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Background: In clinical practice, anaerobic threshold (AT) is used to guide training and rehabilitation programs, to define risk of major thoracic or abdominal surgery, and to assess prognosis in heart failure (HF). AT of oxygen uptake (V.O2; V.O2AT) has been reported as an absolute value (V.O2ATabs), as a percentage of predicted peak V.O2 (V.O2AT%peak_pred), or as a percentage of observed peak V.O2 (V.O2AT%peak_obs). A direct comparison of the prognostic power among these different ways to report AT is missing. Research question: What is the prognostic power of these different ways to report AT? Study design and methods: In this observational cohort study, we screened data of 7,746 patients with HF with a history of reduced ejection fraction (< 40%) recruited between 1998 and 2020 and enrolled in the Metabolic Exercise Combined With Cardiac and Kidney Indexes register. All patients underwent a maximum cardiopulmonary exercise test, executed using a ramp protocol on an electronically braked cycle ergometer. Results: This study considered 6,157 patients with HF with identified AT. Follow-up was median, 4.2 years (25th-75th percentiles, 1.9-5.0 years). Both V.O2ATabs (mean ± SD, 823 ± 305 mL/min) and V.O2AT%peak_pred (mean ± SD, 39.6 ± 13.9%), but not V.O2AT%peak_obs (mean ± SD, 69.2 ± 17.7%), well stratified the population regarding prognosis (composite end point: cardiovascular death, urgent heart transplant, or left ventricular assist device). Comparing area under the receiver operating characteristic curve (AUC) values, V.O2ATabs (0.680) and V.O2AT%peak_pred (0.688) performed similarly, whereas V.O2AT%peak_obs (0.538) was significantly weaker (P < .001). Moreover, the V.O2AT%peak_pred AUC value was the only one performing as well as the AUC based on peak V.O2 (0.710), with an even a higher AUC (0.637 vs 0.618, respectively) in the group with severe HF (peak V.O2 < 12 mL/min/kg). Finally, the combination of V.O2AT%peak_pred with peak V.O2 and V. per CO2 production shows the highest prognostic power. Interpretation: In HF, V.O2AT%peak_pred is the best way to report V.O2 at AT in relationship to prognosis, with a prognostic power comparable to that of peak V.O2 and, remarkably, in patients with severe HF.

Recently, various possible expressions for the vacuum-to-vacuum superstring amplitudes has been proposed at genus $g=3,4,5$. To compare the different proposals, here we will present a careful analysis of the comparison between the two main technical tools adopted to realize the proposals: the classical theta constants and the lattice theta series. We compute the relevant Fourier coefficients in order to relate the two spaces. We will prove the equivalence up to genus 4. In genus five we will show that the solutions are equivalent modulo the Schottky form and coincide if we impose the vanishing of the cosmological constant., 21 pages