spectral asymptotics for the third order operator with periodic coefficients abstract we consider the self adjoint third order operator with 1 periodic coefficients on the real line the spectrum of the operator is absolutely continuous and covers the real line we determine the high energy asymptotics of the periodic antiperiodic eigenvalues and of the branch points of the lyapunov function furthermore in the case of small coefficients we show that either whole spectrum has multiplicity one or the spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three in the last case the asymptotics of this small interval is determined

a second wronskian formulation of the boussinesq equation a wronskian formulation leading to rational solutions is presented for the boussinesq equation it involves third order linear partial differential equations whose representative systems are systematically solved the resulting solutions formulas provide a direct but powerful approach for constructing rational solutions positon solutions and complexiton solutions to the boussinesq equation various examples of exact solutions of those three kinds are computed the newly presented wronskian formulation is different from the one previously presented by li et al which does not yield rational solutions

risk sensitive and risk neutral multiarmed bandits for the multiarmed bandit the classic result is probabilistic each state of each bandit markov chain with rewards has an index that is determined by an optimal stopping time for that state s bandit and expected discounted income is maximized by playing at each epoch a bandit whose current state has the largest index our approach is analytic not probabilistic it uses pairwise comparison in place of stopping times a simple recursion assigns to each state of each bandit a utility and an amplification of future utility that depend solely on the data for that state s bandit these utilities and amplifications determine whether or not one state dominates another we show that it is optimal to play at each epoch any bandit whose current state is not dominated by the current states of the other bandits we obtain this result by a coherent analysis that encompasses three models one with risk averse exponential utility one with risk seeking exponential utility and one with linear utility and either stopping or discounting we also show that the risk seeking case and a model of nash nash p 1980 a generalized bandit problem j roy statist soc b42 165 169 are equivalent to each other

resonances of third order differential operators abstract we consider resonances for third order ordinary differential operators with compactly supported coefficients on the real line resonance are defined as zeros of a fredholm determinant on a non physical sheet of three sheeted riemann surface we determine upper bounds of the number of resonances in complex discs at large radius we express the trace formula in terms of resonances only

transient policies in discrete dynamic programming linear programming including suboptimality tests and additional constraints this paper investigates the computation of transient optimal policies in discrete dynamic programming the model is quite general it may contain transient as well as nontransient policies and the transition matrices are not necessarily substochastic a functional equation for the so called transient value vector is derived and the concept of superharmonicity is introduced this concept provides the linear program to compute the transientvalue vector and a transient optimal policy we also discuss the elimination of suboptimal actions the solution of problems with additional constraints and the computation of an efficient policy for a multiple objective dynamic programming problem

dynamic recomputation cannot extend the optimality range of priority indices sequencing rules that rely on priority indices are particularly attractive because they are simple to calculate and easy to implement it has been established that there are exactly two classes of delay cost functions for which policies that are determined by time invariant priority indices are capable of producing optimal sequences linear delay costs and discounted linear delay costs we consider index based policies that allow dynamic recalculation of priority indices and show that this class does not enlarge the set of delay cost functions for which index based rules can produce optimal policies our analysis relies on the argument that index based rules must induce a transitive ranking of the tasks we show that the classes of linear and discounted linear functions are the only ones that can be associated with such rankings

the knowledge gradient algorithm for a general class of online learning problems we derive a one period look ahead policy for finite and infinite horizon online optimal learning problems with gaussian rewards our approach is able to handle the case where our prior beliefs about the rewards are correlated which is not handled by traditional multiarmed bandit methods experiments show that our kg policy performs competitively against the best known approximation to the optimal policy in the classic bandit problem and it outperforms many learning policies in the correlated case

a generalized gittins index for a class of multiarmed bandits with general resource requirements we generalise classical multiarmed bandits to allow for the distribution of a fixed amount of a divisible resource among the constituent bandits at each decision point bandit activation consumes amounts of the available resource which may vary by bandit and state any collection of bandits may be activated at any decision epoch provided they do not consume more resource than is available we propose suitable bandit indices that reduce to those proposed by gittins gittins j c 1979 bandit processes and dynamic allocation indices with discussion j r statist soc b41 148 177 for the classical models the index that emerges is an elegant generalization of the gittins index which measures in a natural way the reward earnable from a bandit per unit of resource consumed the paper discusses both how such indices may be computed and how they may be used to construct heuristics for resource distribution we also describe how to develop bounds on the closeness to optimality of index heuristics and demonstrate a form of asymptotic optimality for a greedy index heuristic in a class of simple models a numerical study testifies to the strong performance of a weighted index heuristic

the boussinesq equation revisited the continuous spectrum and soliton solutions for the boussinesq equation are investigated using the dressing method solitons demonstrate quite extraordinary behavior they may decay or form a singularity in a finite time formation of singularity collapse of solitons for the boussinesq equation was discovered several years ago systematic study of the solitonic sector is presented

the non local delta problem and 2 1 dimensional soliton equations a method of constructing 2 1 dimensional non linear integrable equations and their solutions by means of the non local delta problem is developed a basic set of equations is obtained by using different normalisations of the non local delta problem and the lagrangian of the set is found other integrable equations which are degenerate cases of the basic set are also lagrangian

index policies for stochastic search in a forest with an application to r d project management this paper concerns a stochastic search problem in a forest as motivation consider the issue of investing in a research and development project each activity that could be undertaken in the project is represented as an edge in a forest each edge has a cost of being attempted and a probability of success an edge can be attempted if each of its predecessors has been attempted and if each of those attempts has succeeded the overall project succeeds if a path is found from a stem to a leaf of the forest all of whose edges are successful overall success yields an economic benefit the problem is to find an investment strategy that maximizes expected utility either with a linear or an exponential utility function this problem will be shown to have a simple solution each edge will be assigned an index such that expected utility is maximized by attempting at each opportunity an edge whose index is most positive terminating the search when no edges remain whose indices are positive these indices are nested in a way that makes them quick to compute

polynomial degree robust a posteriori estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed discretizations we present equilibrated flux a posteriori error estimates in a unified setting for conforming nonconforming discontinuous galerkin and mixed finite element discretizations of the two dimensional poisson problem relying on the equilibration by the mixed finite element solution of patchwise neumann problems the estimates are guaranteed locally computable locally efficient and robust with respect to polynomial degree maximal local overestimation is guaranteed as well numerical experiments suggest asymptotic exactness for the incomplete interior penalty discontinuous galerkin scheme

solutions to the kaup broer system and its 2 1 dimensional integrable generalization via the dressing method abstract in this paper we formulate the nonlocal dbar problem dressing method of manakov and zakharov zakharov and manakov 1984 1985 zakharov 1989 for the 4 scaling classes of the 1 1 dimensional kaup broer system broer 1975 kaup 1975 the applications of the method for the 1 1 dimensional kaup broer systems are reductions of a method for a complex valued 2 1 dimensional completely integrable partial differential equation first introduced in rogers and pashaev 2011 this method allows computation of solutions to all scaling classes of the kaup broer system we then consider the case of non capillary waves with gravitational forcing and use the dressing method to compute n soliton solutions and more general solutions in the closure of the n soliton solutions in the topology of uniform convergence in compact sets called primitive solutions these more general solutions are analogous to the solutions derived in dyachenko and zakharov 2016 zakharov and dyachenko 2016 zakharov et al 2016 for the kdv equation we derive dressing functions for finite gap solutions and compute counter propagating dispersive shockwave type solutions numerically

a simple approach to reliable and robust a posteriori error estimation for singularly perturbed problems abstract a simple flux reconstruction for finite element solutions of reaction diffusion problems is shown to yield fully computable upper bounds on the energy norm of error in an approximation of singularly perturbed reaction diffusion problem the flux reconstruction is based on simple independent post processing operations over patches of elements in conjunction with standard raviart thomas vector fields we prove that the corresponding local error indicators are locally efficient and robust up to a nonstandard oscillation term with respect to any mesh size and any size of the reaction coefficient including the singularly perturbed limit

whitham modulation equations coalescing characteristics and dispersive boussinesq dynamics abstract whitham modulation theory with degeneracy in wave action is considered the case where all components of the wave action conservation law when evaluated on a family of periodic travelling waves have vanishing derivative with respect to wavenumber is considered it is shown that whitham modulation equations morph on a slower time scale into the two way boussinesq equation both the 1 1 and 2 1 cases are considered the resulting boussinesq equation arises in a universal form in that the coefficients are determined from the abstract properties of the lagrangian and do not depend on particular equations one curious by product of the analysis is that the theory can be used to confirm that the two way boussinesq equation is not a valid model in shallow water hydrodynamics modulation of nonlinear travelling waves of the complex klein gordon equation is used to illustrate the theory

korn s inequalities and their applications in continuum mechanics korn s inequalities have played a central role in the development of linear elasticity not only in connection with the basic theoretical issues such as existence and uniqueness but also in a variety of applications the korn inequalities and other related inequalities for integrals of quadratic functionals also arise in the analysis of viscous incompressible fluid flow the dimensionless optimal constants appearing in these inequalities the korn constants depend only on the shape of the domains of concern information on the geometric dependence of these constants is essential in applications in this review article we summarize the major results on korn s inequalities for bounded domains in two and three dimensions with emphasis on results concerning the korn constants some applications in continuum mechanics are also described

equilibrated residual error estimator for edge elements reliable a posteriori error estimates without generic constants can be obtained by a comparison of the finite element solution with a feasible function for the dual problem a cheap computation of such functions via equilibration is well known for scalar equations of second order we simplify and modify the equilibration such that it can be applied to the curl curl equation and edge elements the construction is more involved for edge elements since the equilibration has to be performed on subsets with different dimensions for this reason raviart thomas elements are extended in the spirit of distributions

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

asymptotic behavior of semidiscrete finite element approximations of biot s consolidation problem error estimates for spatially discrete galerkin finite element approximations of biot s model for consolidation of saturated porous media are presented the short and long time behaviors of such approximations based on both stable and unstable combinations of finite element spaces of displacement and pore pressure fields are discussed

resonant triads for two bidirectional equations in 1 1 dimensions the two soliton solution of the kdv equation is known not to degenerate into a solution describing resonant triads of solitons in 1 1 dimensions however we show that two integrable coupled kdv systems of the drinfeld sokolov class possess solutions which may degenerate into resonant triads these solutions are associated with a resonance relation which generalizes the usual one previously considered by hirota and ito

constrained undiscounted stochastic dynamic programming in this paper we investigate the computation of optimal policies in constrained discrete stochastic dynamic programming with the average reward as utility function the state space and action sets are assumed to be finite constraints which are linear functions of the state action frequencies are allowed in the general multichain case an optimal policy will be a randomized nonstationary policy an algorithm to compute such an optimal policy is presented furthermore sufficient conditions for optimal policies to be stationary are derived there are many applications for constrained undiscounted stochastic dynamic programming e g in multiple objective markovian decision models

a least squares mixed finite element method for biot s consolidation problem in porous media a least squares mixed finite element method for the coupled problem of flow and deformation is presented and analyzed in this paper for the analysis we restrict ourselves to fully saturated conditions for the flow process and to a linearly elastic material law for the deformation process this is known in the literature as biot s consolidation problem for simplicity the analysis is presented for the problem in two space dimensions our least squares approach is motivated by the fact that all process variables i e fluid pressure and flux as well as displacement field and stress tensor are approximated directly by suitable finite element spaces ellipticity of the corresponding variational formulation is proven for the stationary case as well as for the subproblems arising at each step of an implicit time discretization in the general time dependent case standard h1 conforming piecewise linear and quadratic finite elements are used for the fluid pressure and for each component of the displacement respectively for the flux and stress components the h div conforming raviart thomas spaces of lowest order are used computational results are presented for some two dimensional test problems

locking free finite element methods for poroelasticity we propose a new formulation along with a family of finite element schemes for the approximation of the interaction between fluid motion and linear mechanical response of a porous medium known as biot s consolidation problem the steady state version of the system is recast in terms of displacement pressure and volumetric stress and both continuous and discrete formulations are analyzed as compact perturbations of invertible problems employing a fredholm argument in particular the error estimates are derived independently of the lame constants numerical results indicate the satisfactory performance and competitive accuracy of the introduced methods

equilibrated residual error estimates are p robust abstract equilibrated residual error estimators applied to high order finite elements are analyzed the estimators provide always a true upper bound for the energy error we prove that also the efficiency estimate is robust with respect to the polynomial degrees a complete theory is provided for tensor product elements in the case of simplicial elements the theorem was first based on a conjecture but the conjecture was quite recently proved by costabel and mcintosh m costabel a mcintosh on bogovskii and regularized poincare integral operators for de rham complexes on lipschitz domains irmar preprint 08 40 rennes august 2008 numerical evidence for the p robustness is provided

maximum stopping value policies in finite markov population decision chains this paper considers infinite horizon finite state and action markov population decision chains mpdcs in which the goal is to find a stationary stopping policy with maximum stopping value that is with maximum value over all deterministic markov stopping policies a policy is stopping if the resulting expected population size in a period diminishes to zero as the period converges to infinity the paper shows that the following are equivalent a there is a stationary maximum stopping value policy b the maximum stopping value is finite c there is a stopping policy and an excessive point of the optimal return operator and d the maximum stopping value is the least excessive resp fixed point of the optimal return operator the paper shows how to use linear programming policy improvement and successive approximations to solve the problem the problem arises in stopping a markov chain as in optimally eradicating a pest or disease the problem is one of two key subproblems used repeatedly to f

parameter robust discretization and preconditioning of biot s consolidation model biot s consolidation model in poroelasticity has a number of applications in science medicine and engineering the model depends on various parameters and in practical applications these parameters range over several orders of magnitude a current challenge is to design discretization techniques and solution algorithms that are well behaved with respect to these variations the purpose of this paper is to study finite element discretizations of this model and construct block diagonal preconditioners for the discrete biot systems the approach taken here is to consider the stability of the problem in nonstandard or weighted hilbert spaces and employ the operator preconditioning approach we derive preconditioners that are robust with respect to both the variations of the parameters and the mesh refinement the parameters of interest are small time step sizes large bulk and shear moduli and small hydraulic conductivity

recurrence conditions for average and blackwell optimality in denumerable state markov decision chains in a previous paper dekker and hordijk 1988 presented an operator theoretical approach for multichain markov decision processes with a countable state space compact action sets and unbounded rewards conditions were presented guaranteeing the existence of a laurent series expansion for the discounted rewards the existence of average and blackwell optimal policies and the existence of solutions for the average and blackwell optimality equations while these assumptions were operator oriented and formulated as conditions for the deviation matrix we will show in this paper that the same approach can also be carried out under recurrence conditions these new conditions seem easier to check in general and are especially suited for applications in queueing models

bifurcation from rolls to multi pulse planforms via reduction to a parabolic boussinesq model a mechanism is presented for the bifurcation from one dimensional spatially periodic patterns rolls into two dimensional planar states planforms the novelty is twofold the planforms are solutions of a boussinesq partial differential equation pde on a periodic background and secondly explicit formulas for the coefficients in the boussinesq equation are derived based on a form of planar conservation of wave action flux the boussinesq equation is integrable with a vast array of solutions and an example of a new planform bifurcating from rolls which appears to be generic is presented adding in time leads to a new time dependent pde which models the nonlinear behaviour emerging from a generalization of eckhaus instability the class of pdes to which the theory applies is evolution equations whose steady part is a gradient elliptic pde examples are the 2 1 ginzburg landau equation with real coefficients and the 2 1 planar swift hohenberg equation 2014 elsevier b v all rights reserved

optimal stopping problems for multiarmed bandit processes with arms independence abstract this paper deals with the optimal stopping problem for multiarmed bandit processes under the assumption of independence of arms we show that optimal strategies and stopping times are expressed by the dynamic allocation indices for each arm this paper reduces this problem to several independent one parameter optimal stopping problems on the basis of these results we characterize optimal strategies and stopping times moreover this paper also extends those to the case allowing time constraints in the case where arm s state evolve according to markov chains with finite state linear programming calculation of optimal strategies and stopping times is discussed

explicit solutions for a semidiscrete boussinesq system a semidiscrete boussinesq system has been introduced by the means of recombination flows from a 4 4 spectral problem we construct the darboux transformation and explicit solutions for the semidiscrete boussinesq system we can find that the 1 fold darboux transformation solution of the system possesses more than one traveling waves especially one of solutions comes into being two solitary waves after a period of time which means that properties of solutions to semidiscrete integrable systems are richer than the ones in continuous integrable systems

linear programming for finite state multi armed bandit problems we consider the multi armed bandit problem we show that when the state space is finite the computation of the dynamic allocation indices can be handled by linear programming methods

the multi armed bandit problem decomposition and computation this paper is dedicated to our friend and mentor cyrus derman on the occasion of his 60th birthday the multi armed bandit problem arises in sequentially allocating effort to one of n projects and sequentially assigning patients to one of n treatments in clinical trials gittins and jones gittins j c jones d m 1974 a dynamic allocation index for the sequential design of experiments j gani k sarkadi l vince eds progress in statistics european meeting of statisticians 1972 1 north holland amsterdam 241 266 have shown that one optimal policy for the n project problem an n dimensional discounted markov decision chain is determined by the following largest index rule there is an index for each state of each given project that depends only on the data of that project in each period one allocates effort to a project with largest current index the purpose of this paper is to give a short proof of this result and a new characterization of the index of a project in state i viz as the maximum expected present value in state i for the restart in i problem in which in each state and period one either continues allocating effort to the project or immediately restarts the project in state i moreover it is shown that an approximate largest index rule yields an approximately optimal policy these results lead to more efficient methods of computing the indices on line and or for sparse transition matrices in large state spaces than have been suggested heretofore by using a suitable implementation of successive approximations a policy whose expected present value is within 100e of the maximum possible range of values of the indices can be found on line with at most n t 1tm operations where m is the number of operations required to calculate one approximation t is the least integer majorizing the ratio ln e ln a and 0 a 1 is the discount factor

stochastic scheduling with priority classes we consider controlled stochastic systems in which multiple job types customers projects jobs etc may require processing by one or more servers the job types are all members of priority classes and admissible controls must respect the constraints imposed by these for systems which when unconstrained satisfy generalised conservation laws we obtain the performance space and show that linear objectives are optimised by admissible controls which choose job types within each priority class according to a set of gittins indices various degrees of decomposition of the problem are described and illustrated in the special case of discounted and undiscounted branching bandits the theory is developed further to yield index based suboptimality bounds for general policies

a note on m n katehakis and y r chen s computation of the gittins index in a recent paper katehakis and chen propose a sequence of linear programs for the computation of the gittins indices if there are n projects and project v has kv states then v 1nkv linear programs have to be solved in this note it is shown that instead of the kv linear programs for project v also one parametric linear program with the same dimensions can be solved

a class of bayesian models for optimal exploration each of several locations contains an unknown number of objects of value a single search of a location will discover some of these objects which are then removed discoveries yield rewards a distribution of effort problem is posed concerning how to explore the locations optimally i e to maximize the expected return from discoveries made this is formulated as a bayes sequential decision problem for which index policies are optimal a natural simple case is one in which conditionally on the number of undiscovered objects at a location n the number of discoveries made in a single search is binomial b n p where p is a detection rate for this case we can gain considerable insight into how model structure relates to policy structure the tail behaviour of the priors for the number of objects at each location plays an important role

a posteriori error estimates by weakly symmetric stress reconstruction for the biot problem abstract a posteriori error estimates are constructed for the three field variational formulation of the biot problem involving the displacements the total pressure and the fluid pressure the discretization under focus is the h 1 conforming taylor hood finite element combination consisting of polynomial degrees k 1 for the displacements and the fluid pressure and k for the total pressure an a posteriori error estimator is derived on the basis of h div conforming reconstructions of the stress and flux approximations the symmetry of the reconstructed stress is allowed to be satisfied only weakly the reconstructions can be performed locally on a set of vertex patches and lead to a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation particular emphasis is given to nearly incompressible materials and the error estimates hold uniformly in the incompressible limit numerical results on the l shaped domain confirm the theory and the suitable use of the error estimator in adaptive strategies

discounted multiarmed bandit problems on a collection of machines with varying speeds this paper is the first to consider general multiarmed bandit problems on parallel machines working at different speeds block allocation policies make a once for all allocation of bandits to machines at time zero in this class we describe how to achieve blackwell optimality under given conditions the block allocation policy identified allocates the bandits with the largest guaranteed reward rates to the machines operating at greatest speed this policy is shown to be average reward optimal in the class of general nonanticipative nonidling policies

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available

optimal adaptive policies for markov decision processes in this paper we consider the problem of adaptive control for markov decision processes we give the explicit form for a class of adaptive policies that possess optimal increase rate properties for the total expected finite horizon reward under sufficient assumptions of finite state action spaces and irreducibility of the transition law a main feature of the proposed policies is that the choice of actions at each state and time period is based on indices that are inflations of the right hand side of the estimated average reward optimality equations

solitonic interactions and double wronskian type solutions for a variable coefficient variant boussinesq model in the long gravity water waves abstract under investigation in this paper is a variable coefficient variant boussinesq vcvb model for the nonlinear and dispersive long gravity waves in shallow water traveling in two horizontal directions with varying depth connection between the vcvb model and a variable coefficient ablowitz kaup newell segur system is revealed under certain constraints with the help of the symbolic computation multi solitonic solutions in terms of the double wronskian determinant for the vcvb model are derived interactions among the vcvb solitons are discussed a novel dynamic property is observed i e the coexistence of elastic inelastic interactions