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Max-plus algebra is a kind of idempotent semiring over $mathbb{R}_{max}:=mathbb{R}cup{-infty}$ with two operations $oplus := max$ and $otimes := +$.In this paper, we introduce a new model of a walk on one dimensional lattice on $mathbb{Z}$, as an analogue of the quantum walk, over the max-plus algebra and we call it max-plus walk. In the conventional quantum walk, the summation of the $ell^2$-norm of the states over all the positions is a conserved quantity. In contrast, the summation of eigenvalues of state decision matrices is a conserved quantity in the max-plus walk.Moreover, spectral analysis on the total time evolution operator is also given.
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We find a formula to compute the number of the generators, which generate the $n$-filtered space of Hopf algebra of rooted trees, i.e. the number of equivalent classes of rooted trees with weight $n$. Applying Hopf algebra of rooted trees, we show that the analogue of Andruskiewitsch and Schneiders Conjecture is not true. The Hopf algebra of rooted trees and the enveloping algebra of the Lie algebra of rooted trees are two important examples of Hopf algebras. We give their representation and show that they have not any nonzero integrals. We structure their graded Drinfeld doubles and show that they are local quasitriangular Hopf algebras.
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A parametrization of 3x3 unitary matrices is presented. This mathematical approach is inspired on polarization algebra and is formulated through the identification of a set of three orthonormal three-dimensional Jones vectors representing the respective pure polarization states. This approach leads to the representation of a 3x3 unitary matrix as an orthogonal similarity transformation of a particular type of unitary matrix that depends on six independent parameters, while the remaining three parameters correspond to the orthogonal matrix of the said transformation. The results obtained are applied to determine the structure of the second component of the characteristic decomposition of a 3x3 positive semidefinite Hermitian matrix.
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