A Lower Bound on the Constant in the Fourier Min-Entropy/Influence Conjecture

We describe a new construction of Boolean functions. A specific instance of our construction provides a 30-variable Boolean function having min-entropy/influence ratio to be $128/45 \approx 2.8444$ which is presently the highest known value of this ratio that is achieved by any Boolean function. Correspondingly, $128/45$ is also presently the best known lower bound on the universal constant of the Fourier min-entropy/influence conjecture

Influence of a Set of Variables on a Boolean Function

The influence of a variable is an important concept in the analysis of Boolean functions. The more general notion of influence of a set of variables on a Boolean function has four separate definitions in the literature. In the present work, we introduce a new definition of influence of a set of variables which is based on the auto-correlation function and develop its basic theory. Among the new results that we obtain are generalisations of the Poincar\'{e} inequality and the edge expansion property of the influence of a single variable. Further, we obtain new characterisations of resilient and bent functions using the notion of influence. We show that the previous definition of influence due to Fischer et al. (2002) and Blais (2009) is half the value of the auto-correlation based influence that we introduce. Regarding the other prior notions of influence, we make a detailed study of these and show that each of these definitions do not satisfy one or more desirable properties that a notion of influence may be expected to satisfy

Influence of a Set of Variables on a Boolean Function

The influence of a set of variables on a Boolean function has three separate definitions in the literature, the first due to Ben-Or and Linial (1989), the second due to Fischer et al. (2002) and Blais (2009) and the third due to Tal (2017). The goal of the present work is to carry out a comprehensive study of the notion of influence of a set of variables on a Boolean function. To this end, we introduce a definition of this notion using the auto-correlation function. A modification of the definition leads to the notion of pseudo-influence. Somewhat surprisingly, it turns out that the auto-correlation based definition of influence is equivalent to the definition introduced by Fischer et al. (2002) and Blais (2009) and the notion of pseudo-influence is equivalent to the definition of influence considered by Tal (2017). Extensive analysis of influence and pseduo-influence as well as the Ben-Or and Linial notion of influence is carried out and the relations between these notions are established

Separation Results for Boolean Function Classes

We show (almost) separation between certain important classes of Boolean functions. The technique that we use is to show that the total influence of functions in one class is less than the total influence of functions in the other class. In particular, we show (almost) separation of several classes of Boolean functions which have been studied in the coding theory and cryptography from classes which have been studied in combinatorics and complexity theory

Problems faced by ASHA workers for malarial services under NVBDCP: a cross sectional study

Background: About 95% of the country’s population resides in malaria endemic areas. The NVBDCP is multi-faceted public health programme in the country. The NVBDCP became an integral part of the NRHM launched in 2005. In NRHM, ASHA worker should be placed at grass root level. So role of ASHA is critical at grass root level. Problems faced by ASHA, it will affect the performance. So, ASHA workers positioned at village, Population catered by them, education, work experience, problems related to incentive and any other problems should be considered. The primary objective was to know the problems faced by ASHA workers for malarial services under NVBDCP in Jamnagar district.Methods: A cross-sectional study was conducted in rural areas of the district by using multistage sampling.Results: There was vacant post of ASHA in one village of low performing sub-centre. ASHA having work experience of less than one year was concentrated in poor performing sub-centre villages. More than one third ASHA served > 2000 population and more than half of them belonged to low performing sub centres. Around one sixth ASHA workers had difficulty in getting incentive or any problems. Majority of them belonged to low performing sub centres-villages.Conclusions: Main identified problems were some areas were far to reach by them, getting incentive for work and populations catered by them were too high. Most of them belonged to low performing sub centre-villages. All of above findings had affected the performance of ASHA workers.