Intermediate convergents and a metric theorem of Khinchin

A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function $f$ defined on the positive integers and a real number $x$, and form the partial sums $s_n$ of $f$ evaluated at the partial quotients $a_1,..., a_n$ in the continued fraction expansion for $x$. Does the sequence $\{s_n/n\}$ have a limit as n\rar\infty? In 1935 A. Y. Khinchin proved that the answer is yes for almost every $x$, provided that the function $f$ does not grow too quickly. In this paper we are going to explore a natural reformulation of this problem in which the function $f$ is defined on the rationals and the partial sums in question are over the intermediate convergents to $x$ with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above

Correlation Between Anti-gp41 Antibodies and Virus Infectivity Decay During Primary HIV-1 Infection

Recent experiments have suggested that the infectivity of simian immunodeficiency virus (SIV) and human immunodeficiency virus type-1 (HIV-1) in plasma decreases over time during primary infection. Because anti-gp41 antibodies are produced early during HIV-1 infection and form antibody-virion complexes, we studied if such early HIV-1 specific antibodies are correlated with the decay in HIV-1 infectivity. Using a viral dynamic model that allows viral infectivity to decay and frequent early viral load data obtained from 6 plasma donors we estimate that HIV-1 infectivity begins to decay after about 2 weeks of infection. The length of this delay is consistent with the time before antibody-virion complexes were detected in the plasma of these donors and is correlated (p = 0.023, r = 0.87) with the time for antibodies to be first detected in plasma. Importantly, we identify that the rate of infectivity decay is significantly correlated with the rate of increase in plasma anti-gp41 IgG concentration (p = 0.046, r = 0.82) and the increase in IgM+IgG anti-gp41 concentration (p = 8.37 × 10−4, r = 0.98). Furthermore, we found that the viral load decay after the peak did not have any significant correlation with the rate of anti-gp41 IgM or IgG increase. These results indicate that early anti-gp41 antibodies may cause viral infectivity decay, but may not contribute significantly to controlling post-peak viral load, likely due to insufficient quantity or affinity. Our findings may be helpful to devise strategies, including antibody-based vaccines, to control acute HIV-1 infection

A philosophical analysis of the evidence-based medicine debate

BACKGROUND: The term "evidence-based medicine" (or EBM) was introduced about ten years ago, and there has been considerable debate about the value of EBM. However, this debate has sometimes been obscured by a lack of conceptual clarity concerning the nature and status of EBM. DISCUSSION: First, we note that EBM proponents have obscured the current debate by defining EBM in an overly broad, indeed almost vacuous, manner; we offer a clearer account of EBM and its relation to the alternative approaches to medicine. Second, while EBM proponents commonly cite the philosophical work of Thomas Kuhn and claim that EBM is a Kuhnian 'paradigm shift,' we argue that such claims are seriously mistaken and unduly polarize the EBM debate. Third, we suggest that it is much more fruitful to understand the relationship between EBM and its alternatives in light of a different philosophical metaphor: W.V. Quine's metaphor of the web of belief. Seen in this way, we argue that EBM is an approach to medical practice that is indeed importantly different from the alternatives. SUMMARY: We can have a more productive debate about the value of EBM by being clearer about the nature of EBM and its relationship to alternative approaches to medicine

Genetic Signatures in the Envelope Glycoproteins of HIV-1 that Associate with Broadly Neutralizing Antibodies

A steady increase in knowledge of the molecular and antigenic structure of the gp120 and gp41 HIV-1 envelope glycoproteins (Env) is yielding important new insights for vaccine design, but it has been difficult to translate this information to an immunogen that elicits broadly neutralizing antibodies. To help bridge this gap, we used phylogenetically corrected statistical methods to identify amino acid signature patterns in Envs derived from people who have made potently neutralizing antibodies, with the hypothesis that these Envs may share common features that would be useful for incorporation in a vaccine immunogen. Before attempting this, essentially as a control, we explored the utility of our computational methods for defining signatures of complex neutralization phenotypes by analyzing Env sequences from 251 clonal viruses that were differentially sensitive to neutralization by the well-characterized gp120-specific monoclonal antibody, b12. We identified ten b12-neutralization signatures, including seven either in the b12-binding surface of gp120 or in the V2 region of gp120 that have been previously shown to impact b12 sensitivity. A simple algorithm based on the b12 signature pattern was predictive of b12 sensitivity/resistance in an additional blinded panel of 57 viruses. Upon obtaining these reassuring outcomes, we went on to apply these same computational methods to define signature patterns in Env from HIV-1 infected individuals who had potent, broadly neutralizing responses. We analyzed a checkerboard-style neutralization dataset with sera from 69 HIV-1-infected individuals tested against a panel of 25 different Envs. Distinct clusters of sera with high and low neutralization potencies were identified. Six signature positions in Env sequences obtained from the 69 samples were found to be strongly associated with either the high or low potency responses. Five sites were in the CD4-induced coreceptor binding site of gp120, suggesting an important role for this region in the elicitation of broadly neutralizing antibody responses against HIV-1

Effect of a Nutrition Supplement and Physical Activity Program on Pneumonia and Walking Capacity in Chilean Older People: A Factorial Cluster Randomized Trial

Alan Dangour and colleagues report results from the CENEX (Cost-effectiveness Evaluation of a Nutritional supplement and EXercise program for older people) trial, which evaluates a nutritional and exercise program aiming to prevent pneumonia and physical decline in Chilean people