interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations we present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three a suitable weak solution is regular near an interior point z if the scaled lx tp q norm of the velocity with 1 3 p 2 q 2 1 q is sufficiently small near z and if the scaled lx tl m norm of the magnetic field with 1 3 l 2 m 2 1 m is bounded near z similar results are also obtained for the vorticity and for the gradient of the vorticity furthermore with the aid of the regularity criteria we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations

remarks on the singular set of suitable weak solutions for the three dimensional navier stokes equations abstract in this study s denotes the possible interior singular set of suitable weak solutions for the three dimensional navier stokes equations we improve the known upper box counting dimension of this set from 360 277 1 300 given by 24 to 975 758 1 286 we also show that s r log e r 0 0 27 113 which extends the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor given by choe and lewis 3 j funct anal 175 348 369 2000 and by choe and yang 4 comm math phys 336 171 198 2015 the proof is inspired by a new e regularity criterion which was proved by guevara and phuc 7 calc var 56 68 2017

finite dimensional approximation in infinite dimensional mathematical programming we consider the problem of approximating an optimal solution to a separable doubly infinite mathematical program p with lower staircase structure by solutions to the programs p n obtained by truncating after the first n variables and n constraints of p viewing the surplus vector variable associated with the nth constraint as a state and assuming that all feasible states are eventually reachable from any feasible state we show that the efficient set of all solutions optimal to all possible feasible surplus states for p n converges to the set of optimal solutions to p a tie breaking algorithm which selects a nearest point efficient solution for p n is shown for convex programs to converge to an optimal solution to p a stopping rule is provided for discovering a value of n sufficiently large to guarantee any prespecified level of accuracy the theory is illustrated by an application to production planning

hausdorff measure of boundary singular points in the magnetohydrodynamic equations abstract we study boundary singular points of suitable weak solutions to the three dimensional incompressible magnetohydrodynamic equations by using the generalised hausdorff measure we estimate the size of boundary singular points and present the improved range of powers of logarithmic factors

rolling planning horizons error bounds for the dynamic lot size model a new and tight bound is placed on the error induced by imposing a finite time horizon t on the dynamic lot size model this bound equals the value of perfect information about the data beyond the time horizon t with this bound one can assess the value of using the dynamic lot size model repeatedly on a rolling basis with a time horizon of t periods when this bound is low the time horizon is long enough when this bound is not low it may pay to invest in forecasts of demands for later periods thereby increasing the time horizon

on the singular set in the navier stokes equations we consider suitably weak solutions u p to the incompressible navier stokes equations and under various assumptions on u obtain estimates for the size of its singular set one of our results improves a well known theorem of caffarelli kohn and nirenberg

a regularity criterion at one scale without pressure for suitable weak solutions to the navier stokes equations abstract in this paper we continue our work in 15 to derive e regularity criteria at one scale without pressure for suitable weak solutions to the navier stokes equations we establish an e regularity criterion below of suitable weak solutions for any 0 q 1 u 5 2 d x d t e as an application we extend the previous corresponding results concerning the improvement of the classical caffarelli kohn nirenberg theorem by a logarithmic factor

hausdorff measure of the singular set in the incompressible magnetohydrodynamic equations we derive a local energy inequality for weak solutions of the three dimensional magnetohydrodynamic equations combining biot savart law interpolation inequalities and the local energy inequality we prove a partial regularity theorem for suitable weak solutions furthermore we obtain an improved estimate for the logarithmic hausdorff dimension of the singular set of suitable weak solutions

solution and forecast horizons for infinite horizon nonhomogeneous markov decision processes we consider a nonhomogeneous infinite horizon markov decision process mdp problem with multiple optimal first period policies we seek an algorithm that given finite data delivers an optimal first period policy such an algorithm can thus recursively generate within a rolling horizon procedure an infinite horizon optimal solution to the original problem however it can happen that no such algorithm exists i e the mdp is not well posed equivalently it is impossible to solve the problem with a finite amount of data assuming increasing marginal returns in actions with respect to states and stochastically increasing state transitions with respect to actions we provide an algorithm that is guaranteed to solve the given mdp whenever it is well posed this algorithm determines in finite time a forecast horizon for which an optimal solution delivers an optimal first period policy as an application we solve all well posed instances of the time varying version of the classic asset selling problem

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

technical note identifying forecast horizons in nonhomogeneous markov decision processes a procedure for identifying forecast horizons in nonhomogeneous markov decision processes based on convergence results for relative value functions is developed two different algorithmic implementations of this procedure are discussed and a closed form expression for computing sufficiently long horizons to guarantee epsilon optimality is presented

partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations in this paper we study the local behavior of the solutions to the three dimensional magnetohydrodynamic equations we are interested in both the uniform gradient estimates for smooth solutions and regularity of weak solutions it is shown that in some neighborhood of x0 t0 the gradients of the velocity field u and the magnetic field b are locally uniformly bounded in l norm as long as that either the scaled local l2 norm of the gradient or the scaled local total energy of the velocity field is small and the scaled local total energy of the magnetic field is uniformly bounded these estimates indicate that the velocity field plays a more dominant role than that of the magnetic field in the regularity theory as an immediately corollary we can derive an estimates of hausdorff dimension on the possible singular set of a suitable weak solution as in the case of pure fluid various partial regularity results are obtained as consequences of our blow up estimates

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

on the number of singular points of weak solutions to the navier stokes equations we consider a suitable weak solution to the three dimensional navier stokes equations in the space time cylinder 0 t let be the set of singular points for this solution and t x t for a given open subset and for a given moment of time t 0 t we obtain an upper bound for the number of points of the set t 2001 john wiley sons inc

solving nonstationary infinite horizon dynamic optimization problems monotonicity of optimal solutions to finite horizon dynamic optimization problems is used to prove the existence of a forecast horizon i e a long enough planning horizon that ensures that a first period optimal action for the infinite horizon and the finite horizon problem agree regardless of problem parameter changes in the tail the existence of extremal monotone optimal solutions motivates a stopping rule that is ensured to detect the minimal forecast horizon to be used in a rolling horizon procedure to exactly solve the problem

conditions for the discovery of solution horizons we present necessary and sufficient conditions for discrete infinite horizon optimization problems with unique solutions to be solvable these problems can be equivalently viewed as the task of finding a shortest path in an infinite directed network we provide general forward algorithms with stopping rules for their solution the key condition required is that of weak reachability which roughly requires that for any sequence of nodes or states it must be possible from optimal states to reach states close in cost to states along this sequence moreover the costs to reach these states must converge to zero applications are considered in optimal search undiscounted markov decision processes and deterministic infinite horizon optimization

a forecast horizon and a stopping rule for general markov decision processes in this paper we are concerned with stochastic optimal control problems over an infinite horizon when the reward is discounted there is only one optimality criterion and standard dynamic programming can be applied now in the undiscounted case the total reward might be unbounded for any feasible policy and there is not a unique optimahty criterion any more greatest average reward overtaking optimality average overtaking optimality and i optimality are such criteria 4 lo the most used of them is the greatest average reward and several technics have been proposed to get an average optimal policy at least for finite state space markov decision processes mdp i 1 these results are all based on the existence of a fixed point solution to the optimality equations for a relative value function in the case of denumerable state space weak conditions have been recently exhibited under which a fixed point solution exists 3 in the case of finite state space non homogeneous mdp the relative value function is also shown to converge under so called weak ergodicity con ditions s while the latter assumptions are stronger one can easily check them by inspection of the one step transition probability matrices and also

degeneracy in infinite horizon optimization we consider sequential decision problems over an infinite horizon the forecast or solution horizon approach to solving such problems requires that the optimal initial decision be unique we show that multiple optimal initial decisions can exist in general and refer to their existence as degeneracy we then present a conceptual cost perturbation algorithm for resolving degeneracy and identifying a forecast horizon we also present a general near optimal forecast horizon

concepts of forecast and decision horizons applications to dynamic stochastic optimization problems in this paper we develop a rigorous new framework for the concepts of forecast and decision horizons these concepts are conditional in nature and in turn enable us to unify the existing concepts of strong and weak horizons moreover we are able to precisely state the questions of existence of forecast decision horizons finally we are able to further develop the theory of existence of forecast decision horizons in the discounted case and procedures to compute these horizons

a new optimality criterion for nonhomogeneous markov decision processes we propose a new definition of optimality for nonhomogeneous markov decision processes called periodic forecast horizon pfh optimality using measures of discounting and ergodicity we establish conditions under which pfh optimal strategies exist and pfh optimality implies the more conventional notions of optimality and average optimality finally we use this definition of optimality as the basis for a direct development of forecast horizon results for discounted and undiscounted nonhomogeneous markov decision processes

conditions for the existence of planning horizons we consider the general class of discounted problems involving sequential decision making over an unbounded horizon under the assumptions of a finite set of policy alternatives at each decision time and costs that are eventually uniformly bounded by some exponential existence of an optimal infinite horizon strategy is established moreover it is shown that there is a finite horizon version whose first policy decision agrees with an infinite horizon optimal policy under the additional assumption that the infinite horizon optimal strategy is eventually cyclic the stronger result follows that the infinite horizon optimal solution is unique for almost all interest rates and that therefore a planning horizon exists algorithmic implications are explored for a restricted subclass of these problems

weak solutions to fokker planck equations and mean field games we deal with systems of pdes arising in mean field games theory where viscous hamilton jacobi and fokker planck equations are coupled in a forward backward structure we consider the case of local coupling when the running cost depends on the pointwise value of the distribution density of the agents in which case the smoothness of solutions is mostly unknown we develop a complete weak theory proving that those systems are well posed in the class of weak solutions for monotone couplings under general growth conditions and for superlinear convex hamiltonians as a key tool we prove new results for fokker planck equations under minimal assumptions on the drift through a characterization of weak and renormalized solutions the results obtained give new perspectives even for the case of uncoupled equations as far as the uniqueness of weak solutions is concerned

regularity criteria of the magnetohydrodynamic equations in bounded domains or a half space abstract we study three dimensional incompressible magnetohydrodynamic equations in bounded domains or a half space we present new regularity criteria of weak solutions a pair of weak solutions u b become regular if u satisfies serrin s type conditions when we consider no slip or slip boundary conditions for the velocity field u and slip boundary conditions for the magnetic field b in either bounded domains or a half space in addition in the case of a half space with no slip boundary conditions for u and slip boundary conditions for b we demonstrate that if tangential components of u and normal component of b satisfy serrin s type conditions then a pair of weak solutions u b become regular

a shadow simplex method for infinite linear programs we present a simplex type algorithm that is an algorithm that moves from one extreme point of the infinite dimensional feasible region to another not necessarily adjacent extreme point for solving a class of linear programs with countably infinite variables and constraints each iteration of this method can be implemented in finite time whereas the solution values converge to the optimal value as the number of iterations increases this simplex type algorithm moves to an adjacent extreme point and hence reduces to a true infinite dimensional simplex method for the important special cases of nonstationary infinite horizon deterministic and stochastic dynamic programs

a linear programming approach to nonstationary infinite horizon markov decision processes nonstationary infinite horizon markov decision processes mdps generalize the most well studied class of sequential decision models in operations research namely that of stationary mdps by relaxing the restrictive assumption that problem data do not change over time linear programming lp has been very successful in obtaining structural insights and devising solution methods for stationary mdps however an lp approach for nonstationary mdps is currently missing this is because the lp formulation of a nonstationary infinite horizon mdp includes countably infinite variables and constraints and research on such infinite dimensional lps has traditionally faced several hurdles for instance duality results may not hold an extreme point may not be a basic feasible solution and in the context of a simplex algorithm a pivot operation may require infinite data and computations and a sequence of improving extreme points need not converge in value to optimal in this paper we tackle these challenges a

a tie breaking rule for discrete infinite horizon optimization we study discrete infinite horizon optimization problems without the common assumption of a unique optimum a method based on solution set convergence is employed for finding optimal initial decisions by solving finite horizon problems this method is applicable to general discrete decision models that satisfy a weak reachability condition the algorithm together with a stopping rule is applied to production planning and capacity expansion and computational results are reported

denumerable state nonhomogeneous markov decision processes we consider denumerable state nonhomogeneous markov decision processes and extend results from both denumerable state homogeneous and finite state nonhomogeneous problems we show that under weak ergodicity accumulation points of finite horizon optima termed algorithmic optima are average cost optimal we also establish the existence of solution horizons finally an algorithm is presented to solve problems of this class for the case where there is a unique algorithmic optimum

boundary regularity criteria for suitable weak solutions of the magnetohydrodynamic equations abstract we present some new regularity criteria for suitable weak solutions of magnetohydrodynamic equations near boundary in dimension three we prove that suitable weak solutions are holder continuous near boundary provided that either the scaled l x t p q norm of the velocity with 3 p 2 q 2 2 q or the scaled l x t p q norm of the vorticity with 3 p 2 q 3 2 q are sufficiently small near the boundary

metric gradient flows with state dependent functionals the nash mfg equilibrium flows and their numerical schemes we investigate the convergence of a relaxed version of the best reply numerical schemes also known as best response or fictitious play used to find nash mean field games equilibriums this leads us to consider evolution equations in metric spaces similar to gradient flows except that the functional to be differentiated depends on the current point these are called equilibrium flows we give two definitions of solutions and prove that as the time step tends to zero the interpolated a la de giorgi numerical curves converge to equilibrium flows as a by product we obtain a sufficient condition for the uniqueness of a mean field games equilibrium we close with applications to congestion and vaccination mean field games

rolling horizon procedures in nonhomogeneous markov decision processes by far the most common planning procedure found in practice is to approximate the solution to an infinite horizon problem by a series of rolling finite horizon solutions although many empirical studies have been done this so called rolling horizon procedure has been the subject of few analytic studies we provide a cost error bound for a general rolling horizon algorithm when applied to infinite horizon nonhomogeneous markov decision processes both in the discounted and average cost cases we show that a doeblin coefficient of ergodicity acts much like a discount factor to reduce this error in particular we show that the error goes to zero for any fixed rolling horizon as this doeblin measure of control over the future decreases the theory is illustrated through an application to vehicle deployment

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references