This is problematic in practical terms, since we often wish to compare nestedness of matrices that differ in these basic properties; in fact, cases where we compare empirically derived matrices with identical size and fill are an exception.
To further minimize biases that can occur in our comparison between species, we carried out the following: Overlap of tissue types: for all three species, we created data matrices with identical composition of tissues within each set.
The eigenvectors matrix in this case will also form a block diagonal matrix with identical blocks whose first rows are the basis functions of the individual taps.
If these coefficients are not identical, we can set all the elements of SI channel matrix ({tilde {boldsymbol {G}}}_{mathrm {s}}) to be the maximum value β max, which represents the worst case, then (tilde {boldsymbol {G}_{s}}) becomes a matrix with identical large-scale fading.
The end result of this analysis is a binary matrix with identical row and column indices consisting of all possible MoLFunCs (Figure 2).
As we calculated in the manuscript, the Fisher information for any lattice (as long as the support of the tuning curve is contained in the fundamental domain and the neuron number M is sufficiently large) is a diagonal matrix with identical entries.
For any two matrices A,B with identical number of rows (columns), [A,B] [A B]) denotes the matrix formed by concatenating the columns (rows) of B with those of A. tr{A} and λmin{A} denote the trace and the minimum eigenvalue of the square matrix A. I k indicates the identity matrix of order k.
Nucleotide sequence matrices simulated with identical or different evolutionary parameters on an identical tree were compared for M = 2 or 5; s = 0.02 or 0.4; α = 0.06, 0.8168 or 200; CMI = 2 or 5; n = 10, 25, 50 or 100; and L = 1000, 5000, 10 000 or 20 000 bp.
Given that the system in Eq. (13) only depends on j through the term (sum_{l in Q} -1)^{j_{l}} chi_{k,l}) in the A matrix, systems with identical Hamming weights have identical (A, B, C, D) matrices if all (chi_{k,l}) are eQ} -1to a single constant χ: sum_{l in Q}(-1)^{j_{l}} chi_{kigl(vert Q vert - 2h(j) bigr)chi.
Each pair is composed of two matrices with dimensions identical to sX1 and sX2, whose elements are extracted from randomly permuted drug and disease expression matrices based on reduced drug and disease expression matrices rX1 and rX2.
Each pair is composed of two matrices with dimensions identical to sX1 and sX2, whose elements are extracted from randomly permuted gene and miRNA expression matrices based on X1 and X2.
