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We study multidimensional diagrams in independent amalgamation in the framework of abstract elementary classes (AECs). We use them to prove the eventual categoricity conjecture for AECs, assuming a large cardinal axiom. More precisely, we show assuming the existence of a proper class of strongly compact cardinals that an AEC which has a single model of some high enough cardinality will have a single model in any high enough cardinal. Assuming a weak version of the generalized continuum hypothesis, we also establish the eventual categoricity conjecture for AECs with amalgamation. [ABSTRACT FROM AUTHOR]

Objective: Due to a complex set of processes involved with the recording of health information in the Electronic Health Records (EHRs), the truthfulness of EHR diagnosis records is questionable. We present a computational approach to estimate the probability that a single diagnosis record in the EHR reflects the true disease.Materials and Methods: Using EHR data on 18 diseases from the Mass General Brigham (MGB) Biobank, we develop generative classifiers on a small set of disease-agnostic features from EHRs that aim to represent Patients, pRoviders, and their Interactions within the healthcare SysteM (PRISM features).Results: We demonstrate that PRISM features and the generative PRISM classifiers are potent for estimating disease probabilities and exhibit generalizable and transferable distributional characteristics across diseases and patient populations. The joint probabilities we learn about diseases through the PRISM features via PRISM generative models are transferable and generalizable to multiple diseases.Discussion: The Generative Transfer Learning (GTL) approach with PRISM classifiers enables the scalable validation of computable phenotypes in EHRs without the need for domain-specific knowledge about specific disease processes.Conclusion: Probabilities computed from the generative PRISM classifier can enhance and accelerate applied Machine Learning research and discoveries with EHR data. [ABSTRACT FROM AUTHOR]

We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models. We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear notion of superstability in this framework. We also present an application to homogeneous model theory: for D a homogeneous diagram in a first-order theory T , if D is both stable in | T | and categorical in | T | then D is stable in all λ ≥ | T |. [ABSTRACT FROM AUTHOR]

Shelah has provided sufficient conditions for an ${\Bbb L}_{\omega _1 ,\omega } $ -sentence ψ to have arbitrarily large models and for a Morley-like theorem to hold of ψ. These conditions involve structural and set-theoretic assumptions on all the ${\aleph _n}$ 's. Using tools of Boney, Shelah, and the second author, we give assumptions on ${\aleph _0}$ and ${\aleph _1}$ which suffice when ψ is restricted to be universal: Theorem. Assume ${2^{{\aleph _0}}}. Let ψ be a universal ${\Bbb L}_{\omega _1 ,\omega } $ -sentence. (1) If ψ is categorical in ${\aleph _0}$ and $1 \leqslant {\Bbb L}\left({\psi ,\aleph _1 } \right) , then ψ has arbitrarily large models and categoricity of ψ in some uncountable cardinal implies categoricity of ψ in all uncountable cardinals. (2) If ψ is categorical in ${\aleph _1}$ , then ψ is categorical in all uncountable cardinals. The theorem generalizes to the framework of ${\Bbb L}_{\omega _1 ,\omega } $ -definable tame abstract elementary classes with primes. [ABSTRACT FROM AUTHOR]

Abstract: A new case of Shelah's eventual categoricity conjecture is established: Let K be an abstract elementary class with amalgamation. Write μ : = ℶ ( 2 LS ( K ) ) + and H 2 : = ℶ ( 2 μ ) +. Assume that K is H2‐tame and K ≥ H 2 has primes over sets of the form M ∪ { a }. If K is categorical in some λ > H 2, then K is categorical in all λ ′ ≥ H 2. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of the mentioned result is that Shelah's categoricity conjecture holds in the context of homogeneous model theory (this was known, but our proof gives new cases): If D be a homogeneous diagram in a first‐order theory T and D is categorical in a λ > | T |, then D is categorical in all λ ′ ≥ min ( λ , ℶ ( 2 | T | ) + ). [ABSTRACT FROM AUTHOR]

Abstract: In the setup of abstract elementary classes satisfying a local version of superstability, we prove the uniqueness property for μ‐forking, a certain independence notion arising from splitting. This had been a longstanding technical difficulty when constructing forking‐like notions in this setup. As an application, we show that the two versions of forking symmetry appearing in the literature (the one defined by Shelah for good frames and the one defined by VanDieren for splitting) are equivalent. [ABSTRACT FROM AUTHOR]

We show how to build prime models in classes of saturated models of abstract elementary classes (AECs) having a well-behaved independence relation: Let [ABSTRACT FROM AUTHOR]

We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough $$\lambda {:}$$ Our proofs use independence calculus and a generalization of averages to this non first-order context. [ABSTRACT FROM AUTHOR]