Stratification and enumeration of Boolean functions by canalizing depth

Boolean network models have gained popularity in computational systems biology over the last dozen years. Many of these networks use canalizing Boolean functions, which has led to increased interest in the study of these functions. The canalizing depth of a function describes how many canalizing variables can be recursively picked off, until a non-canalizing function remains. In this paper, we show how every Boolean function has a unique algebraic form involving extended monomial layers and a well-defined core polynomial. This generalizes recent work on the algebraic structure of nested canalizing functions, and it yields a stratification of all Boolean functions by their canalizing depth. As a result, we obtain closed formulas for the number of n-variable Boolean functions with depth k, which simultaneously generalizes enumeration formulas for canalizing, and nested canalizing functions

Multi-scale simulation of capillary pores and gel pores in Portland cement paste

The microstructures of Portland cement paste (water to cement ratio is 0.4, curing time is from 1 day to 28 days) are simulated based on the numerical cement hydration model, HUMOSTRUC3D (van Breugel, 1991; Koenders, 1997; Ye, 2003). The nanostructures of inner and outer C-S-H are simulated by the packing of monosized (5 nm) spheres. The pore structures (capillary pores and gel pores) of Portland cement paste are established by upgrading the simulated nanostructures of C-S-H to the simulated microstructures of Portland cement paste. The pore size distribution of Portland cement paste is simulated by using the image segmentation method (Shapiro and Stockman, 2001) to analyse the simulated pore structures of Portland cement paste. The simulation results indicate that the pore size distribution of the simulated capillary pores of Portland cement paste at the age of 1 day to 28 days is in a good agreement with the pore size distribution determined by scanning electron microscopy (SEM). The pore size distribution of the simulated gel pores of Portland cement paste (interlayer gel pores of outer C-S-H and gel pores of inner C-S-H are not included) is validated by the pore size distribution obtained by mercury intrusion porosimetry (MIP). The pores with pore size of 20 nm to 100 nm occupy very small volume fraction in the simulated Portland cement paste at each curing time (0.69% to 1.38%). This is consistent with the experimental results obtained by nuclear magnetic resonance (NMR)