are proved

bi hamiltonian structures for integrable systems on regular time scales a construction of the bi hamiltonian structures for integrable systems on regular time scales is presented the trace functional on an algebra of pseudodifferential operators valid on an arbitrary regular time scale is introduced the linear poisson tensors and the related hamiltonians are derived the quadratic poisson tensors are given by the use of the recursion operators of the lax hierarchies the theory is illustrated by differential counterparts of ablowitz kaup newell segur and kaup broer hierarchies

on the birch and swinnerton dyer conjecture let f be a finite extension of the rational field q if e is an elliptic curve defined over f then the mordell weil group e f of points on e with coordinates in f is a finitely generated abelian group let l e f s be the hasseweil zeta function of e over f which at least in the case where e has complex multiplication is known to be an analytic function on the entire complex plane birch and swinnerton dyer have conjectured that l e f s has a zero at s l of order precisely equal to the rank of e f over z the strongest result known in support of this conjecture is the following theorem of coates and wiles 4

stress integration schemes for novel homogeneous anisotropic hardening model abstract numerical formulations and implementation of stress integration algorithms in the elasto plastic finite element method are provided for the homogeneous yield function based anisotropic hardening hah model this model is able to describe complex material behavior under non monotonic loading conditions two numerical algorithms based on the semi explicit and fully implicit schemes are compared in terms of accuracy to efficiently treat the yield locus distortion when the strain path changes a multi step newton raphson method is proposed to calculate the first and second derivatives of the hah yield surface for the validation of the developed numerical algorithms the r value anisotropy is compared for the conventional yield model with classical isotropic hardening and for the hah model moreover detailed error analysis is presented using iso error maps the results show that the fully implicit stress integration algorithm based on the closet point projection method leads to better accuracy in general however the semi explicit algorithm also provides comparable accuracy if an appropriate time increment is chosen furthermore in spite of the yield surface distortion the developed numerical algorithms can successfully update stress with the equivalent level of the error for the conventional yield model

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

algebraic structure of the gradient holonomic algorithm for lax integrable nonlinear dynamical systems i the algebraic structure of the gradient holonomic algorithm for lax integrable dynamical systems is discussed a generalization of the r structure approach for the case of operator valued affine lie algebras is used to prove the bi hamiltonian formulation of nonlinear integrable dynamical systems in multidimensions the monodromy transfer matrix is constructed to describe the operator manifold in relation to canonical lie poisson bracket on initial affine lie algebra with gauge central extension as an illustration the two dimensional operator benney kaup integrable hierarchy is considered and their bi hamiltonicity is proved

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

nonlinear thin shell analysis using a curved triangular element abstract this paper describes a simple but robust curved triangular thin shell element the element has twelve degrees of freedom namely the displacement components in the corner nodes and the rotations about the element sides the element is based upon the well known facet element with constant membrane strains and constant stress couples and is augmented to take into account the influence of the changes of curvature on the membrane deformations due to the quadratic character of this influence curved geometries can be modelled in order to permit arbitrarily large rotational increments reference vectors are attached to each element side these reference vectors are updated by means of orthogonal transformations the latter are determined uniquely by the subsequent orientations of the associated element sides the changes of curvature are evaluated using these reference vectors and a corotated reference configuration to describe time independent elastoplasticity a layered shell model is applied while besseling s fraction model is adopted for each layer a number of numerical examples illustrate the performance of the element

the τ function of the universal whitham hierarchy matrix models and topological field theories the universal witham hierarchy is considered from the point of view of topological field theories the function for this hierarchy is defined it is proved that the algebraic orbits of whitham hierarchy can be identified with various topological matter models coupled with topological gravity

looped cotangent virasoro algebra and non linear integrable systems in dimension 2 1 we consider a lie algebra generalizing the virasoro algebra to the case of two space variables we study its coadjoint representation and calculate the corresponding euler equations in particular we obtain a bi hamiltonian system that leads to an integrable non linear partial differential equation this equation is an analogue of the kadomtsev petviashvili of type b equation

studies in anisotropic plasticity with reference to the hill criterion algorithms based upon the notion of return mapping have been developed for the hill yield function of anisotropic plasticity the relative accuracy of two algorithms is assessed by means of iso error maps the choice of the algorithm turns out to be much more critical for the orthotropic hill criterion than for the underlying isotropic von mises plasticity model a tangent operator that is consistent with the developed integration algorithm is formulated and its efficiency is assessed compared with the classical continuum tangent operator the model has been applied to three shell plate structures

the extended supersymmetrization of the multicomponent kadomtsev petviashvilli hierarchy we describe three different approaches to the extended n 2 supersymmetrization of the multicomponent kp hierarchy in the first one we utilize only superfermions while in the second only superbosons and in the third superbosons as well as superfermions it is shown that many soliton equations can be embedded in the supersymmetry theory by using the first approach even if we do not change these equations in the bosonic limit of the supersymmetry in the second or third approach we obtain a generalization of the soliton equations in the bosonic limit which remains in the class of the usual commuting functions as a byproduct of our analysis we prove that for the first procedure the bosonic part of the one component supersymmetric kp hierarchy coincides with the usual classical two component kp hierarchy

nonlinear poisson structures and r matrices we introduce quadratic poisson structures on lie groups associated with a class of solutions of the modified yang baxter equation and apply them to the hamiltonian description of lax systems the formal analog of these brackets on associative algebras provides second structures for certain integrable equations in particular the integrals of the toda flow on generic orbits are shown to satisfy recursion relations finally we exhibit a third order poisson bracket for which ther matrix approach is feasible

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

quasiclassical limit of coupled kdv equations riemann invariants and multi hamiltonian structure abstract we consider the quasiclassical limit of the first nontrivial flow in coupled kdv hierarchy written down in riemann invariants it assumes the extremely simple form r i k 1 n r k 2r i r x i i 1 n up to certain natural equivalence this turns out to be the only hydrodynamic system with n 1 compatible purely local hamiltonian structures

extensions and contractions of the lie algebra ofq pseudodifferential symbols on the circle abstract we construct cocycles on the lie algebra of pseudo and q pseudodifferential symbols of one variable and on their close relatives the sine algebra and the poisson algebra on two torus a quantum godbillon vey cocycle on pseudo differential operators appears in this construction as a natural generalization of the gelfand fuchs 3 cocycle on periodic vector fields a nontrivial embedding of the virasoro algebra into a completion of q pseudodifferential symbols is proposed we describe q analogs of the kp and kdv hierarchies admitting an infinite number of conserved charges as well as q deformed gelfand dickey structures

coupled kdv equations with multi hamiltonian structures abstract we present a large class of systems of n equations which possess n 1 purely differential compatible hamiltonian structures our generic equations are isospectral to n 1 0 e i 2 i 2 n 1 0 i 2 i 2 n but our class also includes degenerate nondispersive systems which are unrelated to this linear problem embedded in this class for each n there are n distinct coupled kdv systems when n 2 our class includes 3 known equations dispersive water waves ito s equation and reduced benney s equations these equations are thus tri hamiltonian we also present 2 examples of 3 component quadri hamiltonian systems which generalise the above mentioned dispersive and nondispersive water waves

algebro geometric approach in the theory of integrable hydrodynamic type systems the algebro geometric approach for integrability of semi hamiltonian hydrodynamic type systems is presented the class of symmetric hydrodynamic type systems is defined and the calculation of the associated riemann surfaces is greatly simplified for this class many interesting and physically motivated examples are investigated

supersymmetric extensions of the harry dym hierarchy we study the supersymmetric extensions of the harry dym hierarchy of equations we obtain the susy b extension the doubly susy b extension as well as the n 1 and the n 2 supersymmetric extensions for this system the n 2 supersymmetric extension is particularly interesting since it leads to new classical integrable systems in the bosonic limit we prove the integrability of these systems through the bi hamiltonian formulation of integrable models and through the lax description we also discuss the supersymmetric extension of the hunter zheng equation which belongs to the harry dym hierarchy of equations

meromorphic lax representations of 1 1 dimensional multi hamiltonian dispersionless systems rational lax hierarchies introduced by krichever are generalized a systematic construction of infinite multi hamiltonian hierarchies and related conserved quantities is presented the method is based on the classical r matrix approach applied to poisson algebras a proof that poisson operators constructed near different points of laurent expansion of lax functions are equal is given all results are illustrated by several examples

algebraic structure of the gradient holonomic algorithm for lax integrable nonlinear dynamical systems ii the reduction via dirac and canonical quantization procedure the generalized theory of the r structure on affine operator lie algebras is used to construct a complete theory of lax integrable nonlinear dynamical systems in multidimensions the operator bi hamiltonian structures and their functional reductions are discussed in great detail in the examples of operator korteweg de vries and benney kaup dynamical systems as an important by product of the developed algebraic theory the dirac canonical quantization problem is solved almost completely for the neumann bogoliubov type oscillatory dynamical system on spheres associated via moser with the spectral moment map on an affine associative metrized lie coalgebra with a one parameter gauge two cocycle some remarks are given on the problem of extending the developed algebraic theory to the case of lax integrable dynamical systems on discrete manifolds

dispersionful analog of the whitham hierarchy the transition from the zero genus universal whitham hierarchy to its dispersionful counterpart making use only of the lax representations is presented this is an alternative approach to that of takasaki who has recently shown that the dispersionless limit of the charged multicomponent kadomtsev petviashvili kp hierarchy is the whitham hierarchy the advantage of the presented approach is the construction of finite field reductions which are the main concern in this paper the theory is illustrated by several significant examples

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

a consistent formulation for the integration of combined plasticity and creep within the context of the consistent tangent update this paper outlines a stress update algorithm for combined creep and plasticity the algorithm is implicit providing unconditional stability and utilizes local newton iteration to solve scalar forms of the coupled constitutive equations for the creep and plastic strain increments the tangent for the local iteration is obtained accurately providing quadratic convergence at the gauss point level quadratic convergence of the global iteration procedure is also maintained using an explicitly derived consistent tangent for combined plasticity and creep further combination with an automatic time stepping scheme provides an efficient stable accurate and robust computational algorithm the algorithm has been implemented in the general purpose fe package lusas1

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

cyclic homology of differential operators the virasoro algebra and a q analogue we show how methods from cyclic homology give easily an explicit 2 cocycle on the lie algebra of differential operators of the circle such that restricts to the cocycle defining the virasoro algebra the same methods yield also aq analogue of as well as an infinite family of linearly independent cocycles arising when the complex parameterq is a root of unity we use an algebra ofq difference operators andq analogues of koszul and the rham complexes to construct these quantum cocycles

on a class of constitutive equations in viscoplasticity formulation and computational issues the viscoplastic constitutive model is formulated based on the existence of the dissipation potential which embodies the notion of the gauge minkowski function of the convex set a perturbation method is used for a solution of stiff differential equations characterizing the associated problem of evolution it relies on a discrete formulation of viscoplasticity which results from the regularized version of the principle of maximum plastic dissipation the operator split methodology and the newton raphson method are used to obtain the numerical solution of the discretized equations of evolution the consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discretized evolution equations for several variants of the flow potential function including some representative stiff functional forms numerical tests of the integration algorithm based on iso error maps are provided finally a numerical example is presented to illustrate the robustness and the effectiveness of the proposed approach

elliptic curves with complex multiplication and the conjecture of birch and swinnerton dyer 1 quick review of elliptic curves 2 2 elliptic curves over c 4 3 elliptic curves over local fields 6 4 elliptic curves over number fields 12 5 elliptic curves with complex multiplication 15 6 descent 22 7 elliptic units 27 8 euler systems 37 9 bounding ideal class groups 43 10 the theorem of coates and wiles 47 11 iwasawa theory and the main conjecture 50 12 computing the selmer group 61

poisson algebras associated with constrained dispersionless modified kadomtsev petviashvili hierarchies we investigate the bi hamiltonian structures associated with constrained dispersionless modified kadomtsev petviashvili kp hierarchies which are constructed from truncations of the lax operator of the dmkp hierarchy after transforming their second hamiltonian structures to those of the gelfand dickey type we obtain the poisson algebras of the coefficient functions of the truncated lax operators then we study the conformal property and free field realizations of these poisson algebras some examples are worked out explicitly to illustrate the obtained results

integrable discrete systems on r and related dispersionless systems a general framework for integrable discrete systems on r in particular containing lattice soliton systems and their q deformed analogs is presented the concept of regular grain structures on r generated by discrete one parameter groups of diffeomorphisms in terms of which one can define algebra of shift operators is introduced two integrable hierarchies of discrete chains together with bi hamiltonian structures and their continuous limits are constructed the inverse problem based on the deformation quantization scheme is considered

integrable hydrodynamic chains a new approach for derivation of benney type moment chains and integrable hydrodynamic type systems is presented new integrable hydrodynamic chains are constructed all their hydrodynamical reductions are described and integrated new 2 1 integrable hydrodynamic type systems are found