totally ramified splitting fields of central simple algebras over henselian fields abstract let d be a central division algebra over a henselian field f with algebraically closed residue field f it is proved that if the degree of d is relatively prime to the characteristic of f then every finite extension of f which splits d contains a maximal subfield of d

on nearly optimal controls of hybrid lqg problems the paper is concerned with hybrid control of a class of linear quadratic gaussian lqg systems modulated by a finite state markov chain it develops approximation schemes for systems involving singularly perturbed markov chains with weak and strong interactions which are useful for natural time scale separation and for large scale markovian systems the presentation encompasses three cases the recurrent markov chains inclusion of transient states and inclusion of absorbing states near optimality of the approximation schemes is obtained and a numerical example is presented computational results indicate our approximation schemes perform well

robust jump linear quadratic control a mode stabilizing solution this correspondence is concerned with optimal feedback control of a class of linear stochastic systems for a quadratic performance index the solution to this problem is known but the resulting optimal regulator does not ensure stability in every mode of operation to improve robustness a new mode stabilizing solution is defined an example illustrates the advantages of the proposed controller

control of a hybrid stochastic system abstract we consider in this paper a continuous time stochastic hybrid control system with a finite time horizon the objective is to minimize a linear function of the expected state trajectory the state evolves according to a linear dynamics however the parameters of the state evolution equation may change at discrete times according to a controlled markov chain which has finite state and action spaces we use a procedure similar in form to the maximum principle this determines a control strategy which is asymptotically optimal as the number of transitions during the finite time horizon grows to infinity

definable sets in ordered structures i this paper introduces and begins the study of a well behaved class of linearly ordered structures the minimal structures the definition of this class and the corresponding class of theories the strongly dlminlmal theories is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories theorems 2 1 and 2 3 respectively provide characterizations of 41 minimal ordered groups and rings several other simple results are collected in 3 the primary tool in the analysis of minimal structures is a strong analogue of ii forking symmetry given by theorem 4 2 this result states that any parametri cally definable unary function in an yminimal structure is piecewise either constant or an ore er preserving or reversing bijection of intervals the results that follow include the existence and uniqueness of prime models over sets theorem 5 1 and a characterization of all bto categorical aminimal structures theorem 6 1 1 introduction the class of linearly ordered structures has long been an im portant subject of concern to model theorists impressive results have been obtained in the study of models of several particular theories that extend the theory of linear order among those theories that have been approached successfully are peano arithmetic the theory of ordered abelian groups that of real closed fields and that of linear order itself yet very little has been done in the way of developing a general model theory for ordered structures in this paper we develop the model theory for a class of linearly ordered structures that we isolate by demanding that a structure in this class satisfy a condition whose effect is that the linear ordering and the algebraic part of the structure behave quite well with respect to one another let l be a finitary first order language and w an l structure a set of n tuples a c x is said to be parametrically definable if there is some l formula

hybrid filtering for linear systems with non gaussian disturbances this paper is concerned with hybrid filtering of a non gaussian system coupled by a hidden switching process an optimal control approach is used to derive a finite dimensional recursive filter that is optimal in the sense of the most probable trajectory mpt estimate an approximate scheme is obtained when the probability distribution of the switching process converges to a limiting distribution which is often easier to obtain numerical examples are considered and computational experiments are reported in these examples our schemes outperform popular filtering schemes available in the literature

a new construction of noncrossed product algebras new examples of noncrossed product division algebras are obtained using methods different from all previous noncrossed product constructions the examples are division algebras over intersections of henselian valued fields and they have schur index and exponent p for any prime number p and any integers m n 2 n 3if 2 the basic tools used in the construction are valuation theory and galois cohomology no generic methods are applied and there is no p i theory along the way local global principles are proved for central simple algebras over intersections of p henselian valued fields the first examples of central simple division algebras which are not crossed products were obtained by amitsur in am in 1972 amitsur thereby settled a question that had been one of the outstanding open problems in the theory of algebras for at least thirty years his examples were the generic division algebras ud q n of index n over q the rational numbers for any natural number n such that p2 n p an odd prime or 8 all subsequent constructions of noncrossed products in ss 83 883 r r an ti have heretofore been based on amitsur s specialization argument and they are all generic division algebras or extensions of generic division algebras the centers of these noncrossed product algebras are not known nor are the brauer groups nor the absolute galois groups of the centers we present here a new method of constructing noncrossed product algebras in our approach the noncrossed product is realized as the underlying division algebra d of a tensor product of suitably chosen cyclic algebras over a field f l n l2 where each l is a pth root henselian valued field we prove local global principles relating the splitting fields of d to those of d f lt i 1 2 it is shown that the part of the brauer group of f is completely determined by that of ll and l2 computations for central simple f algebras thus become very tractable for exam ple we not only show that d is not a crossed product but also calculate exactly how large r must be so that the matrix ring mr d is a crossed product and how large s must be so that ms d is a tensor product of cyclic algebras indeed the structure of the brauer group of f is so nice that we were somewhat surprised that noncrossed products could possibly exist over f

totally ramified valuations on finite dimensional division algebras division algebras d with valuation v are studied where d is lsnite dimensional and totally ramified over its center f i e the ramification index of v over wlf equals d f such division algebras have appeared in some important constructions but the structure of these algebras has not been systematically analyzed before when vlf is henselian a full classification of the f subalgebras of d is given when f has a henselian valuation v with separably closed residue field and a is any tame central simple f algebra an algorithm is given for computing the underlying division algebra of a from a suitable subgroup of a f some examples are constructed using this valuation theory including the first example of finitbdimensional f central division algebras d1 and d2 with d1 xf d2 not a division ring but d1 and d2 having no common subfield k d f valuation theory long a basic tool in commutative algebra has been relatively neglected in the study of division algebras until quite recently nontheless valuations are naturally present in a number of division algebras that have been constructed to exhibit special properties particularly algebras over iterated laurent power series fields for example such division algebras have been key ingredients in amitsur s noncrossed product construction am and in platonov s construction p of division algebras d with sk1 d 7 1 valuations are not so prevalent on division algebras as on fields but if a division algebra d does have a valuation this structure contains a substantial amount of information about d which would scarcely be accessible otherwise we consider here valued divisioll algebras d for which d is totqlly ramified and tame over its center f i e for which the ramification index ird rfi equals the dimension d f of d over f and the characteristic char d does not divide d f here rd is the value group of the valuation on d and d is the residue division algebra we assume throughout that d f oo valued division algebras of this type appear e g in amitsur s noncrossed product paper am 2 in saltman s work on indecomposable division algebras sa in certain of the malcevneumann division algebras considered by the first author and amitsur ta2 4 etc however the intrinsic structure of totally ramified tame division algebras has apparently not been examined closely before this may be because most past work on valued division algebras has concentrated on discrete valuations when rd z for such a valuation d is never totally ramified over its center f unless d f cf 3 2 below received by the editors june 26 1986 1980 mathematics subject clmsification 1985 reon primary 16a39 2supported in part by f n r s 2supported in part by the national science foundation 1987 american mathematlcal society 0002 9947 87 00 25 per page

on the structure of pythagorean fields abstract in this paper we study in detail the relationship between fans and valuations first uncovered by brocker see 2 our techniques are somewhat different as our valuations are constructed by brute force from the properties of the fields involved this approach has the advantage in that the valuation theory it yields can be directly related to the structure theory for spaces of orderings as developed by murray marshall see 11 14 we also discuss several applications which include results in k theory

definable sets in ordered structures on introduit la notion de theorie o minimale des structures ordonnees une telle theorie etant telle que les sous ensembles definissables de ses modeles soient particulierement simples

asymptotic expansions of singularly perturbed systems involving rapidly fluctuating markov chains a class of singularly perturbed time varying systems with a small parameter varepsilon 0 is considered in this paper the importance of the study stems from the fact that many problems arise in various applications involve a rapidly fluctuating markov chain to investigate the limit behavior of such systems it is necessary to consider the corresponding singular perturbation problems existing results in singular perturbation of ordinary differential equations cannot be applied since the coefficient matrix of the equation is a generator of a finite state markov chain and as a result it is singular asymptotic properties of the aforementioned systems are developed via matched asymptotic expansion in this paper thanks to the specific structure and the properties of markovian generators it is established that the solution of the system can be approximated as close as possible by a series expansion in terms of the small parameter varepsilon 0

demuskin fields with valuations we study the arithmetic structure of fields f of characteristic neq p containing a primitive pth root of unity for which the maximal pro p galois group of f is a finitely generated demuskin group

division algebras over henselian fields in this chapter we focus on the tame division algebras d with center a field f with henselian valuation v as usual we approach this by first obtaining results for graded division algebras then lifting back from operatorname mathsf gr d to d this is facilitated by results in 8 1 on existence and uniqueness of lifts of tame subalgebras from operatorname mathsf gr d to d in 8 2 we describe four fundamental canonical up to conjugacy subalgebras of d that reflect its valuative structure the rest of the chapter is devoted to brauer group factorizations of d corresponding to the noncanonical direct product decomposition of operatorname mathit br t f given in cor 7 85 the factor operatorname mathit hom c operatorname mathcal g overline f mathbb t gamma f is represented by a type of division algebra n called decomposably semiramified defined in 8 3 and characterized by the property that n contains a maximal subfield inertial over f and another totally ramified over f we show in 8 4 that every tame division algebra d is brauer equivalent to some s f t where s is inertially split and t is tame and totally ramified over f we show further that every inertially split division algebra s is brauer equivalent to some i f n where i is inertial over f and n is decomposably semiramified the classes overline i for the i appearing in the i f n decompositions of s are shown to range over a single coset of mathit dec z overline s overline f in operatorname mathit br overline f called the specialization coset of s in the final subsection 8 4 6 we summarize what happens in the special case that v is discrete of rank 1 where substantial simplifications occur

h infintity control of large scale jump linear systems via averaging and aggregation this paper studies under state feedback policies the h infinity control design for large scale jump linear systems where the form process admits strong and weak interactions through an analysis that covers both finite and infinite horizon cases and using averaging and aggregation techniques an aggregate jump linear system of considerably smaller order has been obtained along with a corresponding compatible cost function this reduced order aggregate problem is another piecewise deterministic h infinity control problem and on the basis of the solution of this problem we obtain the asymptotic limit of the optimal performance level for the full order system as well as an approximate controller that can asymptotically achieve any desired performance level for the full order system a by product of this analysis is a similar decomposition for the piecewise deterministic linear regulator problem

a generalization of marshall s equivalence relation for p prime and for a field f containing a root of unity of order p we generalize marshall s equivalence relation on orderings to arbitrary subgroups of f of index p the equivalence classes then correspond to free pro p factors of the maximal pro p galois group of f we generalize to this setting results of jacob on the maximal pro 2 galois group of a pythagorean field

value functions on central simple algebras in this paper we study noncommutative valuation rings as defined by dubrovin while there is in general no valuation associated to a dubrovin valuation ring we show that there is a value function associated to any dubrovin valuation ring integral over its center by using value functions we obtain information on dubrovin valuation rings in a tensor product both generalizing and giving a much simpler proof of a result about valued division algebras by being able to work directly with central simple algebras we gain new information about division algebras over henselian fields

singularly perturbed discrete time markov chains originating from a wide range of applications in optimization and control of large scale systems such as telecommunications queueing networks and manufacturing systems this work is devoted to a class of singularly perturbed discrete time markov chains the states of the markov chain are naturally decomposable into recurrent and transient classes such that within each class the interactions are strong and among different classes the interactions are weak our study focuses on the difference equations representing the probabilityvector and aims at deriving matched asymptotic expansions of the solutions justification and error analysis are also provided both time homogeneous and time inhomogeneous models are examined

a recursive description of pro p galois groups abstract in this note we study a modified version of the elementary type conjecture for pro p galois groups to be precise let g p f be the galois group of the maximal galois p extension of a field f containing a primitive p th root of unity under some natural assumptions concerning valuation rings of f and also orderings of f when p 2 we prove that g p f can be obtained from suitable closed subgroups using a finite number of free pro p products and semidirect products

asymptotically optimal controls of hybrid linear quadratic regulators in discrete time this work develops asymptotically optimal controls for a class of discrete time hybrid systems involving singularly perturbed markov chains having weak and strong interactions the state space of the underlying markov chain is decomposed into a number of recurrent classes and a group of transient states using a hierarchical control approach by aggregating the states in each recurrent class into a single state a continuous time quadratic limit control problem in which the resulting limit markov chain has much smaller state space is derived using the optimal control for the limit problem a control for the original problem is constructed which is shown to be nearly optimal finally a numerical example is given to demonstrate the effectiveness of the approximation scheme

markowitz s mean variance portfolio selection with regime switching from discrete time models to their continuous time limits we study a discrete time version of markowitz s mean variance portfolio selection problem where the market parameters depend on the market mode regime that jumps among a finite number of states the random regime switching is delineated by a finite state markov chain based on which a discrete time markov modulated portfolio selection model is presented such models either arise from multiperiod portfolio selections or result from numerical solution of continuous time problems the natural connections between discrete time models and their continuous time counterpart are revealed since the markov chain frequently has a large state space to reduce the complexity an aggregated process with smaller state space is introduced and the underlying portfolio selection is formulated as a two time scale problem we prove that the process of interest yields a switching diffusion limit using weak convergence methods next based on the optimal control of the limit process obtained from our recent work we devise portfolio selection strategies for the original problem and demonstrate their asymptotic optimality

optimal adaptive lqg control for systems with finite state process parameters the situation where a totally observed process y t is generated by a stochastic differential equation whose parameters evolve on a finite set 1 n according to a stochastic differential equation is considered the optimal control law is sought with respect to quadratic loss functions on y t and the control u t the auxiliary p d e technique of hijab 6 is used together with a nonlinear filter to obtain the solution whose existence depends upon that of a smooth solution to the auxiliary p d e and strong solutions to the system s d e under the given control inputs

controllability stabilizability and continuous time markovian jump linear quadratic control consideration is given to the control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes the relationship between appropriately defined controllability stabilizability properties and the solution of the infinite time jump linear quadratic jlq optimal control problems is also examined although the solution of the continuous time markov jlq problem with finite or infinite time horizons is known only sufficient conditions for the existence of finite cost constant stabilizing controls for the infinite time problem appear in the literature in this paper necessary and sufficient conditions are established these conditions are based on new definitions of controllability observability stabilizability and detectability that are appropriate for continuous time markovian jump linear systems these definitions play the same role for the jlq problem as the deterministic properties do for the linear quadratic regulator lqr problem

jump linear quadratic gaussian control in continuous time the optimal quadratic control of continuous time linear systems that possess randomly jumping parameters which can be described by finite state markov processes is addressed the systems are also subject to gaussian input and measurement noise the optimal solution for the jump linear quadratic gaussian jlqc problem is given this solution is based on a separation theorem the optimal state estimator is sample path dependent if the plant parameters are constant in each value of the underlying jumping process then the controller portion of the compensator converges to a time invariant control law however the filter portion of the optimal infinite time horizon jlqc compensator is not time invariant thus a suboptimal filter which does converge to a steady state solution under certain conditions is derived and a time invariant compensator is obtained

stability of markov modulated discrete time dynamic systems motivated by various control applications this paper develops stability analysis of discrete time systems with regime switching in which the dynamics are modulated by markov chains with two time scales using the high contrast of the different transition rates among the markovian states singularly perturbed markov chains are used in the formulations taking into consideration of the regime changes and time scale separation makes the stability analysis a difficult task in this work asymptotic stability analysis is carried out using perturbed liapunov function techniques it is demonstrated that if the limit system is stable then the original system is also stable in addition we examine path excursion derive bounds on mean recurrence time and obtain the associated probability bounds

how to avoid a compact set abstract a first order expansion of the r vector space structure on r does not define every compact subset of every r n if and only if topological and hausdorff dimension coincide on all closed definable sets equivalently if a r k is closed and the hausdorff dimension of a exceeds the topological dimension of a then every compact subset of every r n can be constructed from a using finitely many boolean operations cartesian products and linear operations the same statement fails when hausdorff dimension is replaced by packing dimension

algorithms for singularly perturbed limiting average markov control problems a singularly perturbed markov decision process with the limiting average reward criterion is considered it is assumed that the underlying process is composed of n separate irreducible processes and that the small perturbation is such that it unites these processes into a single irreducible process two algorithms for the solution of the underlying limit markov control problem are presented the first of these is a linear program possessing the wolfe dantzig structure inherited from the ergodic nearly decomposable assumption in the model the second is an aggregation disaggregation policy improvement algorithm

structures having o minimal open core the open core of an expansion of a dense linear order is its reduct in the sense of definability generated by the collection of all of its open definable sets in this paper expansions of dense linear orders that have o minimal open core are investigated with emphasis on expansions of densely ordered groups the first main result establishes conditions under which an expansion of a densely ordered group has an o minimal open core specifically the following is proved let r be an expansion of a densely ordered group r that is definably complete and satisfies the uniform finiteness property then the open core of r is o minimal two examples of classes of structures that are not o minimal yet have o minimal open core are discussed dense pairs of o minimal expansions of ordered groups and expansions of o minimal structures by generic predicates in particular such structures have open core interdefinable with the original o minimal structure these examples are differentiated by the existence of definable unary functions whose graphs are dense in the plane a phenomenon that can occur in dense pairs but not in expansions by generic predicates the property of having no dense graphs is examined and related to uniform finiteness definable completeness and having o minimal open core