When does the Bombieri-Vinogradov Theorem hold for a given multiplicative function?

Let $f$ and $g$ be $1$-bounded multiplicative functions for which $f*g=1_{.=1}$. The Bombieri-Vinogradov Theorem holds for both $f$ and $g$ if and only if the Siegel-Walfisz criterion holds for both $f$ and $g$, and the Bombieri-Vinogradov Theorem holds for $f$ restricted to the primes.Comment: 19 page

Bombieri-Vinogradov for multiplicative functions, and beyond the x1/2x^{1/2}-barrier

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for particular examples of multiplicative functions, there has not previously been headway on a general theory; seemingly none of the different proofs of the Bombieri-Vingradov Theorem for primes adapt well to this situation. In this article we find out why such a result has been so elusive, and discover what can be proved along these lines and develop some limitations. For a fixed residue class $a$ we extend such averages out to moduli $\leq x^{\frac {20}{39}-\delta}$.Comment: 54 page

Smooth-supported multiplicative functions in arithmetic progressions beyond the x1/2x^{1/2}-barrier

We show that smooth-supported multiplicative functions $f$ are well-distributed in arithmetic progressions $a_1a_2^{-1} \pmod q$ on average over moduli $q\leq x^{3/5-\varepsilon}$ with $(q,a_1a_2)=1$.Comment: 23 page

A Pulse-Gated, Predictive Neural Circuit

Recent evidence suggests that neural information is encoded in packets and may be flexibly routed from region to region. We have hypothesized that neural circuits are split into sub-circuits where one sub-circuit controls information propagation via pulse gating and a second sub-circuit processes graded information under the control of the first sub-circuit. Using an explicit pulse-gating mechanism, we have been able to show how information may be processed by such pulse-controlled circuits and also how, by allowing the information processing circuit to interact with the gating circuit, decisions can be made. Here, we demonstrate how Hebbian plasticity may be used to supplement our pulse-gated information processing framework by implementing a machine learning algorithm. The resulting neural circuit has a number of structures that are similar to biological neural systems, including a layered structure and information propagation driven by oscillatory gating with a complex frequency spectrum.Comment: This invited paper was presented at the 50th Asilomar Conference on Signals, Systems and Computer

Role of Meal Replacements on Weight Management, Health and Nutrition

Meal replacements are a safe and effective tool for weight loss and weight management and beyond. Recent research and concepts suggest that the category can provide additional metabolic and nutritional benefits not previously recognized. Recent studies indicate that use of protein‐enriched meal replacements helps maintain lean body mass during weight loss, providing additional metabolic benefits in the form of improved insulin sensitivity and reduced inflammation. Depending on the formulation, meal replacements can have a low glycemic index and have a high nutrient density relative to energy density, the latter being an important aspect highlighted in government dietary guidance. While well defined in some markets, there is a need to establish clear regulatory standards in other key markets to ensure a level playing field and proper recognition of the category

A new method of accurate broken rotor bar diagnosis based on modulation signal bispectrum analysis of motor current signals

Motor current signature analysis (MCSA) has been an effective way of monitoring electrical machines for many years. However, inadequate accuracy in diagnosing incipient broken rotor bars (BRB) has motivated many studies into improving this method. In this paper a modulation signal bispectrum (MSB) analysis is applied to motor currents from different broken bar cases and a new MSB based sideband estimator (MSB-SE) and sideband amplitude estimator are introduced for obtaining the amplitude at (1±2s)fs(1±2s)fs (s is the rotor slip and fsfs is the fundamental supply frequency) with high accuracy. As the MSB-SE has a good performance of noise suppression, the new estimator produces more accurate results in predicting the number of BRB, compared with conventional power spectrum analysis. Moreover, the paper has also developed an improved model for motor current signals under rotor fault conditions and an effective method to decouple the BRB current which interferes with that of speed oscillations associated with BRB. These provide theoretical supports for the new estimators and clarify the issues in using conventional bispectrum analysis

Study protocol for a randomised controlled trial evaluating the effect of prenatal omega-3 LCPUFA supplementation to reduce the incidence of preterm birth: The ORIP trial

Introduction: Preterm birth accounts for more than 85% of all perinatal complications and deaths. Seventy-five per cent of early preterm births (EPTBs) occur spontaneously and without identifiable risk factors. The need for a broadly applicable, effective strategy for primary prevention is paramount. Secondary outcomes from the docosahexaenoic acid (DHA) to Optimise Mother Infant Outcome trial showed that maternal supplementation until delivery with omega-3 (ω-3) long chain polyunsaturated fatty acid (LCPUFA), predominantly as DHA, resulted in a 50% reduction in the incidence of EPTB and an increase in the incidence of post-term induction or post-term prelabour caesarean section due to extended gestation. We aim to determine the effectiveness of supplementing the maternal diet with ω-3 LCPUFA until 34 weeks’ gestation on the incidence of EPTB. Methods and analysis: This is a multicentre, parallel group, randomised, blinded and controlled trial. Women less than 20 weeks’ gestation with a singleton or multiple pregnancy and able to give informed consent are eligible to participate. Women will be randomised to receive high DHA fish oil capsules or control capsules without DHA. Capsules will be taken from enrolment until 34 weeks’ gestation. The primary outcome is the incidence of EPTB, defined as delivery before 34 completed weeks’ gestation. Key secondary outcomes include length of gestation, incidence of post-term induction or prelabour caesarean section and spontaneous EPTB. The target sample size is 5540 women (2770 per group), which will provide 85% power to detect an absolute reduction in the incidence of preterm birth of 1.16% (from 2.45% to 1.29%) between the DHA and control group (two sided α=0.05). The primary analysis will be based on the intention-to-treat principle. Trial registration number: Australia and New Zealand Clinical Trial Registry Number: 2613001142729; Pre-results