Coordinate sum and difference sets of dd-dimensional modular hyperbolas

Many problems in additive number theory, such as Fermat's last theorem and the twin prime conjecture, can be understood by examining sums or differences of a set with itself. A finite set $A \subset \mathbb{Z}$ is considered sum-dominant if $|A+A|>|A-A|$. If we consider all subsets of ${0, 1, ..., n-1}$, as $n\to\infty$ it is natural to expect that almost all subsets should be difference-dominant, as addition is commutative but subtraction is not; however, Martin and O'Bryant in 2007 proved that a positive percentage are sum-dominant as $n\to\infty$. This motivates the study of "coordinate sum dominance". Given $V \subset (\Z/n\Z)^2$, we call $S:={x+y: (x,y) \in V}$ a coordinate sumset and $D:=\{x-y: (x,y) \in V\}$ a coordinate difference set, and we say $V$ is coordinate sum dominant if $|S|>|D|$. An arithmetically interesting choice of $V$ is $\bar{H}_2(a;n)$, which is the reduction modulo $n$ of the modular hyperbola $H_2(a;n) := {(x,y): xy \equiv a \bmod n, 1 \le x,y < n}$. In 2009, Eichhorn, Khan, Stein, and Yankov determined the sizes of $S$ and $D$ for $V=\bar{H}_2(1;n)$ and investigated conditions for coordinate sum dominance. We extend their results to reduced $d$-dimensional modular hyperbolas $\bar{H}_d(a;n)$ with $a$ coprime to $n$.Comment: Version 1.0, 14 pages, 2 figure

Distinguishing schemes and tasks in children's development of multiplicative reasoning

We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children’s thinking, and learning situations must be organized to (a) build on children’s available schemes, (b) promote the next scheme in the sequence, and (c) link to intended mathematical concepts

Simplified HIV Testing and Treatment in China: Analysis of Mortality Rates Before and After a Structural Intervention.

BackgroundMultistage stepwise HIV testing and treatment initiation procedures can result in lost opportunities to provide timely antiretroviral therapy (ART). Incomplete patient engagement along the continuum of HIV care translates into high levels of preventable mortality. We aimed to evaluate the ability of a simplified test and treat structural intervention to reduce mortality.Methods and findingsIn the "pre-intervention 2010" (from January 2010 to December 2010) and "pre-intervention 2011" (from January 2011 to December 2011) phases, patients who screened HIV-positive at health care facilities in Zhongshan and Pubei counties in Guangxi, China, followed the standard-of-care process. In the "post-intervention 2012" (from July 2012 to June 2013) and "post-intervention 2013" (from July 2013 to June 2014) phases, patients who screened HIV-positive at the same facilities were offered a simplified test and treat intervention, i.e., concurrent HIV confirmatory and CD4 testing and immediate initiation of ART, irrespective of CD4 count. Participants were followed for 6-18 mo until the end of their study phase period. Mortality rates in the pre-intervention and post-intervention phases were compared for all HIV cases and for treatment-eligible HIV cases. A total of 1,034 HIV-positive participants (281 and 339 in the two pre-intervention phases respectively, and 215 and 199 in the two post-intervention phases respectively) were enrolled. Following the structural intervention, receipt of baseline CD4 testing within 30 d of HIV confirmation increased from 67%/61% (pre-intervention 2010/pre-intervention 2011) to 98%/97% (post-intervention 2012/post-intervention 2013) (all p &lt; 0.001 [i.e., for all comparisons between a pre- and post-intervention phase]), and the time from HIV confirmation to ART initiation decreased from 53 d (interquartile range [IQR] 27-141)/43 d (IQR 15-113) to 5 d (IQR 2-12)/5 d (IQR 2-13) (all p &lt; 0.001). Initiation of ART increased from 27%/49% to 91%/89% among all cases (all p &lt; 0.001) and from 39%/62% to 94%/90% among individuals with CD4 count ≤ 350 cells/mm3 or AIDS (all p &lt; 0.001). Mortality decreased from 27%/27% to 10%/10% for all cases (all p &lt; 0.001) and from 40%/35% to 13%/13% for cases with CD4 count ≤ 350 cells/mm3 or AIDS (all p &lt; 0.001). The simplified test and treat intervention was significantly associated with decreased mortality rates compared to pre-intervention 2011 (adjusted hazard ratio [aHR] 0.385 [95% CI 0.239-0.620] and 0.380 [95% CI 0.233-0.618] for the two post-intervention phases, respectively, for all newly diagnosed HIV cases [both p &lt; 0.001], and aHR 0.369 [95% CI 0.226-0.603] and 0.361 [95% CI 0.221-0.590] for newly diagnosed treatment-eligible HIV cases [both p &lt; 0.001]). The unit cost of an additional patient receiving ART attributable to the intervention was US$83.80. The unit cost of a death prevented because of the intervention was US$234.52.ConclusionsOur results demonstrate that the simplified HIV test and treat intervention promoted successful engagement in care and was associated with a 62% reduction in mortality. Our findings support the implementation of integrated HIV testing and immediate access to ART irrespective of CD4 count, in order to optimize the impact of ART

Distinción de esquemas y tareas en el desarrollo del razonamiento multiplicativo de los niños

We present a synthesis of findings from constructivist teaching experiments regarding six schemes children construct for reasoning multiplicatively and tasks to promote them. We provide a task-generating platform game, depictions of each scheme, and supporting tasks. Tasks must be distinguished from children’s thinking, and learning situations must be organized to (a) build on children’s available schemes, (b) promote the next scheme in the sequence, and (c) link to intended mathematical concepts.Presentamos una síntesis de hallazgos de experimentos de enseñanza constructivistas en relación con seis esquemas que los niños construyen para razonar multiplicativamente y tareas para promoverlos. Proveemos una plataforma de juego generadora de tareas, descripciones de cada esquema y tareas para apoyarlos. Las tareas deben distinguirse del pensamiento de los niños, y las situaciones de aprendizaje deben organizarse para que (a) se basen en los esquemas que los niños tienen disponibles, (b) promuevan el siguiente esquema en la secuencia y (c) se relacionen con los conceptos matemáticos pretendidos.This research was supported by the US National Science Foundation under grant DRL 0822296, which funded the activities of the Nurturing Multiplicative Reasoning in Students with Learning Disabilities in a Computerized Conceptual-Modeling Environment (NMRSD) project. The opinions expressed in this article do not necessarily reflect the views of the Foundation

Some Additive Combinatorics Problems in Matrix Rings

We study the distribution of singular and unimodular matrices in sumsets in matrix rings over finite fields. We apply these results to estimate the largest prime divisor of the determinants in sumsets in matrix rings over the integers

An increase in albuminuria is associated with a higher incidence of malignancies

Background. A single albuminuria measurement is reported to be an independent predictor of cancer risk. Whether change in albuminuria is also independently associated with cancer is not known. Methods. We included 64 303 subjects of the Stockholm CREAtinine Measurements( SCREAM) project without a history of cancer and with at least two urine albumin-creatinine ratio( ACR) tests up to 2 years apart. Albuminuria changes were quantified by the fold-change in ACR over 2 years, and stratified into the absence of clinically elevated albuminuria( i.e. never) , albuminuria that remained constant, and albuminuria that increased or decreased. The primary outcome was overall cancer incidence. Secondary outcomes were site-specific cancer incidences. Results. During a median follow-up of 3.7( interquartile range 3.6-3.7) years, 5126 subjects developed de novo cancer. After multivariable adjustment including baseline estimated glomerular filtration rate and baseline ACR, subjects with increasing ACR over 2 years had a 19%( hazard ratio 1.19; 95% confidence interval 1.08-1.31) higher risk of overall cancer compared with those who never had clinically elevated ACR. No association with cancer risk was seen in the groups with decreasing or constant ACR. Regarding site-specific cancer risks, subjects with increasing ACR or constant ACR had a higher risk of developing urinary tract and lung cancer. No other associations between 2-year ACR changes and site-specific cancers were found. Conclusions. Increases in albuminuria over a 2-year period are associated with a higher risk of developing overall, urinary tract and lung cancer, independent of baseline kidney function and albuminuria. These data add important weight to the link that exists between albuminuria and cancer incidence.</p

An increase in albuminuria is associated with a higher incidence of malignancies

Background. A single albuminuria measurement is reported to be an independent predictor of cancer risk. Whether change in albuminuria is also independently associated with cancer is not known. Methods. We included 64 303 subjects of the Stockholm CREAtinine Measurements( SCREAM) project without a history of cancer and with at least two urine albumin-creatinine ratio( ACR) tests up to 2 years apart. Albuminuria changes were quantified by the fold-change in ACR over 2 years, and stratified into the absence of clinically elevated albuminuria( i.e. never) , albuminuria that remained constant, and albuminuria that increased or decreased. The primary outcome was overall cancer incidence. Secondary outcomes were site-specific cancer incidences. Results. During a median follow-up of 3.7( interquartile range 3.6-3.7) years, 5126 subjects developed de novo cancer. After multivariable adjustment including baseline estimated glomerular filtration rate and baseline ACR, subjects with increasing ACR over 2 years had a 19%( hazard ratio 1.19; 95% confidence interval 1.08-1.31) higher risk of overall cancer compared with those who never had clinically elevated ACR. No association with cancer risk was seen in the groups with decreasing or constant ACR. Regarding site-specific cancer risks, subjects with increasing ACR or constant ACR had a higher risk of developing urinary tract and lung cancer. No other associations between 2-year ACR changes and site-specific cancers were found. Conclusions. Increases in albuminuria over a 2-year period are associated with a higher risk of developing overall, urinary tract and lung cancer, independent of baseline kidney function and albuminuria. These data add important weight to the link that exists between albuminuria and cancer incidence.</p