stability of time stepping methods for abstract time dependent parabolic problems we consider an abstract nonautonomous parabolic problem u t a t u t quad u t 0 u 0 where a t d t subset x to x t 0 le t le t 1 is a family of sectorial operators in a banach space x this problem is discretized in time by means of either an a theta stable runge kutta or an a theta stable linear multistep method we prove that the resulting discretization is stable under some natural assumptions on the relative total variation of a t with respect to t for strongly a theta stable runge kutta methods stability holds even for variable time step sizes our results are applicable to the analysis of time dependent parabolic problems in the lp norms

a general procedure for the adaptation of multistep algorithms to the integration of oscillatory problems this paper introduces a general technique for the construction of multistep methods capable of integrating without local truncation error homogeneous linear odes with constant coefficients including those in particular that result in oscillatory solutions moreover these methods can be further adapted through coefficient modification for the exact integration of forced oscillations in one or more frequencies even confluent ones that occur from nonhomogeneous terms in the differential equation our procedure allows the derivation of many of the existing codes with similar properties as well as the improvement of others that in their original design were only able to integrate oscillations in a single frequency the properties of the methods are studied within a general framework and numerical examples are presented these demonstrate the way in which the new algorithms perform distinctly better than the general purpose codes particularly when integrating the class of equations with perturbed oscillatory solutions the methods developed are mainly applicable to the accurate and efficient integration of problems for which the oscillation frequencies are known as occurs in satellite orbit propagation the underlying ideas have already been applied to the improvement of some chebyshev methods that are not multistep

zero stability properties of the three ordinate variable stepsize variable formula methods consider the systemy f x y y a x a b y r n wheref is continuous and lipschitzian with respect to the second argument very often linear multistep variable stepsize variable formula methods lm vsvfm s are used to computey k y xk on the points of the grid a x 0 x1 x2 xn b the general lm vsvfm is based on formulae of the following type y k sum limits i 1 s k alpha i bar h k sk y k i sum limits i 0 s k h k i beta i bar h k s k f x k i y k i whereh k xk xk 1 bar h k h k h k 1 ldots h k s k s k k k 1 1 n the coefficients i and i depend on the lasts k 1 stepsizes and on the formula used at stepk only the zero stability properties of some special classes of lm vsvfm s as for example those based on adams formulae were investigated in the literature a class of three ordinate lm vsvfm s is defined in this paper some results concerning the zero stability properties of these methods are proved it is shown that some well known results are simple corollaries of the results found for the three ordinate lm vsvfm s it is easily seen that similar results hold for the corresponding one leg vsvfm s finally the use of the theoretical results in the practical implementations is briefly discussed

on the stability and performance of discrete event methods for simulating continuous systems this paper establishes a link between the stability of a first order explicit discrete event integration scheme and the stability criteria for the explicit euler method the paper begins by constructing a time varying linear system with bounded inputs that is equivalent to the first order discrete event integration scheme the stability of the discrete event system is shown to result from the fact that it automatically adjusts its time advance to lie below the limit set by the explicit euler stability criteria moreover because it is not necessary to update all integrators at this rate a significant performance advantage is possible our results confirm and explain previously reported studies where it is demonstrated that a reduced number of updates can provide a significant performance advantage compared to fixed step methods these results also throw some light on stability requirements for discrete event simulation of spatially extended systems

quantitative stability analysis of stochastic quasi variational inequality problems and applications we consider a parametric stochastic quasi variational inequality problem sqvip for short where the underlying normal cone is defined over the solution set of a parametric stochastic cone system we investigate the impact of variation of the probability measure and the parameter on the solution of the sqvip by reformulating the sqvip as a natural equation and treating the orthogonal projection over the solution set of the parametric stochastic cone system as an optimization problem we effectively convert stability of the sqvip into that of a one stage stochastic program with stochastic cone constraints under some moderate conditions we derive holder outer semicontinuity and continuity of the solution set against the variation of the probability measure and the parameter the stability results are applied to a mathematical program with stochastic semidefinite constraints and a mathematical program with sqvip constraints

logarithmic barrier decomposition based interior point methods for stochastic symmetric programming abstract we introduce and study two stage stochastic symmetric programs with recourse to handle uncertainty in data defining deterministic symmetric programs in which a linear function is minimized over the intersection of an affine set and a symmetric cone we present a benders decomposition based interior point algorithm for solving these problems and prove its polynomial complexity our convergence analysis proved by showing that the log barrier associated with the recourse function of stochastic symmetric programs behaves a strongly self concordant barrier and forms a self concordant family on the first stage solutions since our analysis applies to all symmetric cones this algorithm extends zhao s results g zhao a log barrier method with benders decomposition for solving two stage stochastic linear programs math program ser a 90 2001 507 536 for two stage stochastic linear programs and mehrotra and ozevin s results s mehrotra m g ozevin decomposition based interior point methods for two stage stochastic semidefinite programming siam j optim 18 1 2007 206 222 for two stage stochastic semidefinite programs

computation of minimum volume covering ellipsoids we present a practical algorithm for computing the minimum volume n dimensional ellipsoid that must contain m given points a1 am n this convex constrained problem arises in a variety of applied computational settings particularly in data mining and robust statistics its structure makes it particularly amenable to solution by interior point methods and it has been the subject of much theoretical complexity analysis here we focus on computation we present a combined interior point and active set method for solving this problem our computational results demonstrate that our method solves very large problem instances m 30 000 and n 30 to a high degree of accuracy in under 30 seconds on a personal computer

probabilistic set covering with correlations we consider two variants of a probabilistic set covering psc problem the first variant assumes that there is uncertainty regarding whether a selected set can cover an item and the objective is to determine a minimum cost combination of sets so that each item is covered with a prespecified probability the second variant seeks to maximize the minimum probability that a selected set can cover all items to date literature on this problem has focused on the special case in which uncertainties are independent in this paper we formulate deterministic mixed integer programming models for distributionally robust psc problems with correlated uncertainties by exploiting the supermodularity of certain substructures and analyzing their polyhedral properties we develop strong valid inequalities to strengthen the formulations computational results illustrate that our modeling approach can outperform formulations in which correlations are ignored and that our algorithms can significantly reduce overall comput

on the zero stability of variable stepsize multistep methods the spectral radius approach in this paper we illustrate a novel approach for studying the asymptotic behaviour of the solutions of linear difference equations with variable coefficients in particular we deal with the zero stability of the 3 step bdf method on grids with variable stepsize for the numerical solution of ivps for odes our approach is based on the theory of the spectral radius of a family of matrices and yields almost optimal results which give a slight improvement to the best results already known from the literature the success got on the chosen example suggests that our approach has a good potential for more general and harder stability analyses of numerical methods

endoproperties of modules and local duality abstract let r be any ring and n i i n i be a direct sum of finitely presented left r modules n i suppose that d n and d n i are the local duals of n and n i for each i i we prove that the lattice of endosubmodules of n is anti isomorphic to the lattices of matrix subgroups of d n and of m i i d n i as consequences n is endoartinian if and only if m or d n is endonoetherian and n is endonoetherian if and only if m or d n is pure injective we obtain in particular that if r is a krull schmidt ring and m is an indecomposable pure injective endonoetherian right r module which is the source of a left almost split morphism in mod r then m is endofinite as an application a ring r is of finite representation type if and only if every pure injective right r module is endonoetherian

the starting procedure in variable stepsize variable order pece codes abstract an existing starting procedure used in codes for solving ordinary differential equation initial value problems by variable stepsize variable order pece formulas is studied the actual error propagation and the estimate formed in the code are analyzed and compared based on the behavior of a bound on error propagation we present a modified starting procedure which improves efficiency significantly for stringent error tolerances

numerical solution by lmms of stiff delay differential systems modelling an immune response we consider the application of linear multistep methods lmms for the numerical solution of initial value problem for stiff delay differential equations ddes with several constant delays which are used in mathematical modelling of immune response for the approximation of delayed variables the nordsieck s interpolation technique providing an interpolation procedure consistent with the underlying linear multistep formula is used an analysis of the convergence for a variable stepsize and structure of the asymptotic expansion of global error for a fixed stepsize is presented some absolute stability characteristics of the method are examined implementation details of the code difsub dde being a modification of the gear s difsub are given finally an efficiency of the code developed for solution of stiff ddes over a wide range of tolerances is illustrated on biomedical application model

classes of modules with the exchange property in searching for isomorphic refinements of direct sum decompositions of modules9 crawley and jonsson introduced an exchange property for modules that is strongly reminiscent of steinitz exchange lemma for vector spaces an r module m where r is an associative ring with identity is said to have the finite exchange property if m can be fitted into any finite sum grid as follows whenever a4 occurs as a direct summand of a finite direct sum a bielai then a m ci for suitable submodules ci of the ai 2 this concept led to very general extensions of the classical refinement theorems see e g 2 3 171 many of the problems suggested by crawley and jonsson in their pioneering paper are still open the most salient one does the finite exchange property imply the unrestricted exchange property we show that the answer is yes for modules which possess decompositions into indecomposable summands in the background of this observation stands the following

a splitting technique of higher order for the navier stokes equations this article presents a splitting technique for solving the time dependent incompressible navier stokes equations using nested finite element spaces which can be interpreted as a postprocessing step the splitting method is of more than second order accuracy in time the integration of adaptive methods in space and time in the splitting are discussed in this algorithm a gradient recovery technique is used to compute boundary conditions for the pressure and to achieve a higher convergence order for the gradient at different points of the algorithm results on the flow around a cylinder s and the driven cavity s problem are presented

models for minimax stochastic linear optimization problems with risk aversion we propose a semidefinite optimization sdp model for the class of minimax two stage stochastic linear optimization problems with risk aversion the distribution of second stage random variables belongs to a set of multivariate distributions with known first and second moments for the minimax stochastic problem with random objective we provide a tight sdp formulation the problem with random right hand side is np hard in general in a special case the problem can be solved in polynomial time explicit constructions of the worst case distributions are provided applications in a productiontransportation problem and a single facility minimax distance problem are provided to demonstrate our approach in our experiments the performance of minimax solutions is close to that of data driven solutions under the multivariate normal distribution and better under extremal distributions the minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity

on the stability of interpolatory variable stepsize adams methods in nordsieck form this paper considers the stability of adams methods that vary the stepsize using the interpolation technique of nordsieck sufficient conditions sometimes necessary on the stepsize ratios h n 1 h n that imply the stability of adams methods with different strategies of stepsize variation are obtained this study extends previous results on this subject by gear and tu siamj numer anal 11 1974 pp 1025 1043 and skeel and jackson siam j numer anal 20 1983 pp 840 853

data driven chance constrained stochastic program in this paper we study data driven chance constrained stochastic programs or more specifically stochastic programs with distributionally robust chance constraints dccs in a data driven setting to provide robust solutions for the classical chance constrained stochastic program facing ambiguous probability distributions of random parameters we consider a family of density based confidence sets based on a general phi divergence measure and formulate dcc from the perspective of robust feasibility by allowing the ambiguous distribution to run adversely within its confidence set we derive an equivalent reformulation for dcc and show that it is equivalent to a classical chance constraint with a perturbed risk level we also show how to evaluate the perturbed risk level by using a bisection line search algorithm for general phi divergence measures in several special cases our results can be strengthened such that we can derive closed form expressions for the perturbed risk levels in addition we show that the conservatism of dcc vanishes as the size of historical data goes to infinity furthermore we analyze the relationship between the conservatism of dcc and the size of historical data which can help indicate the value of data finally we conduct extensive computational experiments to test the performance of the proposed dcc model and compare various phi divergence measures based on a capacitated lot sizing problem with a quality of service requirement

two high order multistep methods with offstep points applied with a variable stepsize abstract in this paper we derive new generalized multistep methods with variable stepsize for initial value problems in ordinary differential equations some numerical examples are given

discrete approximation and quantification in distributionally robust optimization discrete approximation of probability distributions is an important topic in stochastic programming in this paper we extend the research on this topic to distributionally robust optimization dro

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

the order of accuracy for deferred corrections using uncentered end formulas a result of christiansen and russell siam j numer math 35 1980 pp 21 33 on the order of accuracy of an iterated deferred corrections algorithm of lentini and pereyra siam j numer math 14 1977 pp 91 111 is improved at the same time providing a realistic example for a theoretical framework developed by the author for proving accuracy results for deferred corrections

constructing risk measures from uncertainty sets we illustrate the correspondence between uncertainty sets in robust optimization and some popular risk measures in finance and show how robust optimization can be used to generalize the concepts of these risk measures we also show that by using properly defined uncertainty sets in robust optimization models one can construct coherent risk measures and address the issue of the computational tractability of the resulting formulations our results have implications for efficient portfolio optimization under different measures of risk

time integration for ale simulations of fluid structure interaction problems stepsize and order selection based on the bdf abstract we present an adaptive algorithm for time integration of fluid structure integration problems the method relies on a fully coupled procedure to solve fsi problems in which a naturally gcl compliant ale formulation for the finite element spatial discretization is used the main originality of the proposed solution procedure is that time integration is performed using automatic order and stepsize selections hp adaptivity based on the backward differentiation formulas bdf the stepsize selection is based on a local error estimate an error controller and a step rejection mechanism it guarantees that the solution precision is within the user targeted tolerance the order selection is based on a stability test and a quarantine mechanism the selection is performed to ensure that no other methods within the family of 0 stable bdf methods would produce a solution of the targeted precision for a larger stepsize and thus a lower computational time to improve efficiency the time integration procedure also relies on a modified newton method and a predictor the time adaptive algorithm behaviors and performances are assessed on the vortex induced translational and rotational vibrations of a square cylinder and on the wake induced vibrations of 3 cylinders in an in line arrangement the algorithm yields substantial cpu time savings compared to constant stepsize and order integration while delivering solutions of prescribed accuracies

quantitative stability analysis for minimax distributionally robust risk optimization this paper considers distributionally robust formulations of a two stage stochastic programming problem with the objective of minimizing a distortion risk of the minimal cost incurred at the second stage we carry out a stability analysis by looking into variations of the ambiguity set under the wasserstein metric decision spaces at both stages and the support set of the random variables in the case when the risk measure is risk neutral the stability result is presented with the variation of the ambiguity set being measured by generic metrics of zeta structure which provides a unified framework for quantitative stability analysis under various metrics including total variation metric and kantorovich metric when the ambiguity set is structured by a zeta ball we find that the hausdorff distance between two zeta balls is bounded by the distance of their centers and difference of their radii the findings allow us to strengthen some recent convergence results on distributionally robust optimization where the center of the wasserstein ball is constructed by the empirical probability distribution

stability of multistep methods on variable grids this paper is concerned with the stability of multistep methods for ordinary initial value problems on grids with variable mesh sizes a necessary and sufficient condition for stability is given from which generalizations of recent results by gear et al and by zlatev can be obtained as special cases as an application the stability of the variable bdf formulas is treated

on the stability of variable stepsize nordsieck bdf methods in this paper the stability of backward differentiation methods that vary the stepsize using the interpolation technique of nordsieck is considered conditions for the stability of the methods are obtained the study is an extension of previous results given by gear and tu 3 and skeet and jackson 14

partial coxeter functors and right pure semisimple hereditary rings an important problem in the representation theory of artinian rings is to obtain a charcterization of rings of finite representation type see 2 7 16 201 we recall that a ring r is of finite representation type if r is both left and right artinian and there is only a finite number of isomorphism classes of indecomposable finitely generated r modules the main tools for studying the representation theory of hereditary artinian rings are partial coxeter functors and coxeter functors see 4 8 10 141 they are successfuly applied in the investigation of hereditary rings of finite representation type 8 9 12 141 in the present paper we study partial coxeter functors for hereditary artinian rings and we use them as a tool for studying right pure semisimple hereditary rings which are close to rings of finite representation type we recall that for any ring r the right pure global dimension r p gl dim r and the left pure global dimension l p gl dim r are defined 19 21 261 there is an interesting problem how to characterize rings r with r p gl dim r 0 which are exactly those right artinian rings r all of whose right r modules are algebraically compact 19 20 261 or equivalently all of whose right r modules are direct sums of modules of finite length a ring r with r p gl dim r 0 is called right pure semisimple 25 271 it is well known that any ring of finite representation type is both left and right pure semisimple 12 20 23 251 and the converse implication also holds true 2 15 201 however it is still an open question if a right pure semisimple ring r is of finite representation type or equivalently if r p gl dim r 0 implies l p gl dim r 0 see 29 a positive solution of this problem is given by auslander 3 for artin algebras he also discusses the problem for arbitrary left artinian rings see also 1181 we recall that a 195 0021 8693 81 070195 24 02 00 0

a family of multistep methods to integrate orbits on spheres a trajectory problem is an initial value problem where the interest lies in obtaining the curve traced by the solution rather than in finding the actual correspondence between the values of the parameter and the points on that curve this paper introduces a family of multi stage multi step numerical methods to integrate trajectory problems whose solution is on a spherical surface it has been shown that this kind of algorithms has good numerical properties consistency stability convergence and others that are not standard the latest ones make them a better choice for certain problems

the endomorphism ring of a localized coherent functor abstract letcbe a commutative artinian ring and an artinc algebra the category of coherent additive functorsa mod ab on the finitely presented right modules will be denoted by ab this category is equivalent to the free abelian category over the ring if s 0 ab is the serre subcategory of the finite length objects of ab anda ab it is proved that the endomorphism ring endab s 0 a s 0of the localized objecta s 0is a locally artinc algebra this is used to show that the krull gabriel dimension of the category ab cannot equal 1 in particular this holds for finite rings

the convergence of variable stepsize variable formula multistep methods in this paper variable stepsize variable formula multistep methods for the numerical solution of ordinary differential equations are considered some more general results on stability than are given in the papers of gear tu siam j numer anal 11 1974 pp 1025 1043 and gear watanabe siam j numer anal 11 1974 pp 1044 1058 are obtained further the asymptotic behavior of the error in the variable stepsize fixed formula case is studied finally the above study is extended to predictor corrector methods

numerical treatment of large scale air pollution models abstract studying air pollution phenomena over a large space domain by a fairly general mathematical model is discussed the space discretization of such a model leads to huge system of odes the integration algorithm used in the solution of these systems must be efficient if a 3 d model is considered then a vector processor is to be used moreover the most time consuming parts of the code are to be vectorized the calculated results concentrations of different air pollutants must be reliable because they have to be used by other specialists therefore different checks concerningthe reliability of the results are carried out in this paper it is shown that an efficient and reliable code for studying both sulphur and nitrogen pollution has been developed at the danish agency of environmental protection some simulation processes are also described breifly

robust solutions of optimization problems affected by uncertain probabilities in this paper we focus on robust linear optimization problems with uncertainty regions defined by divergences for example chi squared hellinger kullback leibler we show how uncertainty regions based on divergences arise in a natural way as confidence sets if the uncertain parameters contain elements of a probability vector such problems frequently occur in for example optimization problems in inventory control or finance that involve terms containing moments of random variables expected utility etc we show that the robust counterpart of a linear optimization problem with divergence uncertainty is tractable for most of the choices of typically considered in the literature we extend the results to problems that are nonlinear in the optimization variables several applications including an asset pricing example and a numerical multi item newsvendor example illustrate the relevance of the proposed approach this paper was accepted by gerard p cachon optimization

highly continuous interpolants for one step ode solvers and their application to runge kutta methods we suggest a general method for the construction of highly continuous interpolants for one step methods applied to the numerical solution of initial value problems of odes of arbitrary order for the construction of these interpolants one uses along with the numerical data of the discrete solution of a problem provided by a typical one step method at endstep points high order derivative approximations of this solution this approach has two main advantages it allows an easy way of construction of high order runge kutta and nystrom interpolants with reduced cost in additional function evaluations that also preserve the one step nature of the underlying discrete ode solver moreover for problems which are known to possess a solution of high smoothness the approximating interpolant resembles this characteristic a property that on occasion might be desirable an analysis of the stability behavior of such interpolatory processes is carried out in the general case a new numerical technique concerning the accurate determination of the stability behavior of numerical schemes involving higher order derivatives and or approximations of the solution from previous grid points over nonequidistant meshes is presented this technique actually turns out to be of a wider interest as it allows us to infer in certain cases more accurate results concerning the stability of for example the bdf formulas over variable stepsize grids moreover it may be used as a framework for analyzing more complex and supposedly more promising types of methods as they are the general linear methods for first and second order differential equations many particular variants of the new method for first order differential equations that have good prospects of finding a practical implementation are fully analyzed with respect to their stability characteristics a detailed application concerning the construction of c 2 and c 3 continuous extensions for some fifth and sixth order runge kutta pairs supplemented by a detailed study of the local truncation error characteristics of a class of interpolants of this type is also provided various numerical examples show in these cases several advantages of the newly proposed technique with respect to function evaluation cost and global error behavior in comparison with others currently in use

quantitative stability analysis for distributionally robust optimization with moment constraints in this paper we consider a broad class of distributionally robust optimization dro problems where the probability of the underlying random variables depends on the decision variables and the ambiguity set is defined through parametric moment conditions with generic cone constraints under some moderate conditions including slater type conditions of a cone constrained moment system and holder continuity of the underlying random functions in the objective and moment conditions we show local holder continuity of the optimal value function of the inner maximization problem with respect to w r t the decision vector and other parameters in moment conditions and local holder continuity of the optimal value of the whole minimax dro w r t the parameter moreover under the second order growth condition of the lagrange dual of the inner maximization problem we demonstrate and quantify the outer semicontinuity of the set of optimal solutions of the minimax dro w r t variation of the parameter finally w

covers of modules over a fixed ring abstract in this note we find conditions under which it is possible to prove the existence of relative injective covers of any module over the fixed ring r g by means of relative injective covers of modules over the base ring r the same problem is treated for flat covers

the stability of variable stepsize nordsieck methods conditions are given for the stability of multistep methods that vary the stepsize using the interpolation technique of nordsieck if the stepsize selection function is variation bounded then these methods are stable alternatively there exist constants a b c depending on the method and satisfying a leqq b 1 c such that if the stepsize ratio h n 1 h n satisfies either 0 h n 1 h n leqq a or b h n 1 h n leqq c then the method is stable as the frequency of stepsize changing decreases the methods become more stable in the sense that the intervals defined by a b and c are larger tables and graphs are given for the adams and backward differentiation methods indicating how stability changes with frequency of changing stepsize in particular it is shown that k 1 value adams formulas k leqq 6 are stable for 0 h n 1 h n leqq c that is it is shown that stability is maintained whenever the stepsize is decreased by an arb

two clifford s theorems for strongly group graded rings abstract let a be a strongly group graded ring in this paper we obtain the main results as following 1 in a a similar result to clifford s theorem of the decomposition of induced modules from normal subgroups it corrects the result of 4 2 frobenius reciprocity for a 3 we extend the theory of stable modules in two ways one of which is the localization theory of rings and modules in some special cases we can obtain the corresponding results of 1 2 3

elements of a general theory of composite integration methods this contribution is a survey of results on general linear methods after an introduction into the concept of a methods which is a slight generalization of general linear methods and its main results rk methods are presented in the a method notation which offers the following advantages a the order conditions are obtained from a simple recursion without relying on rooted trees or elementary differentials b they appear in a transparent form as orthogonality relations in dependence of the error constants of the stages of the rk method c their orthogonal structure permits elegant existence proofs and yields new auxiliary conditions d the error accumulation within the stages of the method is completely understood the proof of the main result on rk methods theorem 5 is simplified

distributionally robust reward risk ratio optimization with moment constraints reward risk ratio optimization is an important mathematical approach in finance we revisit the model by considering a situation where an investor does not have complete information on the distribution of the underlying uncertainty and consequently a robust action is taken to mitigate the risk arising from ambiguity of the true distribution we consider a distributionally robust reward risk ratio optimization model varied from the ex ante sharpe ratio where the ambiguity set is constructed through prior moment information and the return function is not necessarily linear we transform the robust optimization problem into a nonlinear semi infinite programming problem through standard lagrange dualization and then use the well known entropic risk measure to construct an approximation of the semi infinite constraints we solve the latter by an implicit dinkelbach method finally we apply the proposed robust model and numerical scheme to a portfolio optimization problem and report some preliminary numerical

numerical investigations on global error estimation for ordinary differential equations four techniques of global error estimation which are richardson extrapolation rs zadunaisky s technique zd solving for the correction sc and integration of principal error equation ipee have been compared in different integration codes dopri5 dvode dstep theoretical aspects concerning their implementations and their orders are first given second a comparison of them based on a large number of tests is presented in terms of cost and precision sc is a method of choice for one step methods it is much more precise and less costly than rs and leads to the same precision as zd for half its cost ipee can provide the order of the error for a cheap cost in codes based on one step methods in multistep codes only rs and ipee have been implemented since they are the only ones whose theoretical justification has been extended to this case there rs still provides a more reliable estimation than ipee however as these techniques are based on variations of the global error irrespective of the numerical method used they fail to provide any more usefull information once the numerical method has reached its limit of accuracy due to the finite arithmetic

on the sparsity of representations of rings of pure global dimension zero it is shown that the rings r all of whose left modules are direct sums of finitely generated modules satisfy the following finiteness condition for each positive integer n there are only finitely many isomorphism types of a indecomposable left r modules of length n b finitely presented indecomposable right r modules of length n c indecomposable right r modules having minimal projective resolutions with n relations moreover our techniques yield a very elementary proof for the fact that the presence of the above decomposability hypothesis for both left and right r modules entails finite representation type

quantitative stability of two stage distributionally robust risk optimization problem with full random linear semi definite recourse abstract in this paper we study a distributionally robust risk optimization drro problem where the information on the probability distribution of the underlying random variables is incomplete but it is possible to use partial information to construct an ambiguity set of probability distributions in some cases decision vector x may have a direct impact on the likelihood of the underlying random events that occur after the decision is taken which motivates us to propose an ambiguity set to be parametric and decision dependent to conduct quantitative stability analysis of the optimal value function and the optimal solution mapping of the drro problem we derive error bounds results for the parametrized ambiguity set under the total variation metric and investigate lipschitz continuity of the objective function of the drro problem under some conditions as an application we demonstrate that the two stage stochastic linear semi definite programs satisfy these conditions and then apply results obtained to it

variational theory for optimization under stochastic ambiguity stochastic ambiguity provides a rich class of uncertainty models that includes those in stochastic robust risk based and semi infinite optimization and that accounts for uncertainty about parameter values as well as incompleteness of the description of uncertainty we provide a novel unifying perspective on optimization under stochastic ambiguity that rests on two pillars first ambiguity is formulated in terms of the cumulative probability distribution associated with the random elements more specifically ambiguity is expressed by letting this distribution belong to a subfamily of distributions that might or might not depend on the decision variable we derive a series of estimates by introducing a metric for the space of distribution functions based on the hypo distance between upper semicontinuous functions in the process we show that this metric is consistent with convergence in distribution weak star convergence of the associated probability measures second we rely on the theory

multistep numerical methods based on the scheifele g functions with application to satellite dynamics in 1971 scheifele designed a method for the integration of perturbed oscillators by refining the classical method of power series this method consists of defining a certain sequence of entire functions and finding the sought solution in the form of a linear combination of them using coefficients obtained through recurrence formulas in spite of the good behavior of the scheifele method it is only practical for use in a few particular cases in which the force function is very simple given the complexity of the preliminary calculations needed to obtain the recurrence formulas this problem is resolved in this article by converting the g functions method into a multistep scheme the key property is that those algorithms are able to integrate without truncation error harmonic oscillators involving various frequencies and that for perturbed problems the local error contains the small perturbation parameter as a factor therefore the methods are useful for finding highly accurate solutions to problems whose solutions are close to quasi periodic ones an application is performed in investigating the behavior of the proposed schemes for the integration of the orbital motion of earth satellites

on the zero stability of the variable order variable stepsize bdf formulas this paper contains bounds for the mesh size ratiosh j 1 h j of nonequidistant grids such that any combination of the widely used backward differentiation formulas with ordersl 1 l m m 3 4 5 performs zerostable

endofinite modules and pure semisimple rings let r be a right pure semisimple ring i e a ring r such that every right r module is a direct sum of finitely generated modules it is proved that r is of finite representation type if and only if every finitely presented indecomposable right r module is endofinite if and only if every finitely presented right r module has a left artinian endomorphism ring as applications we obtain an alternative proof of the pure semisimplicity conjecture for pi rings and new criteria for a right pure semisimple ring to be of finite representation type

decomposition algorithms for two stage distributionally robust mixed binary programs in this paper we introduce and study a two stage distributionally robust mixed binary problem tsdr mbp in which the random parameters follow the worst case distribution belonging to an uncertain

group gradings on full matrix rings we study g gradings of the matrix ring mn k k a field and give a complete description of the gradings where all the elements ei j are homogeneous called good gradings among these we determine the ones that are strong gradings or crossed products if g is a finite cyclic group and k contains a primitive g th root of 1 we show how all g gradings of mn k can be produced in particular we give a precise description of all c2 gradings of m2 k and show that for algebraically closed k any such grading is isomorphic to one of the two good gradings

decomposition based interior point methods for two stage stochastic semidefinite programming we introduce two stage stochastic semidefinite programs with recourse and present an interior point algorithm for solving these problems using bender s decomposition this decomposition algorithm and its analysis extend zhao s results math program 90 2001 pp 507 536 for stochastic linear programs the convergence results are proved by showing that the logarithmic barrier associated with the recourse function of two stage stochastic semidefinite programs with recourse is a strongly self concordant barrier on the first stage solutions the short step variant of the algorithm requires o sqrt p kr ln mu 0 epsilon newton iterations to follow the first stage central path from a starting value of the barrier parameter mu 0 to a terminating value e the long step variant requires o p kr ln mu 0 epsilon damped newton iterations the calculation of the gradient and hessian of the recourse function and the first stage newton direction decomposes across the second stage scenarios

indecomposable modules over pure semisimple hereditary rings abstract if r is a hereditary left artinian ring then r is left pure semisimple if and only if the family r ind of all finitely generated indecomposable left r modules has a unique ext injective partition r ind u this partition is used to give a complete description of the distribution of all indecomposable modules over a left pure semisimple hereditary indecomposable ring r of infinite representation type more precisely r ind is the disjoint union of the countable set of all preinjective modules and the finite set of all preprojective modules and countable sets of auslander reiten components of the form k u k for all limit ordinals constructed from the ext injective partition of r ind in particular we show that an indecomposable left r module m is not the source of a left almost split morphism in r mod if and only if m belongs to u where is an infinite limit ordinal and the direct sum of modules in u is not endofinite for each infinite limit ordinal moreover the endomorphism ring of each indecomposable left r module is a division ring

multiderivative variable stepsize variable formula methods abstract in this paper we present a study of consistency stability and convergence properties of linear multiderivative multistep variable stepsize variable formula methods

a 0 stability of variable stepsize bdf methods abstract in this paper the a0 stability of interpolatory variable coefficient and fixed leading coefficient bdf formulas is considered a comparison of the stability properties of the three methods based on their spectral radius is presented in addition we study the effect of some strategies of stepsize variation employed in usual codes on the stability of these methods

variable step implementation of etd methods for semilinear problems abstract spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations exponential integrators solve exactly the linear part which is assumed to be responsible of the stiffness of the problem while the nonlinear terms are approximated numerically in this paper a variable step implementation of the exponential time differencing methods of multistep type is considered a local error estimate of exponential type whose computation is for free is introduced and some implementation issues for diagonalizable matrices are addressed numerical results showing the behaviour of the proposed variable step implementation are provided too

analysis of variable stepsize linear multistep methods with special emphasis on symmetric ones in this paper we deal with several issues concerning variable stepsize linear multistep methods first we prove their stability when their fixed stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients then we prove asymptotic expansions on the considered tolerance for the global error committed using them we study the growth of error with time when integrating periodic orbits we consider strongly and weakly stable linear multistep methods for the integration of first order differential systems as well as those designed to integrate special second order ones we place special emphasis on the latter which are also symmetric because of their suitability when integrating moderately eccentric orbits of reversible systems for these types of methods we give a characterization for symmetry of the coefficients which allows their construction and provide some numerical results for them

computable explicit bounds for the discretization error of variable stepsize multistep methods abstract this paper deals with the achievement of explicit computable bounds for the global discretization error of variable stepsize multistep methods which are perturbation of strongly stable fixed stepsize methods the approach is based on the study of the growth of solutions of certain variable coefficient difference equations satisfied by the global discretization error

advance in the theory of variable stepsize variable formula methods for ordinary differential equations the theory of the constant stepsize constant formula methods cscfm s for solving initial value problems for ordinary differential equations ode s is well developed however the integration methods are often used as variable stepsize variable formula methods vsvfm s in the codes for automatic solution of ode s this fact explains why it is necessary to develop a rigorous theory for the vsvfm s also results obtained until now in the efforts to develop a theory for the vsvfm s are presented in a systematic way and compared with the well known results for cscfm s the comparison is used to outline the directions in which further efforts in the attempt to improve the theoretical results and to obtain new results are most needed the integration methods can be separated into two large groups implicit methods and explicit methods including here the predictor corrector schemes it is discussed what kind of results are needed for each of these two groups assume that a set f of cscfm s is given from a practical point of view it is highly desirable that the cscfm s belonging to set f have the following property if every cscfm from set f is consistent zero stable and convergent then any vsvfm induced by the cscfm s of set f is also consistent zero stable and convergent for a sufficiently large class of grids a theorem concerning a class of cscfm s with this property is presented the application of this theorem in the selection of methods for solving large systems of ode s arising after the space discretization of air pollution models is discussed it is explained why the vsvfm s based on this class provide a very efficient computational process in the treatment of air pollution models

a key module over pure semisimple hereditary rings abstract let r be a hereditary indecomposable left pure semisimple ring inspired by i reiten c m ringel infinite dimensional representations of canonical algebras canad j math 58 2006 180 224 we investigate the perfect cotorsion pair c d in r mod generated by the preinjective component q we show that there is a finitely generated product complete tilting and cotilting left r module w such that c cogen w and d gen w the finite subcategory w of r mod given by the indecomposable summands of w stores important information on r for example if we assume r of infinite representation type then by b zimmermann huisgen strong preinjective partitions and representation type of artinian rings proc amer math soc 109 1990 309 322 there are non preinjective indecomposable modules occurring as direct summands of products of preinjective modules and it turns out that w is precisely the class of such modules moreover we prove that r has finite representation type if and only if every module in w is source of a left almost split map in r mod finally we address the question when w is endofinite

auslander reiten components over pure semisimple hereditary rings abstract let r be a hereditary indecomposable left pure semisimple ring we show that r has finite representation type if and only if a certain finitely presented module is endofinite namely the tilting and cotilting module w studied in l angeleri hugel 2007 2 we then apply the tilting and the cotilting functors to study the endomorphism ring of w and its auslander reiten components finally we transfer this information to the category of right r modules

convergence analysis for distributionally robust optimization and equilibrium problems in this paper we study distributionally robust optimization approaches for a one stage stochastic minimization problem where the true distribution of the underlying random variables is unknown but it is possible to construct a set of probability distributions which contains the true distribution and optimal decision is taken on the basis of the worst possible distribution from that set we consider the case when the distributional set which is also known as the ambiguity set varies and its impact on the optimal value and the optimal solutions a typical example is when the ambiguity set is constructed through samples and we need to look into the impact of increasing the sample size the analysis provides a unified framework for convergence of some problems where the ambiguity set is approximated in a process with increasing information on uncertainty and extends the classical convergence analysis in stochastic programming the discussion is extended briefly to a stochastic nash equilibrium problem wh