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Creators/Authors contains: "Kaplan, Elliot"

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  1. ; (, Model Theory)
    Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields and assume that T is power bounded. Let K be a model of T equipped with a T-convex valuation ring O and a T-derivation ∂ such that ∂ is monotone, i.e., weakly contractive with respect to the valuation induced by O. We show that the theory of monotone T-convex T-differential fields, i.e., the common theory of such K, has a model completion, which is complete and distal. Among the axioms of this model completion, we isolate an analogue of henselianity that we call T∂-henselianity. We establish an Ax-Kochen/Ershov theorem and further results for monotone T-convex T-differential fields that are T∂-henselian. 
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    Free, publicly-accessible full text available January 22, 2026
  2. ; (, Pacific Journal of Mathematics)
    We consider a tuple Φ = (φ_1,...,φ_m) of commuting maps on a finitary matroid X. We show that if φ satisfies certain conditions, then for any finite set A⊆X, the rank of {φ_1^{r_1}···φ_m^{r_m}(a): a ∈ A and r_1+···+r_m = t} is eventually a polynomial in t (we also give a multivariate version of the polynomial). This allows us to easily recover Khovanskii's theorem on the growth of sumsets, the existence of the classical Hilbert polynomial, and the existence of the Kolchin polynomial. We also prove some new Kolchin polynomial results for differential exponential fields and derivations on o-minimal fields, as well as a new result on the growth of Betti numbers in simplicial complexes. 
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    Free, publicly-accessible full text available December 28, 2025
  3. (, Journal of algebra)
    Let T be a complete, model complete o-minimal theory extending the theory of real closed ordered fields. An HT-field is a model K of T equipped with a T-derivation ∂ such that the underlying ordered differential field of (K,∂) is an H-field. We study HT-fields and their extensions. Our main result is that if T is power bounded, then every HT-field K has either exactly one or exactly two minimal Liouville closed HT-field extensions up to K-isomorphism. The assumption of power boundedness can be relaxed to allow for certain exponential cases, such as T = Th(Ran,exp). 
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    Accepted Manuscript Full Text Available
  4. ; (, Mathematical Logic Quarterly)
    Abstract We prove a dichotomy for o‐minimal fields , expanded by a ‐convex valuation ring (where is the theory of ) and a compatible monomial group. We show that if is power bounded, then this expansion of is model complete (assuming that is), it has a distal theory, and the definable sets are geometrically tame. On the other hand, if defines an exponential function, then the natural numbers are externally definable in our expansion, precluding any sort of model‐theoretic tameness. 
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  5. ; (, Notre Dame Journal of Formal Logic)
    null (Ed.)
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  6. ; ; ; (, Fundamenta Mathematicae)
    null (Ed.)
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  7. ; ; ; ; ; ; ; (, Logical methods in computer science)
    Following Chaudhuri, Sankaranarayanan, and Vardi, we say that a function f:[0,1]→[0,1] is r-regular if there is a Büchi automaton that accepts precisely the set of base r∈N representations of elements of the graph of f. We show that a continuous r-regular function f is locally affine away from a nowhere dense, Lebesgue null, subset of [0,1]. As a corollary we establish that every differentiable r-regular function is affine. It follows that checking whether an r-regular function is differentiable is in PSPACE. Our proofs rely crucially on connections between automata theory and metric geometry developed by Charlier, Leroy, and Rigo. 
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