primal dual aggregation and disaggregation for stochastic linear programs a multistage stochastic linear program can be approximated by replacing the underlying information structure by a coarser structure given the ordinary lagrangian for the original problem this amounts to restricting both the primal and dual variables to be adapted to the coarser information structure the resulting primal problem has the same form as the original but with aggregated constraints and decisions mutual upper and lower bounds on the optimal values of the original and aggregated problems are obtained via partially aggregated problems formed by restricting only the primal variables or only the dual variables to the coarser structure the introduction of the partially aggregated problems leads to a general disaggregation framework for developing heuristic approaches to solving the original problem

accelerating the regularized decomposition method for two stage stochastic linear problems practical improvements of the regularized decomposition algorithm for two stage stochastic problems are presented they are associated with the primal simplex method for solving subproblems a penalty formulation of the subproblems is used which facilitates crash and warm starts and allows more freedom when creating the model the computational results are highly encouraging

stochastic network interdiction using limited assets an interdictor attempts to destroy parts of a capacitated network through which an adversary will subsequently maximize flow we formulate and solve a stochastic version of the interdictor s problem minimize the expected maximum flow through the network when interdiction successes are binary random variables extensions are made to handle uncertain arc capacities and other realistic variations these two stage stochastic integer programs have applications to interdicting illegal drugs and to reducing the effectiveness of a military force moving materiel troops information etc through a network in wartime two equivalent model formulations allow jensen s inequality to be used to compute both lower and upper bounds on the objective and these bounds are improved within a sequential approximation algorithm successful computational results are reported on networks with over 100 nodes 80 interdictable arcs and 180 total arcs

distribution of energy levels of quantum free particle on the liouville surface and trace formulae we consider the weyl asymptotic formula left e n leqq r 2 right frac area q 4 pi r 2 n r for eigenvalues of the laplace beltrami operator on a two dimensional torusq with a liouville metric which is in a sense the most general case of an integrable metric we prove that if the surfaceq is non degenerate then the remainder termn r has the formn r r1 2 r where r is an almost periodic function of the besicovitch classb1 and the fourier amplitudes and the fourier frequencies of r can be expressed via lengths of closed geodesics onq and other simple geometric characteristics of these geodesics we prove then that if the surfaceq is generic then the limit distribution of r has a densityp t which is an entire function oft possessing an asymptotics on a real line logp t c t4 ast an explicit expression for the fourier transform ofp t via fourier amplitudes of r is also given we obtain the analogue of the guillemin duistermaat trace formula for the liouville surfaces and discuss its accuracy

mixed l p l 2 norms of the lattice point discrepancy we estimate some mixed lp l2 norms of the discrepancy between the volume and the number of integer points in r x a dilation by a factor r and a translation by a vector x of a convex body in r d with smooth boundary with nonvanishing gaussian curvature int t d 1 h int r r h sum kin z d r x k r d dr p 2 dx 1 p we obtain estimates for fixed values of h and r and also asymptotic estimates when h

on the generalized circle problem for a random lattice in large dimension abstract in this note we study the error term r n l x in the generalized circle problem for a ball of volume x and a random lattice l of large dimension n our main result is the following functional central limit theorem fix an arbitrary function f z r satisfying lim n f n and f n o e e e n for every e 0 then the random function t 1 2 f n r n l t f n on the interval 0 1 converges in distribution to one dimensional brownian motion as n the proof goes via convergence of moments and for the computations we develop a new version of rogers mean value formula from 18 for the individual kth moment of the variable 2 f n 1 2 r n l f n we prove convergence to the corresponding gaussian moment more generally for functions f satisfying f n o e c n for any fixed c 0 c k where c k is a constant depending on k whose optimal value we determine

a regularized decomposition method for minimizing a sum of polyhedral functions a problem of minimizing a sum of many convex piecewise linear functions is considered in view of applications to two stage linear programming where objectives are marginal values of lower level problems it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise linear induced feasibility constraints we propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component the master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts the method goes through nonbasic points in general and is finitely convergent without any nondegeneracy assumptions next we present a special technique for solving the regularized master problem that uses an active set strategy and qr factorization and exploits the structure of the master finally some numerical evidence is given

order and chaos in some trigonometric series curious adventures of a statistical mechanic this paper tells the story how a maple assisted quest for an interesting undergraduate problem in trigonometric series led some amateurs to the discovery that the one parameter family of deterministic trigonometric series p 1 exhibits both order and apparent chaos and how this has prompted some professionals to offer their expert insights as to order an elementary undergraduate proof is given that t with explicitly computed constant p as to chaos the seemingly erratic fluctuations about this overall trend are discussed experts commentaries are reproduced as to why the fluctuations of are presumably not gaussian inspired by a central limit type theorem of marc kac a well motivated conjecture is formulated to the effect that the fluctuations of the t 1 p 1 th partial sum of when properly scaled do converge in distribution to a standard gaussian when t though provided that p is chosen so that the frequencies n p n are rationally linear independent no conjecture has been forthcoming for rationally dependent n p n moreover following other experts tip offs the interesting relationship of the asymptotics of to properties of the riemann function is exhibited using the mellin transform

new variants of bundle methods in this paper we describe a number of new variants of bundle methods for nonsmooth unconstrained and constrained convex optimization convex concave games and variational inequalities we outline the ideas underlying these methods and present rate of convergence estimates

sign changes of the error term in weyl s law for heisenberg manifolds let r t be the error term in weyl s law for the 2l 1 dimensional heisenberg manifold hl gl in this paper several results on the sign changes and odd moments of r t are proved in particular it is proved that for some sufficiently large constant c r t changes sign in the interval t t c t for all large t moreover for a small constant c1 there exist infinitely many subintervals in t 2t of length c1 t log 5 t such that r t c1tl 1 4 holds on each of these subintervals

an adaptive partition based approach for solving two stage stochastic programs with fixed recourse we study an adaptive partition based approach for solving two stage stochastic programs with fixed recourse a partition based formulation is a relaxation of the original stochastic program and we study a finitely converging algorithm in which the partition is adaptively adjusted until it yields an optimal solution a solution guided refinement strategy is developed to refine the partition by exploiting the relaxation solution obtained from a partition in addition to refinement we show that in the case of stochastic linear programs it is possible to merge some components in a partition without weakening the corresponding relaxation bound thus allowing the partition size to be kept small we also show that for stochastic linear programs with simple recourse there exists a small partition that yields an optimal solution the size of this partition is independent of the number of scenarios used in the model our computational results show that the proposed adaptive partition based approach converges v

thermomechanical response of solids at high strains natural approach abstract the paper develops in pursuance of 4 the theoretical framework appertaining to coupled thermomechanical deformations of solids subject to large as well as inelastic deformations the essential feature of the analysis is a consistent natural formulation which encompasses also all thermodynamic aspects on the other hand it must be admitted that current concepts on mechanical constitutive assumptions 5 6 are not sufficiently sophisticated to permit a precise generalisation for the purpose of incorporating thermomechanical effects and delivering a fully compatible theory of the coupled problem uniqueness and bifurcation of the solutions as well as stable postcritical deformation paths are also considered in this account furthermore algorithms for the solution of the nonlinear algebraic equations representing the discretised problem and in particular quasi newton techniques are discussed the paper concludes with a number of applications of technical relevance

applying oracles of on demand accuracy in two stage stochastic programming a computational study abstract traditionally two variants of the l shaped method based on benders decomposition principle are used to solve two stage stochastic programming problems the aggregate and the disaggregate version in this study we report our experiments with a special convex programming method applied to the aggregate master problem the convex programming method is of the type that uses an oracle with on demand accuracy we use a special form which when applied to two stage stochastic programming problems is shown to integrate the advantages of the traditional variants while avoiding their disadvantages on a set of 105 test problems we compare and analyze parallel implementations of regularized and unregularized versions of the algorithms the results indicate that solution times are significantly shortened by applying the concept of on demand accuracy

static and dynamic analysis of space frames using simple timoshenko type elements in this paper a finite element method for geometrically and materially non linear analyses of space frames is described beams with both solid and thin walled open cross sections are considered the equations of equilibrium are formulated using an updated incremental lagrangian description the elements developed can undergo large displacements and rotations but the incremental rotations are assumed to be small the material behaviour is described by elastoplastic temperature dependent elastoplastic and viscoplastic models with special reference to metals computationally more economical formulations based on the relationship between stress resultants and generalized strain quantities are also presented in the case of thin walled beams the torsional behaviour is modelled using a two parameter warping model where the angle of twist and the axial variation of warping have independent approximations this approach yields average warping shear strains directly from the displacement assumptions and no discrepancy between stress and strain fields exists

aggregation bounds in stochastic linear programming stochastic linear programs become extremely large and complex as additional uncertainties and possible future outcomes are included in their formulation row and column aggregation can significantly reduce this complexity but the solutions of the aggregated problem only provide an approximation of the true solution in this paper error bounds on the value of the optimal solution of the original problem are obtained from the solution of the aggregated problem these bounds apply for aggregation of both random variables and time periods

tight bounds for stochastic convex programs variable and row aggregation as a technique of simplifying a mathematical program is utilized to develop bounds for two stage stochastic convex programs with random right hand sides if one is able to utilize the problem structure along with only first moment information a tighter bound than the usual mean model bound based on jensen s inequality may be obtained moreover it is possible to construct examples for which the mean model bound will be arbitrarily poor consequently one can tighten jensen s bound for stochastic programs when the distribution has a compact support this bound may be improved further by partitioning the support using conditional first moments with regard to first moment upper bounds the gassmann ziemba inequality is used for the stochastic convex program to seek a model which can be solved using standard convex programming techniques moreover it allows one to easily construct upper bounds using the solution of the lower bounding problem finally the results are extended to multistage stochastic convex programming problems

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

the variance of the error function in the shifted circle problem is a wild function of the shift we prove that the variance of the error function in the shifted circle problem as a function of the shift is a continuous function which has a sharp local maximum with infinite derivatives at every rational point on a plane

risk averse optimization in two stage stochastic models computational aspects and a study we extend the on demand accuracy approach of oliveira and sagastizabal to constrained convex optimization the resulting method is applied to risk averse two stage stochastic programming problems we present a survey of risk averse models the appropriate oracle is formulated for the case of a conditional value at risk constraint we discuss computational aspects and compare different approaches in a study

some mathematical problems in the theory of quantum chaos abstract we discuss the general problem of distribution of spacings between levels in quantum problems for the quantum kicked rotator model we formulate the results on the poisson distribution for spacings some general remarks concerning the reasons for the appearance of this distribution are given

nonlinear finite element analysis of elastic systems under nonconservative loading natural formulation part ii dynamic problems abstract the paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces attention is focused on the stability behaviour of such systems this leads mathematically to non self adjoint boundary value problems which are of great theoretical and practical interest in particular in connection with alternative modes of instability like divergence of flutter only quasistatic effects are however considered in the present part the methodology of our theory is general but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement dependent nonconservative follower forces in a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives in general rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix as a result the total tangent stiffness matrix becomes unsymmetric an indication of the non self adjoint character of the problem our theory is based on the natural mode technique 1 2 3 and permits i a a simple but elegant derivation of the load correction matrix the application of the general theory as evolved in this paper is demonstrated on the beam element in space a number of numerical examples are considered including divergence and flutter types of instability for which exact analytic solutions are known the problems demonstrate the efficiency of the present finite element formulation the paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in 2 3

distribution of the error term for the number of lattice points inside a shifted circle define begin array 20 c n x r left n in z d x n leqslant r right x in 0 1 d end array 1 1 and f x r r frac d 1 2 n x r r d text vol d b 1 1 2 where b 1 is the unit ball in r d thus n x r is the number of lattice points in z d in the r ball centered at x and f x r is the normalized difference with the volume of the ball we are interested in the problem of the distribution of f x r as a function of r for given x

the number of integral points on arcs and ovals integral lattice points and that the exponent and constant are best possible however swinnerton dyer 10 showed that the preceding result can be substantially improved if we start with a fixed c strictly convex arc and consider the number of lattice points on t the dilation of by a factor t t 1 this of course is the same as asking for rational points mn n n on as n in fact swinnerton dyer proves a bound of type t zz c e t 3 5 e

energy level statistics of model quantum systems universality and scaling in a lattice point problem we investigate the statistics of the numbern r s of lattice pointsn z 2 in an annular domain r w r w a ra wherer w 0 herea is a fixed convex set with smooth boundary andw is chosen so that the area of r w iss the statistics comes fromr being taken as random with a smooth density in some interval c 1 t c 2 t c 2 c 1 0 we find that in the limitt the variance and distribution of n n r s s depend strongly on hows grows witht there is a saturation regimes t ast in which the fluctuations in n coming from the two boundaries of are independent then there is a scaling regime s t z 0 z in which the distribution depends onz in an almost periodic way going to a gaussian asz 0 the variance in this limit approachesz for generic a but can be larger for degenerate cases the former behavior is what one would expect from the poisson limit of a distribution for annuli of finite area

static and dynamic stability of nonlinear elastic systems under nonconservative forces natural approach abstract the paper develops a nonlinear large displacement small strain finite element analysis of elastic structures subject to arbitrary nonconservative external forces both in the static and dynamic realms special attention is focused on the stability behaviour of such systems the methodology of the theory is general covering displacement dependent circulatory velocity dependent dissipative and acceleration dependent stationary as well as instationary explicitly on time dependent nonconservative forces the finite element formulation selected here is based on the natural mode technique permitting i a a simple but elegant as well as concise derivation of the contributory nonsymmetric matrices in conjunction with a clear physical interpretation some typical examples of coalescence and single mode flutter are presented most attention is focused on the evolution of highly nonlinear post flutter response due to various kinds of disturbances in both the stable and the unstable domains the discussion demonstrates the efficiency of the present finite element formulation

mixed models and reduction techniques for large rotation nonlinear problems abstract mixed models and reduction techniques are presented for the large rotation nonlinear analysis of curved beams a total lagrangian description of the beam deformation is used and the analytical formulation is based on a form of reissner s large deformation theory with the effects of transverse shear deformation and extensibility of the centerline included only planar deformations are considered and the fundamental unknowns consist of the three internal forces and the three generalized displacements of the beam the element characteristic arrays are obtained by using the hellinger reissner mixed variational principle the polynomial interpolation or shape functions used in approximating the internal forces are in general of different degree than those used for approximating the generalized displacements and the forces are discontinuous at interelement boundaries the use of reduction methods in conjunction with the mixed models is outlined and the advantages of mixed models over displacement models are delineated numerical results are presented to demonstrate the high accuracy and effectiveness of both the mixed models and the reduction techniques

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

on spectral statistics of classically integrable systems this work is an extensive study of the spectral statistics of three representative classically integrable systems namely rectangle torus and circle billiards we analyse the statistics and focus on the related level spacing distribution p s and the delta statistics the agreement with the poissonian model is typically found to be perfect up to the outer unfolded energy scale beyond which the saturation is observed in agreement with berry s dynamical theory of the spectral rigidity where as the untypical systems are those where for example p s is not a smooth distribution but a sum of the delta functions due to the granularity of the energy scale for example in the rectangle with the rational squared sides ratio however even there we find reasonable trend towards poissonian statistics for large ranges l but we describe theoretically and numerically the broadening of the delta spikes when the rectangular billiard is slightly distorted away from a rational to an irrational shape and find excellent agreement also in irrational rectangle billiards we show and explain the existence of large fluctuations by one order of magnitude bigger than the statistical ones whose origin is in the closeness to some rational billiard shape these fluctuations and their amplitude are independent of the energy if the bin size shrinks inversely with energy finally we tested the mode distribution i e distribution of the reduced mode fluctuation number w and found that it was not poissonian in agreement with steiner s conjecture and in fact follows the prediction by bleher et al that its tail behaves as the general reason for non universal behaviour of p w is that at largest energy scales we are always in the saturation regime

discrete analogues in harmonic analysis spherical averages in this paper we prove an analogue in the discrete setting of d o fthe spherical maximal theorem for d the methods used are two fold the application of certain sampling techniques and ideas arising in the study of the number of representations of an integer as a sum of d squares in particular the circle method the results we obtained are by necessity limited to d 5 and moreover the range of p for the l p estimates differs from its analogue in d

spectral form factors of rectangle billiards the berry tabor conjecture asserts that local statistical measures of the eigenvalues j of a generic integrable quantum system coincide with those of a poisson process we prove that in the case of a rectangle billiard with random ratio of sides the sum behaves for random and n large like a random walk in the complex plane with a non gaussian limit distribution the expectation value of the distribution is zero its variance which is essentially the average pair correlation function is one in accordance with the berry tabor conjecture but all higher moments 4 diverge the proof of the existence of the limit distribution uses the mixing property of a dynamical system defined on a product of hyperbolic surfaces the berry tabor conjecture and the existence of the limit distribution for a fixed generic rectangle are related to an equidistribution conjecture for long horocycles on this product space

aggregation and disaggregation techniques and methodology in optimization a fundamental issue in the use of optimization models is the tradeoff between the level of detail and the ease of using and solving the model aggregation and disaggregation techniques have proven to be valuable tools for manipulating data and determining the appropriate policies to employ for this tradeoff furthermore aggregation and disaggregation techniques offer promise for solving large scale optimization models supply a set of promising methodologies for studying the underlying structure of both univariate and multivariate data sets and provide a set of tools for manipulating data for different levels of decision makers in this paper we develop a general framework for aggregation and disaggregation methodology survey previous work regarding aggregation and disaggregation techniques for optimization problems illuminate the appropriate role of aggregation and disaggregation methodology for optimization applications and propose future research directions

inexact bundle methods for two stage stochastic programming stochastic programming problems arise in many practical situations in general the deterministic equivalents of these problems can be very large and may not be solvable directly by general purpose optimization approaches for the particular case of two stage stochastic programs we consider decomposition approaches akin to a regularized l shaped method that can handle inexactness in the subproblem solution from a nonsmooth optimization perspective these variants amount to applying a proximal bundle method to an oracle that gives inaccurate values for the objective function and a subgradient rather than forcing early termination of the subproblem optimization to define inexact oracles we select a small subset of scenarios for which the subproblem solution is exact and we replace the information for the remaining scenarios by a fast procedure that does not involve solving an optimization problem the inaccurate oracle information creates inexact cuts in the master program which are well handled by th