informational dynamics of censored observations the analysis of stochastic models is often greatly complicated if there are censored observations of the random variables this paper characterizes families of distributions which help keep tractable the analysis of such models our primary motivation is to provide guidance to practitioners in the selection of distributions if a modeler feels that no member of the families we characterize is a reasonable approximation then he will almost surely encounter serious analytic and computational problems if his data include censored observations we characterize a family of distributions for which there exist fixed dimensional sufficient statistics of purely censored observations we also characterize an important subset of this family appropriate for situations where data include both censored and exact observations we derive the corresponding predictive distributions using arbitrary priors and present some general results relating stochastic dominance among predictive distributions to the parameters of the

an optimal approximate dynamic programming algorithm for the lagged asset acquisition problem we consider a multistage asset acquisition problem where assets are purchased now at a price that varies randomly over time to be used to satisfy a random demand at a particular point in time in the future we provide a rare proof of convergence for an approximate dynamic programming algorithm using pure exploitation where the states we visit depend on the decisions produced by solving the approximate problem the resulting algorithm does not require knowing the probability distribution of prices or demands nor does it require any assumptions about its functional form the algorithm and its proof rely on the fact that the true value function is a family of piecewise linear concave functions

noncommutative differential geometry and new models of gauge theory the noncommutative differential geometry of the algebra c v mn c of smooth mn c valued functions on a manifold v is investigated for n 2 the analog of maxwell s theory is constructed and interpreted as a field theory on v it describes a u n yang mills field minimally coupled to a set of fields with values in the adjoint representation that interact among themselves through a quartic polynomial potential the euclidean action which is positive vanishes on exactly two distinct gauge orbits which are interpreted as two vacua of the theory in one of the corresponding vacuum sectors the su n part of the yang mills field is massive for the case n 2 analogies with the standard model of electroweak theory are pointed out finally a brief description is provided of what happens if one starts from the analog of a general yang mills theory instead of maxwell s theory which is a particular case

comment on the solutions of the graded yang baxter equation in this comment we obtain two types of solutions of the graded yang baxter equation whose symmetry group is glqs 1 1

numerical comparisons of high order nonlinear solvers for the transient navier stokes equations based on homotopy and perturbation techniques efficient solvers for the unsteady navier stokes equations are presented a classic time stepping scheme is combined with high order nonlinear solvers coupling homotopy and a perturbation technique polynomial and rational representations are used to approximate the unknowns of the problem a pseudo residual criterion is proposed to improve the efficiency of the solvers the numerical example considered in this paper is the time periodic two dimensional flow around a circular cylinder comparisons with the classical first order newton raphson solver are performed numerical results reveal that a lower number of matrix factorization is needed with the proposed methods decreasing the computational effort

model order reduction in hyperelasticity a proper generalized decomposition approach this paper deals with the extension of proper generalized decomposition methods to non linear problems in particular to hyperelasticity among the different approaches that can be considered for the linearization of the doubly weak form of the problem we have implemented a new one that uses asymptotic numerical methods in conjunction with proper generalized decomposition to avoid complex consistent linearization schemes necessary in newton strategies this approach results in an approximation of the problem solution in the form of a series expansion each term of the series is expressed as a finite sum of separated functions the advantage of this approach is the presence of only one tangent operator identical for every term in the series the resulting approach has proved to render very accurate results that can be stored in the form of a meta model in a very compact format this opens the possibility to use these results in real time reaching khz feedback rates or to be used in deployed handheld devices such as smartphones and tablets

sample path derivatives for s s inventory systems for s s inventory systems we derive sample path derivatives of performance measures with respect to the two parameters s and s these derivatives yield derivative estimators which can be estimated from a single sample path or simulation of the inventory system in some cases not even requiring actual knowledge of the underlying demand distribution such derivative estimates would be useful in sensitivity analysis or in gradient based optimization techniques we consider the nondiscounted periodic review system with general independent and identically distributed i i d continuous demands full backlogging and general holding and shortage costs for the infinite horizon model consistency proofs are given for some special cases although we argue why the estimators should be correct for the more general case

stochastic optimization by simulation convergence proofs for the gi g 1 queue in steady state approaches like finite differences with common random numbers infinitesimal perturbation analysis and the likelihood ratio method have drawn a great deal of attention recently as ways of estimating the gradient of a performance measure with respect to continuous parameters in a dynamic stochastic system in this paper we study the use of such estimators in stochastic approximation algorithms to perform so called single run optimizations of steady state systems under mild conditions for an objective function that involves the mean system time in a gi g 1 queue we prove that many variant of these algorithms converge to the minimizer in most cases however the simulation length must be increased from iteration to iteration otherwise the algorithm may converge to the wrong value one exception is a particular implementation of infinitesimal perturbation analysis for which the single run optimization converges to the optimum even with a fixed and small number of ends of service per iteration as a by product of our convergence proofs we obtain some properties of the derivative estimators that could be of independent interest our analysis exploits the regenerative structure of the system but our derivative estimation and optimization algorithms do not always take advantage of that regenerative structure in a companion paper we report numerical experiments with an m m 1 queue which illustrate the basis convergence properties and possible pitfalls of the various techniques

a priori model reduction through proper generalized decomposition for solving time dependent partial differential equations over the past years model reduction techniques have become a necessary path for the reduction of computational requirements in the numerical simulation of complex models a family of a priori model reduction techniques called proper generalized decomposition pgd methods are receiving a growing interest these methods rely on the a priori construction of separated variables representations of the solution of models defined in tensor product spaces they can be interpreted as generalizations of proper orthogonal decomposition pod for the a priori construction of such separated representations in this paper we introduce and study different definitions of pgd for the solution of time dependent partial differential equations we review classical definitions of pgd based on galerkin or minimal residual formulations and we propose and discuss several improvements for these classical definitions we give an interpretation of optimal decompositions as the solution of pseudo eigenproblems we also introduce a new definition of pgd called minimax pgd which can be interpreted as a petrov galerkin model reduction technique where test and trial reduced basis functions are related by an adjoint problem this new definition improves convergence properties of separated representations with respect to a chosen metric it coincides with a classical pod for degenerate time dependent partial differential equations for the numerical construction of each pgd we propose algorithms inspired from the solution of eigenproblems several numerical examples illustrate and compare the different definitions of pgd on transient advection diffusion reaction equations

an adaptive distribution free algorithm for the newsvendor problem with censored demands with applications to inventory and distribution we consider the problem of optimizing inventories for problems where the demand distribution is unknown and where it does not necessarily follow a standard form such as the normal we address problems where the process of deciding the inventory and then realizing the demand occurs repeatedly the only information we use is the amount of inventory left over rather than attempting to estimate the demand distribution we directly estimate the value function using a technique called the concave adaptive value estimation cave algorithm cave constructs a sequence of concave piecewise linear approximations using sample gradients of the recourse function at different points in the domain since it is a sampling based method cave does not require knowledge of the underlying sample distribution the result is a nonlinear approximation that is more responsive than traditional linear stochastic quasi gradient methods and more flexible than analytical techniques that require distribution information in addition we demonstrate near optimal behavior of the cave approximation in experiments involving two different types of stochastic programs the newsvendor stochastic inventory problem and two stage distribution problems

a best points interpolation method for efficient approximation of parametrized functions we present an interpolation method for efficient approximation of parametrized functions the method recognizes and exploits the low dimensional manifold structure of the parametrized functions to provide good approximation basic ingredients include a specific problem dependent basis set defining a low dimensional representation of the parametrized functions and a set of best interpolation points capturing the spatial parameter variation of the parametrized functions the best interpolation points are defined as solution of a least squares minimization problem which can be solved efficiently using standard optimization algorithms the approximation is then determined from the basis set and the best interpolation points through an inexpensive and stable interpolation procedure in addition an a posteriori error estimator is introduced to quantify the approximation error and requires little additional cost numerical results are presented to demonstrate the accuracy and efficiency of the method copyright 2007 john wiley sons ltd

the latin multiscale computational method and the proper generalized decomposition this paper deals with the synergy between the latin multiscale method and what is called the proper generalized decomposition pgd which is the key of its performances

differential calculi on the quantum group glp q 2 the author presents a systematical study of left covariant differential calculus on the quantum group glp q 2 a two parameter deformation of the general linear group in particular the author explicitly constructs the most general bicovariant calculus it depends on a new parameter s the corresponding lie algebra of left coinvariant vector fields is a two parameter deformation of the classical lie algebra p and q appear only in the combination r pq in the limit p q to 1 one obtains non standard bicovariant differential calculi on the classical lie group

differential calculus on compact matrix pseudogroups quantum groups the paper deals with non commutative differential geometry the general theory of differential calculus on quantum groups is developed bicovariant bimodules as objects analogous to tensor bundles over lie groups are studied tensor algebra and external algebra constructions are described it is shown that any bicovariant first order differential calculus admits a natural lifting to the external algebra so the external derivative of higher order differential forms is well defined and obeys the usual properties the proper form of the cartan maurer formula is found the vector space dual to the space of left invariant differential forms is endowed with a bilinear operation playing the role of the lie bracket commutator generalized antisymmetry relation and jacobi identity are proved

covariant differential calculus on the quantum superplane and generalization of the wess zumino formalism it is shown that the construction of covariant differential calculus on quantum planes developed by wess and zumino 1990 for arbitrary quadratic algebras can generalize to the quantum superplane if one accounts for the differences due to the graduation of the superplane

using stochastic approximation methods to compute optimal base stock levels in inventory control problems in this paper we consider numerous inventory control problems for which the base stock policies are known to be optimal and we propose stochastic approximation methods to compute the optimal base stock levels the existing stochastic approximation methods in the literature guarantee that their iterates converge but not necessarily to the optimal base stock levels in contrast we prove that the iterates of our methods converge to the optimal base stock levels moreover our methods continue to enjoy the well known advantages of the existing stochastic approximation methods in particular they only require the ability to obtain samples of the demand random variables rather than to compute expectations explicitly and they are applicable even when the demand information is censored by the amount of available inventory

dimensional model reduction in non linear finite element dynamics of solids and structures a general approach to the dimensional reduction of non linear finite element models of solid dynamics is presented for the newmark implicit time discretization the computationally most expensive phase is the repeated solution of the system of linear equations for displacement increments to deal with this it is shown how the problem can be formulated in an approximation ritz basis of much smaller dimension similarly the explicit newmark algorithm can be also written in a reduced dimension basis and the computation time savings in that case follow from an increase in the stable time step length in addition the empirical eigenvectors are proposed as the basis in which to expand the incremental problem this basis achieves approximation optimality by using computational data for the response of the full model in time to construct a reduced basis which reproduces the full system in a statistical sense because of this global time viewpoint the basis need not be updated as with reduced bases computed from a linearization of the full finite element model if the dynamics of a finite element model is expressed in terms of a small number of basis vectors the asymptotic cost of the solution with the reduced model is lowered and optimal scalability of the computational algorithm with the size of the model is achieved at the same time numerical experiments indicate that by using reduced models substantial savings can be achieved even in the pre asymptotic range furthermore the algorithm parallelizes very efficiently the method we present is expected to become a useful tool in applications requiring a large number of repeated non linear solid dynamics simulations such as convergence studies design optimization and design of controllers of mechanical systems

a priori hyperreduction method an adaptive approach model reduction methods are usually based on preliminary computations to build the shape function of the reduced order model rom before the computation of the reduced state variables they are a posteriori approaches most of the time these preliminary computations are as complex as the simulation which we want to simplify by the rom the reduction method we propose avoids such preliminary computations it is an a priori approach based on the analysis of some state evolutions such that all the state evolutions needed to perform the model reduction are described by an approximate rom the rom and the state evolution are simultaneously improved by the method thanks to an adaptive strategy obviously an initial set of known shape functions can be used to define the rom to adapt but it is not necessary the adaptive procedure includes extensions of the subspace spanned by the shape functions of the rom and selections of the most relevant shape functions in order to represent the state evolution the hyperreduction is achieved by selecting a part of the integration points of the finite element model to forecast the evolution of the reduced state variables hence both the number of degrees of freedom and the number of integration points are reduced to perform the adaptive procedure different computational strategies can be developed in this paper we propose an incremental algorithm involving adaptive periods during these adaptive periods the incremental computation is restarted until a quality criterion is satisfied this approach is compatible with classical formulations of the equations

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

advanced simulation of models defined in plate geometries 3d solutions with 2d computational complexity abstract many models in polymer processing and composites manufacturing are defined in degenerated three dimensional domains involving plate or shell geometries the reduction of models from 3d to 2d is not obvious when complex physics are involved the hypotheses to be introduced for reaching this dimensionality reduction are sometimes unclear and most of possible proposals will have a narrow interval of validity the only getaway is to explore new discretization strategies able to circumvent or at least alleviate the drawbacks related to mesh based discretizations of fully 3d models defined in plate or shell domains an in plane out of plane separated representation of the involved fields within the context of the proper generalized decomposition allows solving the fully 3d model by keeping a 2d characteristic computational complexity moreover the pgd features allow the introduction of many extra coordinates as for example the orientation of the different laminate plies without affecting the solvability of the resulting multidimensional model

a stochastic programming approach for planning horizons of infinite horizon capacity planning problems planning horizon is a key issue in production planning different from previous approaches based on markov decision processes we study the planning horizon of capacity planning problems within the framework of stochastic programming we first consider an infinite horizon stochastic capacity planning model involving a single resource linear cost structure and discrete distributions for general stochastic cost and demand data non markovian and non stationary we give sufficient conditions for the existence of an optimal solution furthermore we study the monotonicity property of the finite horizon approximation of the original problem we show that the optimal objective value and solution of the finite horizon approximation problem will converge to the optimal objective value and solution of the infinite horizon problem when the time horizon goes to infinity these convergence results together with the integrality of decision variables imply the existence of a planning horizon we also develop a useful formula to calculate an upper bound on the planning horizon then by decomposition we show the existence of a planning horizon for a class of very general stochastic capacity planning problems which have complicated decision structure

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

stochastic lot sizing problem with inventory bounds and constant order capacities abstract in this paper we study the stochastic version of lot sizing problems with inventory bounds and order capacities customer demands inventory bounds and costs are subject to uncertainty and dependent with each other throughout the finite planning horizon two models in stochastic programming are developed the first one has inventory bound constraints and the second one has both inventory bound and order capacity constraints we explore structural properties of the two models and develop o n 2 and o n 2 t log n dynamic programming algorithms for them respectively our model also generalizes the deterministic lot sizing problem with inventory bounds for some cases when applied to the deterministic versions our algorithms outperform existing deterministic algorithms

learning algorithms for separable approximations of discrete stochastic optimization problems we propose the use of sequences of separable piecewise linear approximations for solving nondifferentiable stochastic optimization problems the approximations are constructed adaptively using a combination of stochastic subgradient information and possibly sample information on the objective function itself we prove the convergence of several versions of such methods when the objective function is separable and has integer break points and we illustrate their behavior on numerical examples we then demonstrate the performance on nonseparable problems that arise in the context of two stage stochastic programming problems and demonstrate that these techniques provide near optimal solutions with a very fast rate of convergence compared with other solution techniques

interpolation method for adapting reduced order models and application to aeroelasticity reduced order models are usually thought of as computationally inexpensive mathematical representations that offer the potential for near real time analysis although most reduced order models can operate in near real time their construction can be computationally expensive as it requires accumulating a large number of system responses to input excitations furthermore reduced order models usually lack robustness with respect to parameter changes and therefore must often be rebuilt for each parameter variation together these two issues underline the need for a fast and robust method for adapting precomputed reduced order models to new sets of physical or modeling parameters to this effect this paper presents an interpolation method based on the grassmann manifold and its tangent space at a point that is applicable to structural aerodynamic aeroelastic and many other reduced order models based on projection schemes this method is illustrated here with the adaptation of computational fluid dynamics based aeroelastic reduced order models of complete fighter configurations to new values of the freestream mach number good correlations with results obtained from direct reduced order model reconstruction high fidelity nonlinear and linear simulations are reported thereby highlighting the potential of the proposed reduced order model adaptation method for near real time aeroelastic predictions using precomputed reduced order model databases

on the verification of model reduction methods based on the proper generalized decomposition in this work we introduce a consistent error estimator for numerical simulations performed by means of the proper generalized decomposition pgd approximation this estimator which is based on the constitutive relation error enables to capture all error sources i e those coming from space and time numerical discretizations from the truncation of the pgd decomposition etc and leads to guaranteed bounds on the exact error the specificity of the associated method is a double approach i e a kinematic approach and a unusual static approach for solving the parameterized problem by means of pgd this last approach makes straightforward the computation of a statically admissible solution which is necessary for robust error estimation an attractive feature of the error estimator we set up is that it is obtained by means of classical procedures available in finite element codes it thus represents a practical and relevant tool for driving algorithms carried out in pgd being possibly used as a stopping criterion or as an adaptation indicator numerical experiments on transient thermal problems illustrate the performances of the proposed method for global error estimation

consistent multiparameter quantisation of gl n the author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl n and on its dual the space of polynomials in n2 variables the dimension of the manifold is n2 n 2 2

an error estimator for separated representations of highly multidimensional models fine modeling of the structure and mechanics of materials from the nanometric to the micrometric scales uses descriptions ranging from quantum to statistical mechanics most of these models consist of a partial differential equation defined in a highly multidimensional domain e g schrodinger equation fokker planck equations among many others the main challenge related to these models is their associated curse of dimensionality we proposed in some of our former works a new strategy able to circumvent the curse of dimensionality based on the use of separated representations also known as finite sums decomposition this technique proceeds by computing at each iteration a new sum that consists of a product of functions each one defined in one of the model coordinates the issue related to error estimation has never been addressed this paper presents a first attempt on the accuracy evaluation of such a kind of discretization techniques

a model reduction method for the post buckling analysis of cellular microstructures multiscale finite element models are useful in some applications such as biological tissues or foams to determine the effective properties of the solids in this paper a model reduction method is proposed that is applied in the post buckling regime of cellular microstructures the proposed approach combines a perturbation technique with the proper orthogonal decomposition pod the computation of finite series leads to an efficient continuation method which enables handling limit points and instabilities naturally each term of the series is itself developed using pod the size of the problems to be solved is thus drastically reduced computations involving two and three dimensional microstructures in compression enable to evaluate the potential of the technique in view of applications to multiscale analyses

noncommutative geometry of super jordanian osph 2 1 covariant quantum space extending a recently proposed procedure of construction of various elements of differential geometry on noncommutative algebras we obtain these structures on noncommutative superalgebras as an example a quantum superspace covariant under the action of super jordanian osph 2 1 is studied it is shown that there exist a two parameter family of torsionless connections and the curvature computed from this family of connections is bilinear it is also shown that the connections are not compatible with the metric

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

profit estimation error in the newsvendor model under a parametric demand distribution we consider the newsvendor model in which uncertain demand is assumed to follow a probabilistic distribution with known functional form but unknown parameters these parameters are estimated unbia

asymptotic numerical methods and pade approximants for non linear elastic structures in this paper we apply asymptotic numerical methods for computing non linear equilibrium paths of elastic beam plate and shell structures the non linear branches are sought in the form of asymptotic expansions and they are determined by solving numerically fem several linear problems with a single stiffness matrix a large number of terms of the series can be easily computed by using recurrence formulas in comparison with a more classical step by step procedure the method is rapid and automatic we show with some examples that the choice of the expansion s parameter and the use of pade approximants play an important role in the determination of the size of the domain of convergence

a reduction method for nonlinear structural dynamic analysis abstract a computational algorithm for predicting the nonlinear dynamic response of a structure is presented the nonlinear system of ordinary differential equations resulting from the finite element discretization is highly reduced by means of a rayleigh ritz analysis the basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing only a few basis updatings are required during the whole time integration the truncation error introduced at every change of basis is pointed out as the cause for a divergence type behaviour and some means for eliminating it are discussed results for examples involving large displacements are shown and compared to the results obtained by integrating the complete system of equations

asymptotic numerical method for problems coupling several nonlinearities abstract the objective of this paper is to assess the efficiency of the asymptotic numerical method to solve problems coupling various nonlinearities the 3d hemispherical stretching of a circular sheet that involves geometrical material and red unilateral contact nonlinearities is chosen as an example an elastoplastic model based on the plasticity deformation theory is adopted the structural discretization is performed by a shell finite element well adapted for problems involving large displacements and large rotations the unilateral contact problem is identified to boundary conditions which are replaced by force displacement relations and solved using a special algorithm comparisons with results obtained by the help of an industrial code establish the interest and the performance of the present method

bicovariant differential calculus on quantum groups su q n and so q n following woronowicz s proposal the bicovariant differential calculus on the quantum groupssu q n andso q n is constructed a systematic construction of bicovariant bimodules by using the hat r q matrix is presented the relation between the hopf algebras generated by the linear functionals relating the left and right multiplication of these bicovariant bimodules and theq deformed universal enveloping algebras is given imposing the conditions of bicovariance and consistency with the quantum group structure the differential algebras and exterior derivatives are defined as an application the maurer cartan equations and theq analogue of the structure constants are formulated

differential calculus on the h superplane a noncommutative differential calculus on the h superplane is presented via a contraction of the q superplane an r matrix which satisfies both ungraded and graded yang baxter equations is obtained and a new deformation of the 1 1 dimensional classical phase space the super heisenberg algebra is introduced

stochastic optimization by simulation numerical experiments with the m m 1 queue in steady state this paper gives numerical illustrations of the behavior of stochastic approximation combined with different derivative estimation techniques to optimize a steady state system it is a companion paper to l ecuyer and glynn 1993 which gives convergence proofs for most of the variants experimented here the numerical experiments are made with a simple m m 1 queue which while simple serves to illustrate the basic convergence properties and possible pitfalls of the various techniques

model order reduction for hyperelastic materials in this paper we develop a novel algorithm for the dimensional reduction of the models of hyperelastic solids undergoing large strains unlike standard proper orthogonal decomposition methods the proposed algorithm minimizes the use of the newton algorithms in the search of non linear equilibrium paths of elastic bodies the proposed technique is based upon two main ingredients on one side the use of classic proper orthogonal decomposition techniques that extract the most valuable information from pre computed complete models this information is used to build global shape functions in a ritz like framework on the other hand to reduce the use of newton procedures an asymptotic expansion is made for some variables of interest this expansion shows the interesting feature of possessing one unique tangent operator for all the terms of the expansion thus minimizing the updating of the tangent stiffness matrix of the problem the paper is completed with some numerical examples in order to show the performance of the technique in the framework of hyperelastic kirchhoff saint venant and neo hookean solids

two parameter differential calculus on the h superplane we introduce a noncommutative differential calculus on the two parameter h superplane via a contraction of the p q superplane we manifestly show that the differential calculus is covariant under glh1 h2 1 1 transformations we also give a two parameter deformation of the 1 1 dimensional phase space algebra

optimization of s s inventory systems with random lead times and a service level constraint a major assumption in the analysis of s s inventory systems with stochastic lead times is that orders are received in the same sequence as they are placed even under this assumption much of the work to date has focused on the unconstrained optimization of the system in which a penalty cost for unsatisfied demand is assigned the literature on constrained optimization wherein a service level requirement needs to be met is more sparse in this paper we consider the constrained optimization problem where orders are allowed to cross in time we propose a feasible directions procedure that is simulation based and present computational results for a large number of test cases in the vast majority of cases we come within 5 of estimated optimality