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This study compares the efficacy of GPT-4 and clinalytix Medical AI in predicting the clinical risk of delirium development. Findings indicate that GPT-4 exhibited significant deficiencies in identifying positive cases and struggled to provide reliable probability estimates for delirium risk, while clinalytix Medical AI demonstrated superior accuracy. A thorough analysis of the large language model's (LLM) outputs elucidated potential causes for these discrepancies, consistent with limitations reported in extant literature. These results underscore the challenges LLMs face in accurately diagnosing conditions and interpreting complex clinical data. While LLMs hold substantial potential in healthcare, they are currently unsuitable for independent clinical decision-making. Instead, they should be employed in assistive roles, complementing clinical expertise. Continued human oversight remains essential to ensure optimal outcomes for both patients and healthcare providers.

To figure properties of a curve of form $C_{f,g} = {(x,y)| f(x) - g(y)= 0}$ you must address the genus 0 and 1 components of its projective normalization $\tilde C_{f,g}$. For $f$ and $g$ polynomials with $f$ indecomposable, [Fr73a] distinguished $\tilde C_{f,g}$ with $u=1$ versus $u > 1$ components (Schinzel's problem). For $u = 1$, [Prop. 1, Fr73b] gave a direct genus formula. To complete $u > 1$ required an adhoc genus computation. [Pak22] dropped the indecomposable and polynomial restrictions but added $\tilde C_{f,g}$ is irreducible ($u = 1$). He showed - for fixed $f$ - unless the Galois closure of the cover for $f$ has genus 0 or 1, the genus grows linearly in deg($g$). Method I and Method II extend [Prop. 1, Fr73b}] using Nielsen classes to generalize Pakovich's formulation for $u > 1$. Method I plays on the covers $f$ and $g$ to the $z$-line, $P^1_z$, from which we compute the fiber product. Method II uses the projection to the $y$-line, $P^1_y$, based on explicitly computing branch cycles for this cover. Hurwitz families track the significance of these components. Expanding on [Prop. 2, Fr73a] shows how to approach Pakovich's problem. With no loss, start with ($f^*,g^*$) which have the same Galois closures, and for which their canonical representations are entangled. They, therefore, produce more than one component on the fiber product. Then, we classify the possible component types, $W$, that appear on $\tilde C_{f^*,g^*}$ using the branch cycles for $W$ that come from Method II. The result is a Nielsen class formulation telling explicitly what $g_1\,$s to avoid to assure the growth of the component genuses of $\tilde C_{f*,g*og_1}$ as deg($g_1$) increases. Of particular note: using and expanding on Nielsen classes and the solution of the genus 0 problem (classifying the monodromy groups of indecomposable rational functions)., Comment: Submitted to the memorial volume(s) for Andzej Schinzel to Acta Arithmetica, as of 08/06/22

Using Felgner's problem I revisit a key issue in using the "Galois Stratification Procedure" that first appeared in [FrS76]. The emphasis here is on using arithmetic homotopy to make the production of Poincare; series attached to general diophantine statements canonical. According to work in progress of Michael Benedikt and E. Hrushovski, Galois stratification - over one finite field - is as efficient as is possible: on a statement of length n, it requires time bounded by a stack of exponentials of length linear in n. This doesn't take advantage of problems prepped for using homotopy aspects, Chow Motives, efficiently as in the main example which comes from my paper on the generalization of exceptional covers. That example [FrJ, Chap. 30], simplifies aspects of the original procedure. It combines this with the later theory of Frobenius fields to produce objects over Q whose reductions mod primes give the stratification procedure at the prime. The paper separates two different uses of the Chebotarev non-regular analog. 1. Field crossing to interpret Poincare series coefficients directly from traces on Chow motives (providing valuable statements on variation with the prime p; versus 2. Chebotarev using Lang-Weil to approximate the number of points on an appropriate variety for the Galois stratification procedure. We consider variables taking values in the algebraic closure of Z/p but fixed by respective powers of the Frobenius: we call these Frobenius vectors. For this there is a twisted Chebotarev version stemming from a conjecture of Deligne, and outlined in a preprint of Hrushovski. This paper expands on the work of D. Wan, J. Denef and F. Loeser, J. Nicaise, I. Tomasic and E. Hrushovski, all relevant to taking the Galois stratification procedure beyond the original finite field framework.