discounted dynamic programming with unbounded returns application to economic models in this paper we study discounted markov decision processes on an uncountable state space we allow a utility reward function to be unbounded both from above and below a new feature in our approach is an easily verifiable rate of growth condition introduced for a positive part of the utility function this assumption in turn enables us to prove the convergence of a value iteration algorithm to a solution to the bellman equation moreover by virtue of the optimality equation we show the existence of an optimal stationary policy

optimal allocation for the comparison of proportions optimal allocation strategies for the comparison of proportions p1 and p2 in two groups are discussed allocations which i maximize the precision in the estimation of the difference p1 p2 ii maximize the precision in the estimation of the ratio p1 p2 iii maximize the power to detect a group difference and iv minimize the total cost of a study are compared the recommended allocation varies markedly with the desired objective furthermore there is a wide range of allocation schemes which are effectively optimal with regard to each criterion considered a study of congenital malformations occurring in pregnancies of diabetic women is used to illustrate how these considerations apply to the planning of a study

constrained markov decision models with weighted discounted rewards this paper deals with constrained optimization of markov decision processes both objective function and constraints are sums of standard discounted rewards but each with a different discount factor such models arise e g in production and in applications involving multiple time scales we prove that it a feasible policy exists then there exists an optimal policy which is i stationary nonrandomized from some step onward ii randomized markov before this step but the total number of actions which are added by randomization is bounded by the number of constraints optimality of such policies for multi criteria problems is also established these new policies have the pleasing aesthetic property that the amount of randomization they require over any trajectory is restricted by the number of constraints this result is new even for constrained optimization with a single discount factor where the optimality of randomized stationary policies is known however a randomized stationary policy may req

nonatomic total rewards markov decision processes with multiple criteria we consider a markov decision process with an uncountable state space for which the vector performance functional has the form of expected total rewards under the single condition that initial distribution and transition probabilities are nonatomic we prove that the performance space coincides with that generated by nonrandomized markov policies we also provide conditions for the existence of optimal policies when the goal is to maximize one component of the performance vector subject to inequality constraints on other components we illustrate our results with examples of production and financial problems

allocation of subjects to test null relative risks smaller than one allocating a proportion k 1 1 radicalr 0 of subjects to an intervention is a practical approach to approximately maximize power for testing whether an intervention reduces relative risk of disease below a null ratio r 0 or r 0 0 33 2 3 for 0 33 or r 0 0 17 and 3 4 for 0 17 or r 0 or 0 10 both k and k s are easily calculated and invariant to alterations in disease rate estimates under null and alternative hypotheses when r 0 remains constant in examples that we studied allocating k or k s subjects to intervention achieved close to the minimum possible sample size given test size and power equivalently maximum power given test size and sample size for likelihood score tests compared to equal allocation k and k s reduced sample sizes by amounts ranging from approximately 5 5 per cent for r 0 0 50 to approximately 24 per cent for r 0 0 10 these sample size savings may be particularly important for large studies of prophylactic interventions such as vaccines while k was derived from variance minimization for an arcsine transformation we do not recommend the arcsine test since its true size exceeded the nominal value in contrast the true size for the uncorrected score test was less than the nominal size a skewness correction made the size of the score test very close to the nominal level and slightly increased power we recommend using the score test or the skewness corrected score test for planing studies designed to show a ratio of proportions is less than a prespecified null ratio r 0 1

l infinity pairs and applications to singularities abstract over a field of characteristic zero every deformation problem with cohomology constraints is controlled by a pair consisting of a differential graded lie algebra together with a module unfortunately these pairs are usually infinite dimensional we show that every deformation problem with cohomology constraints is controlled by a typically finite dimensional l pair as a first application we show that for complex algebraic varieties with no weight zero 1 cohomology classes the components of the cohomology jump loci of rank one local systems containing the constant sheaf are tori this imposes restrictions on the fundamental groups the same holds for links and milnor fibers

multiple objective nonatomic markov decision processes with total reward criteria we consider a markov decision process with an uncountable state space and multiple rewards for each policy its performance is evaluated by a vector of total expected rewards under the standard continuity assumptions and the additional assumption that all initial and transition probabilities are nonatomic we prove that the set of performance vectors for all policies is equal to the set of performance vectors for nonrandomized markov policies this result implies the existence of optimal nonrandomized markov policies for nonatomic constrained markov decision processes with total rewards we provide two examples of applications of our results to constrained multiple objective problems in inventory control and finance

subject allocation and study curtailment for fixed event comparative poisson trials comparative poisson trials of prophylactic interventions such as vaccines can be lengthy and costly we evaluate two easily implemented approaches to reduce numbers of disease cases and person years of follow up n u t for comparative poisson trials with fixed numbers of cases t i altering k the portion of n u t allocated to treatment and ii curtailed stopping before t cases if numbers of cases in the treatment or control group indicate that h 0 has already been rejected or will not be rejected at t cases normal and arcsine approximations as well as discrete exact tests are evaluated for studies not stopped early allocating about 1 1 sqrt r a of person years to treatment roughly minimizes t needed for given size and power where r a is the alternative hypothesized relative disease incidence in treated subjects used to power the study this reduces t moderately vs equal allocation k 0 5 by 2 3 cases in our examples with exact tests however the common practice of allocating k 0 5 of the person years to treatment may be the overall best strategy to minimize n u t for studies that are not stopped early for studies analysed by exact test and planned with a one sided alpha ranging from 0 005 to 0 025 beta from 0 1 to 0 2 and r a from 0 2 to 0 5 curtailed stopping reduces both the number of disease cases and n u t by 6 40 per cent depending on true treatment benefit with curtailed stopping allocating k 0 5 person years to treatment approximately minimizes the numbers of cases and person years under most conditions although k as large as 0 6 often performs comparably well if a specific localized k is selected to minimize disease cases for a study analysed by exact test the study may be underpowered should the final allocation deviate even slightly from that k when it is conducted

on the rational homotopy type of function spaces the main result of this paper is the construction of a minimal model for the function space f x y of continuous functions from a finite type finite dimensional space x to a finite type nilpotent space y in terms of minimal models for x and y for the component containing the constant map 7r f x y 0 q 7r y 0 h x q in positive dimensions when x is formal there is a simple formula for the differential of the minimal model in terms of the differential of the minimal model for y and the coproduct of h x q we also give a version of the main result for the space of cross sections of a fibration

a convex analytic approach to markov decision processes this paper develops a new framework for the study of markov decision processes in which the control problem is viewed as an optimization problem on the set of canonically induced measures on the trajectory space of the joint state and control process this set is shown to be compact convex one then associates with each of the usual cost criteria infinite horizon discounted cost finite horizon control up to an exit time a naturally defined occupation measure such that the cost is an integral of some function with respect to this measure these measures are shown to form a compact convex set whose extreme points are characterized classical results about existence of optimal strategies are recovered from this and several applications to multicriteria and constrained optimization problems are briefly indicated

maurer cartan moduli and models for function spaces we set up a formalism of maurer cartan moduli sets for l algebras and associated twistings based on the closed model category structure on formal differential graded algebras a k a differential graded coalgebras among other things this formalism allows us to give a compact and manifestly homotopy invariant treatment of chevalley eilenberg and harrison cohomology we apply the developed technology to construct rational homotopy models for function spaces

andre quillen cohomology and rational homotopy of function spaces we develop a simple theory of andre quillen cohomology for commutative differential graded algebras over a field of characteristic zero we then relate it to the homotopy groups of function spaces and spaces of homotopy self equivalences of rational nilpotent cw complexes this puts certain results of sullivan in a more conceptual framework

constrained discounted markov decision processes with borel state spaces abstract we study discrete time discounted constrained markov decision processes cmdps with borel state and action spaces these cmdps satisfy either weak w continuity conditions that is the transition probability is weakly continuous and the reward function is upper semicontinuous in state action pairs or setwise s continuity conditions that is the transition probability is setwise continuous and the reward function is upper semicontinuous in actions our main goal is to study models with unbounded reward functions which are often encountered in applications e g in consumption investment problems we provide some general assumptions under which the optimization problems in cmdps are solvable in the class of randomized stationary policies and in the class of chattering policies introduced in this paper if the initial distribution and transition probabilities are atomless then using a general purification result of feinberg and piunovskiy we show the existence of a deterministic stationary optimal policy our main results are illustrated by examples

sample size and power for prospective analysis of relative risk in a placebo controlled vaccine efficacy trial or a trial of equivalence of vaccines one may wish to show that relative risk of disease is less than a specified value r0 not equal to one this paper compares three methods for estimating relative risk in the binomial setting based on a logarithmic transformation likelihood scores and a poisson approximation exact power and size of test are calculated by enumeration of possible binomial outcomes and power is approximated from asymptotic formulations although the score method is generally preferable for most studies of practical interest the log and score methods are comparable and the poisson method is also appropriate for small risks up to about 0 05 when true and null relative risks are less than one unequal allocation of study individuals can increase power and the asymptotic formula for the log method may substantially underestimate power in such a study the power approximation for the score method is more reliable even if the log method is used in analysis exact power calculations are helpful in planning studies the log and poisson methods but not the score method apply readily in the case of unequal follow up

on alleged gains in power from lower p values the various suggestions that have been made to increase the power of fisher s test for 2 2 tables are shown to give no real increases one sided tests are examined in detail but two sided test problems are also considered the need for flexibility in use of p values is stressed

statistical approaches to establishing vaccine safety in a vaccine safety trial the primary interest is to demonstrate that the vaccine is sufficiently safe rejecting the null hypothesis that the relative risk of an adverse event attributable to the new vaccine is above a prespecified value greater than one we evaluate the exact probability of type i error of the likelihood score test with sample size determined by normal approximation by enumeration of the binomial outcomes in the rejection region and show that it exceeds the nominal level in the case of rare adverse events we recommend the poisson approximation as an alternative and develop the corresponding conditional and unconditional tests we give sample size and power calculations for these tests we also propose optimal randomization strategies which either i minimize the total number of adverse cases or ii minimize the expected number of subjects when the vaccine is unsafe we illustrate the proposed methods using a hypothetical vaccine safety study copyright 2002 john wiley sons ltd

lie models for the components of sections of a nilpotent fibration we give an explicit lie model for any component of the space of free and pointed sections of a nilpotent fibration and in particular of the free and pointed mapping spaces among the applications presented we obtain a lie model of the exponential law and prove that in many cases the rank of the homotopy groups of the mapping space grows at the same rate as the rank of the homotopy groups of the target space

ergodic control of markov chains with constraints the general case the problem of controlling a markov chain on a countable state space with ergodic or long run average cost is studied in the presence of additional constraints requiring finitely many say m other ergodic costs to satisfy prescribed bounds under extremely general conditions it is proved that an optimal stationary randomized strategy can be found that requires at most m randomizations this generalizes a result of ross

minimal models of quantum homotopy lie algebras via the bv formalism using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral

rational homotopy of the space of sections of a nilpotent bundle we show that an algebraic construction proposed by sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle in his paper uhomologie des espaces fonctionnels r thom studied the homotopy type of the space f of continuous maps of x into f homotopic to a given map starting from a postnikov decomposition of f he built the functional space f step by step he also indicated how one could construct a differential graded algebra describing the rational homotopy type of f later on sullivan gave an algebraic model which mirrors this construction in terms of a dg algebra representing x and the minimal model of f the aim of this paper is to show following the method of thom that the model of sullivan is indeed a model for the functional space under suitable restrictions as in 3 we consider the slightly more general problem of the determination of the rational homotopy type of the space of sections ts of a nilpotent fiber space p y x homotopic to a given section s in 1 we explain thorn s geometric construction in 2 we describe an algebraic model for an abelian galois covering of a nilpotent space in 3 we show how the model of sullivan fits with the geometry i thank the referee for many improvements of this paper 1 a postnikov factorization of the space of sections 1 1 let g be a finitely generated abehan group and let x be a path connected space whose cohomology groups hk x g are finitely generated for each k to avoid difficulties with the topologies cf 4 we can work in the category of simplicial sets proposition thom 4 the space k g m x of continuous maps of x in the eilenberg mac lane complex k g m is homotopically equivalent to the product nr ra fthe eilenberg mac lane spaces ki k hm x g i more precisely let x g hm k g m g be the fundamental class of k g m if e k g m x xx k g m

what do homotopy algebras form abstract in paper 4 we constructed a symmetric monoidal category s lie mc whose objects are shifted and filtered l algebras here we fix a cooperad c and show that algebras over the operad cobar c naturally form a category enriched over s lie mc following 4 we integrate this s lie mc enriched category to a simplicial category hoalg c whose mapping spaces are kan complexes the simplicial category hoalg c gives us a particularly nice model of an 1 category of cobar c algebras we show that the homotopy category of hoalg c is the localization of the category of cobar c algebras and morphisms with respect to quasi isomorphisms finally we show that the homotopy transfer theorem is a simple consequence of the goldman millson theorem

on some formality criteria for dg lie algebras abstract we give some formality criteria for a differential graded lie algebra to be formal for instance we show that a dg lie algebra l is formal if and only if the natural spectral sequence computing the chevalley eilenberg cohomology h ce l l degenerates at e 2

models for classifying spaces and derived deformation theory using the theory of extensions of l l l algebras we construct rational homotopy models for classifying spaces of fibrations giving answers in terms of classical homological functors namely the chevalley eilenberg and harrison cohomology we also investigate the algebraic structure of the chevalley eilenberg complexes of l l l algebras and show that they possess along with the gerstenhaber bracket an l l l structure that is homotopy abelian

disconnected rational homotopy theory we construct two algebraic versions of homotopy theory of rational disconnected topological spaces one based on differential graded commutative associative algebras and the other one on complete differential graded lie algebras as an application of the developed technology we obtain results on the structure of maurer cartan spaces of complete differential graded lie algebras

splitting randomized stationary policies in total reward markov decision processes this paper studies a discrete time total reward markov decision process mdp with a given initial state distribution a randomized stationary policy can be split on a given set of states if the occupancy measure of this policy can be expressed as a convex combination of the occupancy measures of stationary policies each selecting deterministic actions on the given set and coinciding with the original stationary policy outside of this set for a stationary policy necessary and sufficient conditions are provided for splitting it at a single state as well as sufficient conditions for splitting it on the whole state space these results are applied to constrained mdps the results are refined for absorbing including discounted mdps with finite state and actions spaces in particular this paper provides an efficient algorithm that presents the occupancy measure of a given policy as a convex combination of the occupancy measures of finitely many stationary deterministic policies this algorithm generates the splitting policies in a way that each pair of consecutive policies differs at exactly one state the results are applied to constrained problems to efficiently compute an optimal policy by computing and splitting a stationary optimal policy

a version of the goldman millson theorem for filtered l algebras abstract in this paper we consider l algebras equipped with complete descending filtrations we prove that under some mild conditions an l quasi isomorphism u l l induces a weak equivalence between the deligne getzler hinich dgh groupoids corresponding to l and l respectively this paper may be considered as a modest addition to foundational paper 10 by ezra getzler