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This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set:

$$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$

where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$ ) shown by Hussain, Kleinbock, Wadleigh and Wang.

OBJECTIVE

To compare the accuracy of surveillance of severe sepsis using electronic health record clinical data vs claims and to compare incidence and mortality trends using both methods.

DESIGN

We created an electronic health record–based surveillance definition for severe sepsis using clinical indicators of infection (blood culture and antibiotic orders) and concurrent organ dysfunction (vasopressors, mechanical ventilation, and/or abnormal laboratory values). We reviewed 1,000 randomly selected medical charts to characterize the definition’s accuracy and stability over time compared with a claims-based definition requiring infection and organ dysfunction codes. We compared incidence and mortality trends from 2003–2012 using both methods.

SETTING

Two US academic hospitals.

PATIENTS

Adult inpatients.

RESULTS

The electronic health record–based clinical surveillance definition had stable and high sensitivity over time (77% in 2003–2009 vs 80% in 2012, P=.58) whereas the sensitivity of claims increased (52% in 2003–2009 vs 67% in 2012, P=.02). Positive predictive values for claims and clinical surveillance definitions were comparable (55% vs 53%, P=.65) and stable over time. From 2003 to 2012, severe sepsis incidence imputed from claims rose by 72% (95% CI, 57%–88%) and absolute mortality declined by 5.4% (95% CI, 4.6%–6.7%). In contrast, incidence using the clinical surveillance definition increased by 7.7% (95% CI, −1.1% to 17%) and mortality declined by 1.7% (95% CI, 1.1%–2.3%).

CONCLUSIONS

Sepsis surveillance using clinical data is more sensitive and more stable over time compared with claims and can be done electronically. This may enable more reliable estimates of sepsis burden and trends.

Infect. Control Hosp. Epidemiol. 2016;37(2):163–171

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A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

Objective Sepsis Surveillance Using Electronic Clinical Data

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A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

Objective Sepsis Surveillance Using Electronic Clinical Data