A characterization of nested canalyzing functions with maximum average sensitivity
Nested canalyzing functions (NCFs) are a class of Boolean functions which are used to model certain biological phenomena. We derive a complete characterization of NCFs with the largest average sensitivity, expressed in terms of a simple structural property of the NCF. This characterization provides an alternate, but elementary, proof of the tight upper bound on the average sensitivity of any NCF established by Klotz et al. (2013). We also utilize the characterization to derive a closed form expression for the number of NCFs that have the largest average sensitivity.
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NSF-PAR ID:
10067739
Journal Name:
Discrete applied mathematics
ISSN:
1872-6771
This paper aims to quantify how the lowest halo mass that can be detected with galaxy-galaxy strong gravitational lensing depends on the quality of the observations and the characteristics of the observed lens systems. Using simulated data, we measure the lowest detectable NFW mass at each location of the lens plane, in the form of detailed sensitivity maps. In summary, we find that: (i) the lowest detectable mass Mlow decreases linearly as the signal-to-noise ratio (SNR) increases and the sensitive area is larger when we decrease the noise; (ii) a moderate increase in angular resolution (0.07″ versus 0.09″) and pixel scale (0.01″ versus 0.04″) improves the sensitivity by on average 0.25 dex in halo mass, with more significant improvement around the most sensitive regions; (iii) the sensitivity to low-mass objects is largest for bright and complex lensed galaxies located inside the caustic curves and lensed into larger Einstein rings (i.e rE ≥ 1.0″). We find that for the sensitive mock images considered in this work, the minimum mass that we can detect at the redshift of the lens lies between 1.5 × 108 and $3\times 10^{9}\, \mathrm{M}_{\odot }$. We derive analytic relations between Mlow, the SNR and resolution and discuss themore »