computationally feasible bounds for partially observed markov decision processes a partially observed markov decision process pomdp is a sequential decision problem where information concerning parameters of interest is incomplete and possible actions include sampling surveying or otherwise collecting additional information such problems can theoretically be solved as dynamic programs but the relevant state space is infinite which inhibits algorithmic solution this paper explains how to approximate the state space by a finite grid of points and use that grid to construct upper and lower value function bounds generate approximate nonstationary and stationary policies and bound the value loss relative to optimal for using these policies in the decision problem a numerical example illustrates the methodology

existence of some extremal self dual codes two new singly even extremal self dual codes are constructed a 52 26 10 code and a 54 27 10 code

doubly even self dual code of length 96 we prove that 23 11 and 7 do not divide the order of the automorphism group of a binary 96 48 20 doubly even self dual code we construct 25 new inequivalent binary 96 48 16 doubly even self dual codes via an automorphism of order 23

infinite horizon optimization we consider the general problem of choosing a discounted cost minimizing infinite sequence of decisions from a closed subset of the product space formed by a sequence of arbitrary compact metric spaces examples include equipment replacement production planning and more generally infinite stage mathematical programs it is shown that the optimal costs for finite horizon approximating problems converge to the optimal infinite horizon cost as the horizons diverge to infinity moreover the existence of a unique algorithmically optimal i e accumulation point solution is shown to be a necessary and sufficient condition for convergence in the product topology i e policy convergence of all finite horizon optima under the weaker condition of hausdorff convergence of the sets of finite horizon optima to the set of infinite horizon optima we show how to force policy convergence through a natural tie breaking rule finally a forward algorithm is presented which in the presence of a unique infinite horizon optimum is guaranteed to converge

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

limiting average criteria for nonstationary markov decision processes this paper deals with the so called limiting average criteria for nonstationary markov decision processes with possibly unbounded rewards and borel state space a new set of conditions is provided under which the existence of both a solution to the optimality equations and the limiting average epsilon geq 0 optimal markov policies is derived also a rolling horizon algorithm for computing limiting average epsilon 0 optimal markov policies is developed furthermore the results in this paper are illustrated by several examples such as the water regulation problem

a method for registration of 3 d shapes the authors describe a general purpose representation independent method for the accurate and computationally efficient registration of 3 d shapes including free form curves and surfaces the method handles the full six degrees of freedom and is based on the iterative closest point icp algorithm which requires only a procedure to find the closest point on a geometric entity to a given point the icp algorithm always converges monotonically to the nearest local minimum of a mean square distance metric and the rate of convergence is rapid during the first few iterations therefore given an adequate set of initial rotations and translations for a particular class of objects with a certain level of shape complexity one can globally minimize the mean square distance metric over all six degrees of freedom by testing each initial registration one important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model prior to shape inspection experimental results show the capabilities of the registration algorithm on point sets curves and surfaces

optimal average value convergence in nonhomogeneous markov decision processes abstract we address the undiscounted nonhomogeneous markov decision process with average reward criterion and prove two structural results first we establish equivalence of this problem to a discounted markov decision process by means of an ergodic coefficient embedded in the original problem second we prove for the original problem that the optimal finite horizon average values converge to the infinite horizon optimal average value under an ergodic condition

shape analysis of elastic curves in euclidean spaces this paper introduces a square root velocity srv representation for analyzing shapes of curves in euclidean spaces under an elastic metric in this srv representation the elastic metric simplifies to the il2 metric the reparameterization group acts by isometries and the space of unit length curves becomes the unit sphere the shape space of closed curves is the quotient space of a submanifold of the unit sphere modulo rotation and reparameterization groups and we find geodesics in that space using a path straightening approach these geodesics and geodesic distances provide a framework for optimally matching deforming and comparing shapes these ideas are demonstrated using 1 shape analysis of cylindrical helices for studying protein structure 2 shape analysis of facial curves for recognizing faces 3 a wrapped probability distribution for capturing shapes of planar closed curves and 4 parallel transport of deformations for predicting shapes from novel poses

extremal self dual codes with an automorphism of order 2 a method to design binary self dual codes with an automorphism of order two without fixed points is presented extremal self dual codes with lengths 40 42 44 54 58 68 are constructed many of them have weight enumerators for which extremal codes were previously not known to exist

building 3 d statistical shape models by direct optimization statistical shape models are powerful tools for image interpretation and shape analysis a simple yet effective way of building such models is to capture the statistics of sampled point coordinates over a training set of example shapes however a major drawback of this approach is the need to establish a correspondence across the training set in 2 d a correspondence is often defined using a set of manually placed landmarks and linear interpolation to sample the shape in between such annotation is however time consuming and subjective particularly when extended to 3 d in this paper we show that it is possible to establish a dense correspondence across the whole training set automatically by treating correspondence as an optimization problem the objective function we use for the optimization is based on the minimum description length principle which we argue is a criterion that leads to models with good compactness specificity and generalization ability we manipulate correspondence by reparameterizing each training shape we describe an explicit representation of reparameterization for surfaces in 3 d that makes it impossible to generate an illegal i e not one to one correspondence we also describe several large scale optimization strategies for model building and perform a detailed analysis of each approach finally we derive quantitative measures of model quality allowing meaningful comparison between models built using different methods results are given for several different training sets of 3 d shapes which show that the minimum description length models perform significantly better than other approaches

automorphisms of order 2p in binary self dual extremal codes of length a multiple of 24 let c be a binary self dual code with an automorphism g of order 2p where p is an odd prime such that gp is a fixed point free involution if c is extremal of length a multiple of 24 all the involutions are fixed point free except the golay code and eventually putative codes of length 120 connecting module theoretical properties of a self dual code c with coding theoretical ones of the subcode c gp which consists of the set of fixed points of gp we prove that c is a projective f2 g module if and only if a natural projection of c gp is a self dual code we then discuss easy to handle criteria to decide if c is projective or not as an application we consider in the last part extremal self dual codes of length 120 proving that their automorphism group does not contain elements of order 38 and 58

numerical inversion of srnf maps for elastic shape analysis of genus zero surfaces recent developments in elastic shape analysis esa are motivated by the fact that it provides a comprehensive framework for simultaneous registration deformation and comparison of shapes these methods achieve computational efficiency using certain square root representations that transform invariant elastic metrics into euclidean metrics allowing for the application of standard algorithms and statistical tools for analyzing shapes of embeddings of mathbf s 2 in mathbb r 3 jermyn et al 1 introduced square root normal fields srnfs which transform an elastic metric with desirable invariant properties into the mathbb l 2 metric these srnfs are essentially surface normals scaled by square roots of infinitesimal area elements a critical need in shape analysis is a method for inverting solutions deformations averages modes of variations etc computed in srnf space back to the original surface space for visualizations and inferences due to the lack of theory for understanding srnf maps and their inverses we take a numerical approach and derive an efficient multiresolution algorithm based on solving an optimization problem in the surface space that estimates surfaces corresponding to given srnfs this solution is found to be effective even for complex shapes that undergo significant deformations including bending and stretching e g human bodies and animals we use this inversion for computing elastic shape deformations transferring deformations summarizing shapes and for finding modes of variability in a given collection while simultaneously registering the surfaces we demonstrate the proposed algorithms using a statistical analysis of human body shapes classification of generic surfaces and analysis of brain structures

elastic geodesic paths in shape space of parameterized surfaces this paper presents a novel riemannian framework for shape analysis of parameterized surfaces in particular it provides efficient algorithms for computing geodesic paths which in turn are important for comparing matching and deforming surfaces the novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape preserving transformations of surfaces the basic idea is to formulate a space of embedded surfaces surfaces seen as embeddings of a unit sphere in ir3 and impose a riemannian metric on it in such a way that the reparameterization group acts on this space by isometries under this framework we solve two optimization problems one given any two surfaces at arbitrary rotations and parameterizations we use a path straightening approach to find a geodesic path between them under the chosen metric second by modifying a technique presented in 25 we solve for the optimal rotation and parameterization registration between surfaces their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces we illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces

optimization of average rewards of time nonhomogeneous markov chains we study the optimization of average rewards of discrete time nonhomogeneous markov chains in which the state spaces transition probabilities and reward functions depend on time the analysis encounters a few major difficulties 1 notions crucial to homogeneous markov chains such as ergodicity stationarity periodicity and connectivity no longer apply 2 the average reward criterion is under selective i e it does not depend on the decisions in any finite period and thus dynamic programming is not amenable and 3 because of the under selectivity an optimal average reward policy may not be the best in any finite period

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

nonhomogeneous markov decision processes with borel state space the average criterion with nonuniformly bounded rewards this paper deals with nonhomogeneous markov decision processes with borel state space and nonuniformly bounded rewards under the average criterion first under the minorant type assumption we prove the existence of an appropriate solution to the optimality equations second from the optimality equations we also establish the existence of e 0 optimal markov policies under the additional conditions third some sufficient conditions for the validity of the assumptions in this paper and several examples such as inventory production systems are provided finally as an application of the optimality equations a rolling horizon algorithm is given

on the construction of invertible filter banks on the 2 sphere the theories of signal sampling filter banks wavelets and ldquoovercomplete waveletsrdquo are well established for the euclidean spaces and are widely used in the processing and analysis of images while recent advances have extended some filtering methods to spherical images many key challenges remain in this paper we develop theoretical conditions for the invertibility of filter banks under continuous spherical convolution furthermore we present an analogue of the papoulis generalized sampling theorem on the 2 sphere we use the theoretical results to establish a general framework for the design of invertible filter banks on the sphere and demonstrate the approach with examples of self invertible spherical wavelets and steerable pyramids we conclude by examining the use of a self invertible spherical steerable pyramid in a denoising experiment and discussing the computational complexity of the filtering framework

some new extremal self dual 58 29 10 codes the authors discuss extremal self dual 60 30 12 codes and then go on to present new extremal self dual 58 29 10 codes

new extremal self dual codes of lengths 36 38 and 58 we construct new extremal self dual binary codes of lengths 36 38 and 58 we show that there are at least 14 inequivalent extremal self dual 38 18 8 codes and that there are at least 368 inequivalent extremal self dual 38 19 8 codes for length 58 we construct 11 extremal self dual 58 29 10 codes whose weight enumerators were previously unknown

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

extremal binary self dual codes in this correspondence we investigate binary extremal self dual codes numerous extremal self dual codes and interesting self dual codes with minimum weight d 14 and 16 are constructed in particular the first extremal type i 86 43 16 code and new extremal self dual codes with weight enumerators which were not previously known to exist for lengths 40 50 52 and 54 are constructed we also determine the possible weight enumerators for extremal type i codes of lengths 66 100

on the structure of binary self dual codes having an automorphism of order a square of an odd prime we describe a method for constructing binary self dual codes having an automorphism of order p sup 2 for an odd prime p using this method we classify the optimal self dual codes of lengths 36 spl les n spl les 44 and n 54 having an automorphism of order 9 we obtain all self dual 56 28 12 58 29 10 and 60 30 12 codes having an automorphism of order 9 without cycles of length 3 some of the constructed codes of lengths 54 58 and 60 have weight enumerators for which the existence of codes was not known before

state classification of time nonhomogeneous markov chains and average reward optimization of multi chains in a discrete time nonhomogeneous markov chain tnhmc the states spaces transition probabilities and reward functions at different times may be different in this paper with the confluencity previously introduced we show that the states of a tnhmc can be classified into the branching states and a number of classes of confluent states versus the transient and recurrent states in the time homogeneous case the optimization of average reward in tnhmc s consisting of a single confluent class uni chain have been addressed in a previous paper by the author in this paper we show that with confluencity and the state classification and under some bound conditions we can obtain the necessary and sufficient conditions for optimal policies of the average reward of tnhmcs consisting of multiple confluent classes multi chains just like in the uni chain tnhmc case the sufficient condition does not need to hold in any zero frequently visited time sequence this under selectivity makes the problem not amenable to dynamic programming a direct comparison based approach is used to prove the results the results enhance our understanding of state classification and performance optimization with the notion of confluencity

existence of certain extremal self dual codes two new singly even extremal self dual codes are given these are a 42 21 8 code and a 58 29 10 code

existence and discovery of average optimal solutions in deterministic infinite horizon optimization we consider the problem of making a sequence of decisions each chosen from a finite action set over an infinite horizon so as to minimize its associated average cost both the feasibility and cost of a decision are allowed to depend upon all of the decisions made prior to that decision moreover time varying costs and constraints are allowed a feasible solution is said to be efficient if it reaches each of the states through which it passes at minimum cost we show that efficient solutions exist and that under a state reachability condition efficient solutions are also average optimal exploiting the characterization of efficiency via a solution s short run as opposed to long run behavior a forward algorithm is constructed which recursively discovers the first second and subsequent decisions of an efficient and hence average optimal infinite horizon solution

a method for constructing self dual codes with an automorphism of order 2 in this paper we investigate binary self dual codes with an automorphism of order 2 with c cycles and f fixed points a method for constructing such codes using self orthogonal codes of length c and self dual codes of length f is presented we apply this method to construct extremal self dual codes of lengths 40 42 44 52 54 and 58 some of them have weight enumerators for which self dual codes were previously not known to exist we prove that there do not exist self dual 50 25 10 and 96 48 20 codes with an automorphism of order 2 with f fixed points for f 0 in their automorphism groups

approximating extreme points of infinite dimensional convex sets the property that an optimal solution to the problem of minimizing a continuous concave function over a compact convex set in rn is attained at an extreme point is generalized by the bauer minimum principle to the infinite dimensional context the problem of approximating and characterizing infinite dimensional extreme points thus becomes an important problem consider now an infinite dimensional compact convex set in the nonnegative orthant of the product space rinfinity we show that the sets of extreme points en of its corresponding finite dimensional projections onto rn converge in the product topology to the closure of the set of extreme points e of the infinite dimensional set as an application we extend the concept of total unimodularity to infinite systems of linear equalities in nonnegative variables where we show when extreme points inherit integrality from approximating finite systems an application to infinite horizon production planning is considered

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

existence and computation of infinite horizon model predictive control with active steady state input constraints this note addresses the existence and implementation of the infinite horizon controller for the case of active steady state input constraints this case is important because in many practical applications controllers are required to operate at the boundary of the feasible region for instance in order to maximize global economic objectives for this case the usual finite horizon parameterizations with terminal cost cannot be applied and optimal solutions are not generally available we propose here an iterative algorithm that generates two finite horizon approximations to the true infinite horizon problem where the solution to one of the approximations yields an upper bound on the true optimum while the other approximation yields a lower bound we show convergence of both bounding approximations to the optimal solution as the horizon length in the approximations is increased we outline a procedure based on this result to provide a solution to the infinite horizon problem that is exact to within any user specified tolerance finally we present an example that includes a comparison between optimal and suboptimal controllers

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

a new 3d matching method of nonrigid and partially similar models using curve analysis the 3d shape matching problem plays a crucial role in many applications such as indexing or modeling by example here we present a novel approach to matching 3d objects in the presence of nonrigid transformation and partially similar models in this paper we use the representation of surfaces by 3d curves extracted around feature points indeed surfaces are represented with a collection of closed curves and tools from shape analysis of curves are applied to analyze and to compare curves the belief functions are used to define a global distance between 3d objects the experimental results obtained on the tosca and the shrec07 data sets show that the system performs efficiently in retrieving similar 3d models

a method for constructing inequivalent self dual codes with applications to length 56 necessary and sufficient conditions are presented for two doubly even codes with an automorphism of odd prime order to be equivalent under certain assumptions for the codes as an application we obtain all extremal doubly even codes of length 56 with an automorphism of order 13 sixteen such codes are inequivalent

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

dual based methods for solving infinite horizon nonstationary deterministic dynamic programs we develop novel dual ascent and primal dual methods to solve infinite horizon nonstationary deterministic dynamic programs these methods are finitely implementable and converge in value to optimality moreover the dual ascent method produces a sequence of improving dual solutions that pointwise converge to an optimal dual solution while the primal dual algorithm provides a sequence of primal basic feasible solutions with value error bounds from optimality that converge to zero our dual based methods work on a more general class of infinite network flow problems that include the shortest path formulation of dynamic programs as a special case to our knowledge these are the first dual based methods proposed in the literature to solve infinite horizon nonstationary deterministic dynamic programs

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

parameterization invariant shape comparisons of anatomical surfaces we consider 3 d brain structures as continuous parameterized surfaces and present a metric for their comparisons that is invariant to the way they are parameterized past comparisons of such surfaces involve either volume deformations or non rigid matching under fixed parameterizations of surfaces we propose a new mathematical representation of surfaces called q maps such that l2 distances between such maps are invariant to re parameterizations this property allows for removing the parameterization variability by optimizing over the re parameterization group resulting in a proper parameterization invariant distance between shapes of surfaces we demonstrate this method in shape analysis of multiple brain structures for 34 subjects in the detroit fetal alcohol and drug exposure cohort study which results in a 91 classification rate for attention deficit hyperactivity disorder cases and controls this method outperforms some existing techniques such as spherical harmonic point distribution model spharm pdm or iterative closest point icp

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

average optimality in nonhomogeneous infinite horizon markov decision processes we consider a nonhomogeneous stochastic infinite horizon optimization problem whose objective is to minimize the overall average cost per period of an infinite sequence of actions average optimality optimal solutions to such problems will in general be nonstationary moreover a solution that initially makes poor decisions and then selects wisely thereafter can be average optimal however we seek average optimal solutions with optimal short term as well as long term behavior our approach is to first transform our stochastic problem into one that is deterministic using the standard device of formulating the problem as one of choosing a sequence of policies as opposed to actions within this deterministic framework states become probability distributions over the original stochastic states then by weakening the notion of state reachability and strengthening the notion of efficiency traditionally used in the deterministic framework we prove that such efficient solutions exist and are average optimal thus simultaneously exhibiting both optimal long and short run behavior this deterministic view of the property of stochastic ergodicity offers the potential to relax the traditional conditions for average optimality that use coefficients of ergodicity as well as the opportunity to strengthen the criterion of average optimality through the property of efficiency

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

binary self dual codes of lengths 52 to 60 with an automorphism of order 7 or 13 all binary n n 2 optimal self dual codes for length 52 n 60 with an automorphism of order 7 or 13 are classified up to equivalence two of the constructed 54 27 10 codes have weight enumerators that were not previously known to exist there are also some 58 29 10 codes with new values of the parameters in their weight enumerator

a weighted markov decision process the two most commonly considered reward criteria for markov decision processes are the discounted reward and the long term average reward the first tends to neglect the future concentrating on the short term rewards while the second one tends to do the opposite we consider a new reward criterion consisting of the weighted combination of these two criteria thereby allowing the decision maker to place more or less emphasis on the short term versus the long term rewards by varying their weights the mathematical implications of the new criterion include the deterministic stationary policies can be outperformed by the randomized stationary policies which in turn can be outperformed by the nonstationary policies an optimal policy might not exist we present an iterative algorithm for computing an e optimal nonstationary policy with a very simple structure

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

the existence of certain extremal 54 27 10 self dual codes some new extremal binary 54 27 10 self dual codes are constructed using automorphisms of order 7

some new extremal self dual codes with lengths 44 50 54 and 58 we construct extremal self dual codes with lengths 44 50 54 and 58 they have weight enumerators for which extremal codes were previously not known to exist two methods are used for constructing the codes using self dual codes of same or smaller length to obtain the codes we use a combinatorial optimization search

convergence of selections with applications in optimization we consider the problem of finding an easily implemented tie breaking rule for a convergent set valued algorithm i e a sequence of compact non empty subsets of a metric space converging in the hausdorff metric our tie breaking rule is determined by nearest point selections detined by uniqueness points in the space i e points having a unique best approximation in the limit set of the convergent algorithm convergence of the algorithm is shown to be equivalent to convergence of all such nearest point selections under reasonable additional hypotheses all points in the metric space have the uniqueness property consequently all points yield convergent nearest point selections i e tie breaking rules for a convergent algorithm we then show how to apply these results to approximate solutions for the following types of problems infinite systems of inequalities semi infinite mathematical programming non convex optimization and infinite horizon optimization

automorphisms of codes with applications to extremal doubly even codes of length 48 general results on automorphisms of self dual binary codes are given these results are applied to the study of extremal self dual doubly even binary codes of length 48 the main theorem proved is that an extremal self dual doubly even code of length 48 with a nontrivial automorphism of odd order is equivalent to the extended quadratic residue code interesting constructions of the binary extended golay code as well as a conjecture about a possible connection between an extremal self dual doubly even code of length 72 and an extremal quaternary code of length 24 arc yielded by techniques used in the proof

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

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

new extremal binary 44 22 8 codes one of the problems in coding theory is constructing self dual codes whose weight enumerators are not yet known to exist we use the concept of neighbors and construct extremal binary self dual 44 22 8 codes whose weight enumerators are not yet known to exist

new extremal self dual codes of length 62 and related extremal self dual codes new extremal self dual codes of length 62 are constructed with weight enumerators of three different types two of these types were not represented by any known code up till now all these codes possess an automorphism of order 15 some of them are used to construct extremal self dual codes of length 60 by the method of subtracting by additional subtracting an extremal self dual 58 29 10 code was obtained having a weight enumerator which does not correspond to any code known so far

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen