For this matrix, the minimal standard deviation was equal to 0.019792.
On the basis of the form of the electron phonon coupling matrix, the minimal Hamiltonian must necessarily consist of two carbon atoms and two hydrogens atoms.
We reduce this matrix to the minimal payoff matrix by taking into account that the nature of fixed points of evolutionary dynamics (though not their location) remains unaffected under a projective transformation of the relative cells frequencies (Hofbauer and Sigmund, 1998).
It is interesting that the number of all distinct XC matrices with the minimal cardinality (a.k.a. Latin squares) grows very fast with alphabet size [17] as depicted in Table 1.
Then we let ω increase from 1 to 13, combine the binary minimal parity matrices for the minimal polynomials mI(1)(x)…mI x), in order to form H ω by Equation 52, and calculate the LCs of H ω × C r = 0(1 ≤ ω ≤ q) by Equation 48.
where λ ω (1 ≤ ω ≤ q) denotes the index of L m λ ω x in L. Step 3: Let ω increase from 1 to q, combine the binary minimal parity matrices for the minimal polynomials m λ 1 x … m λ ω x, in order to form H ω as follows: H ω = H b min ( m λ 1 x ) H b min ( m λ 2 x ) ⋮ H b min ( m λ ω x ), 1 ≤ ω ≤ q (52).
As before, if we consider a horizontal line that is strictly between the horizontal line that passes through the nodes of rank j and the horizontal line that passes through the nodes of rank (j +1) then these matrices determine the minimal and maximal possible numbers of intersections of this line with branches of subtree i.
Different problems arise such as: and computing the largest C1P sub-matrix; and computing the permutation of columns (and rows) that produce the matrix closest to a C1P matrix; computing the minimal number of elements which can be flipped to obtain a C1P matrix.
Throughout this paper, we denote by the transpose of matrix ; and the minimal and maximal eigenvalues of a real symmetric matrix, respectively; the symmetrical and positive (semipositive, negative, seminegative) definite matrix, and the Euclidian norm of the square matrix.
The resulting path sets are stored in a path matrix, and the minimal weight between each node pair is stored in a distance matrix.
