In this work, we give a rigorous explicit formula for the Lyapunov exponent for some binary infinite products of random 2 × 2 real matrices. All these products are constructed using only two types of matrices, A and B, which are chosen according to a stochastic process. The matrix A is singular, namely its determinant is zero. This formula is derived by using a particular decomposition for the ...

The uniform distribution on matrices with specified row and column sums is often a natural choice of null model when testing for structure in two-way tables (binary or nonnegative integer). Due to the difficulty of sampling from this distribution, many approximate methods have been developed. We will show that by exploiting certain symmetries, exact sampling and counting is in fact possible in ...

We show how to construct the exact factorized S-matrices of 1+1 dimensional quantum field theories whose symmetry charges generate a quantum affine algebra. Quantum affine Toda theories are examples of such theories. We take into account that the Lorentz spins of the symmetry charges determine the gradation of the quantum affine algebras. This gives the S-matrices a non-rigid pole structure. It...

These supplementary notes are provided in an educational spirit. Section 2 shows how the cross-product formula for computing columns of the Jacobian emerges from direct calculus. Section 5 extends to arbitrary Euler angles. Our work is not limited to homogeneous coordinates or Euler angles. However, these representations are well known and chosen merely to present to a wider audience. In fact o...

We analyze properties of general quark mass matrices. The up and down part quark mass matrices are written in terms of six dimensionless parameters and six quark masses. It is shown that two of the former six dimensionless parameters can be chosen to be any value. Once values for these two parameters are chosen, Kobayashi-Maskawa matrix is written in terms of the remaining four parameters. Our ...