We prove that in an arbitrary o-minimal structure, every interpretable group is definably isomorphic to a definable one. We also prove that every definable group lives in a cartesian product of one-dimensional definable group-intervals (or one-dimensional definable groups). We discuss the general open question of elimination of imaginaries in an o-minimal structure.2010 Mathematics Subject Classification. Primary 03C64; Secondary 03C60 22E15 20A15.

We classify definable linear orders in o-minimal structures expanding groups. For example, let (P, ≺) be a linear order definable in the real field. Then (P, ≺) embeds definably in (R n+1 , < lex ), where < lex is the lexicographic order and n is the o-minimal dimension of P . This improves a result of Onshuus and Steinhorn in the o-minimal group context. 1809 Licensed to Kansas St Univ. Prepared on Thu Jun 26 14:54:01 EDT 2014 for download from IP 129.130.252.222. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 1810 JANAK RAMAKRISHNAN Remark 1.1. Although we state the theorem in the group context for simplicity, the proof requires only that M be densely ordered o-minimal, have definable choice, and have a definable order-reversing injection. Moreover, Corollary 1.3 gives a sharper statement when M is a field.Our characterization improves [OS09, Cor. 5.1] in the context of Remark 1.1 since the full order is embedded in a single lexicographic order. This means that the study of definable linear orders in o-minimal structures is just the study of definable subsets of lexicographic orders.Besides [OS09, Cor. 5.1], Theorem A also resembles work done in the general context of embedding ordered sets in lexicographic products [Fle61,CI99]. Seen in that light, Theorem A is a definable version of results in these papers, although the results in the general case are only partial [Fle63]. Knoblauch [Kno00] gives a condition on an order equivalent to its embeddability in a real lexicographic product, but this condition is not easily verifiable for a given class of orders, or even for a specific order.Outside applications. The study of linear orders has also been undertaken in economics. O-minimality results apply in the economics context because the structure under consideration is usually the reals. When the elements are in a finite Cartesian power, and the functions used to define the linear order are (semi-)algebraicexponential, we are in an o-minimal setting [Wil96]. Even when the functions are analytic, if the linear order being considered is contained in a bounded set, then we remain in an o-minimal setting [vdDMM94].In [BCHIM02a], efforts were made to characterize linear orders that are not order-embeddable in the reals. Theorem A says that any linear order definable in an o-minimal group will be order-embeddable in the reals exactly when it is onedimensional, only finitely many elements have unique predecessors or successors in the order, and the group is itself order-embeddable in the reals.The desired g in Theorem A can be constructed, given the ability to construct cell decompositions and definable choice functions. We note here that the uniformity in Theorem A follows from a routine model-theoretic compactness argument. Also, it suffices to prove Theorem A for ∅-definable P .The bound of 2n + 1 is sharp by the following: Example 1.2. Let M = (R, <, +, 0) and let n > 0. Let P = { x 1 , . . . , x 2n+1 ∈ M 2n+1 : x i ∈ {0, 1} for i odd}. Let ≺ be the lexicog...

Let M be an arbitrary o-minimal structure. Let G be a definably compact, definably connected, abelian definable group of dimension n. Here we compute: (i) the new intrinsic o-minimal fundamental group of G; (ii) for each k > 0, the k-torsion subgroups of G; (iii) the o-minimal cohomology algebra over Q of G. As a corollary we obtain a new uniform proof of Pillay's conjecture, an o-minimal analogue of Hilbert's fifth problem, relating definably compact groups to compact real Lie groups, extending the proof already known in o-minimal expansions of ordered fields.

Tracer enrichment data are fitted by multicompartmental models to estimate rate constants and fluxes or transport rates. In apolipoprotein turnover studies, mass measurements are also available, e.g., apolipoprotein B (apoB) levels in VLDL, IDL and LDL, and are often essential to calculate some of the rate constants. The usual method to use mass measurements is to estimate pool masses along with rate constants. A systematic alternative approach is developed to use flux balances around pools to express some rate constants in terms of the other rate constants and the measured masses. The resulting reduction in the number of parameters to be estimated makes the modeling more efficient. In models that would be unidentifiable without mass measurements, the usual approach and the proposed approach yield identical results. In a simple two-pool model, the number of unknown parameters is reduced from four to two. In a published 5-pool model for apoB kinetics with three mass measurements, the number of parameters is reduced from 12 to 9. With m mass measurements, the number of responses to be fitted and the number of parameters to be estimated are each reduced by m, a simplification by 1/4 to 1/3 in a typical pool model. Besides a proportionate reduction in computational effort, there is a further benefit since the dimensionality of the problem is also decreased significantly, which means ease of convergence and a smaller likelihood of suboptimal solutions. While our approach is conceptually straightforward, the dependencies get considerably more complex with increasing model size. To generate dependency definitions automatically, a webaccessible program is available at http://biomath.info/poolfit/constraints.

We show, for various classes of totally ordered structures M = (M, <, . . .), including o-minimal and weakly o-minimal structures, that every definable partial order on a subset of M n extends definably in M to a total order. This extends the result proved in [5] for n = 1 and M o-minimal.