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Three types of Cantor sets are studied.For any integer $m\ge 4$, we show that every real number in $[0,k]$ is the sum of at most $k$ $m$-th powers of elements in the Cantor ternary set $C$ for some positive integer $k$, and the smallest such $k$ is $2^m$.Moreover, we generalize this result to middle-$\frac 1\alpha$ Cantor set for $1

Background A recent community-based disease management (CBDM) pilot study reported a 20.5% prevalence of hypertension and a 0.5 and 3.6% prevalence of stroke and coronary heart disease (CHD), respectively, in an elderly population (mean age 65 years) in the Xin Jiang autonomous region of China. The CBDM was initiated in 2013 as an essential public health service; however, the potential long-term impact of CBDM on cardiovascular (CV: CHD and stroke) events is unknown. The objective of the study was to understand the long-term impact of CBDM interventions on CV risk factors using disease-model simulation based on a single-arm experimental study. Methods A discrete event simulation was developed to evaluate the impact of CBDM on the long-term CV risk among patients with hypertension, in China’s Xin Jiang autonomous region. The model generated pairs of identical patients; one receives CBDM and one does not (control group). Their clinical courses were simulated based on time to CV events (CHD and strokes), which are estimated using published risk equations. The impact of CBDM was incorporated as improvement in systolic blood pressure (SBP) based on observations from the CBDM study. The simulation estimated the number of CV events over patients’ lifetimes. Results During a 2-year follow up, the CBDM led to an average reduction of 8.73 mmHg in SBP from baseline, and a 42% reduction in smoking. The discrete event simulation showed that, in the control group, the model estimated incidence rates of 276, 1789, and 616 per 100,000 individuals for lifetime CHD, stroke, and CV-related death, respectively. The impact of CBDM on SBP translated into reductions of 8, 28, and 23% in CHD, stroke, and CV-related deaths, respectively. Taking into account CBDM’s reduction of both SBP and smoking, deaths from CHD, stroke, and CV-related deaths were reduced by 12, 30, and 26%, respectively. Conclusions The implementation of CBDM in China’s Xinjiang autonomous region is expected to significantly reduce incidences of CHD, strokes, and CV-related deaths.

A battery of logic gates, “YES”, “AND” and “OR”, are constructed using magnetic beads (MBs) modified by DNA which consists of a substrate strand (S) and a signal strand on which the logic operates. Inputs stemming from micro-RNA (which represent three cancer biomarkers) take the place of signal DNA. The released signal strand self-assembles into the hemin-G-quadruplex complex (DNAzyme) that catalyzes a blue-green dye (ABTS+) from the precursor ABTS. This dye (quantified at a wavelength of 414 nm) represents the output signal for the various logic gates. The method allows quantitative detection of microRNA of three kinds of logic gates in the range of 5 nM–500 nM with detection limits of 3.8 nM, 4.9 nM, 5.4 nM. Boolean logic circuitry is also achieved following the principles of multilevel strand displacement. Based on strand displacement and magnetic separation, this work demonstrates the possibility of designing a logic system using micro-RNA in live cell lysate as inputs, and its potential application in DNA computation and cancer diagnosis.