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From an injective map between the mass of the meson 16-plet and the eigenvalue of the right multiplication in the Cayley-Dickson algebra, we obtain the mass formula as 2 m_{D_s} = m_{eta_c} + m_{eta'}, which is in excellent agreement with experiment., Comment: 9 pages, 1 figure, to be published in Nuovo Cimento B

From an injective map between the mass of the meson 16-plet and the eigenvalue of the right multiplication in the Cayley-Dickson algebra, we obtain the mass formula as 2 m_{D_s} = m_{eta_c} + m_{eta'}, which is in excellent agreement with experiment., Comment: 9 pages, 1 figure, to be published in Nuovo Cimento B

We calculate the eigenvalue \rho of the multiplication mapping R on the Cayley-Dickson algebra A_n. If the element in A_n is composed of a pair of alternative elements in A_{n-1}, half the eigenvectors of R in A_n are still eigenvectors in the subspace which is isomorphic to A_{n-1}. The invariant under the reciprocal transformation A_n \times A_{n} \ni (x,y) -> (-y,x) plays a fundamental role in simplifying the functional form of \rho. If some physical field can be identified with the eigenspace of R, with an injective map from the field to a scalar quantity (such as a mass) m, then there is a one-to-one map \pi: m \mapsto \rho. As an example, the electro-weak gauge field can be regarded as the eigenspace of R, where \pi implies that the W-boson mass is less than the Z-boson mass, as in the standard model., Comment: To be published in J. Phys. A: Mathematical and General

We rewrite the Born-Infeld Lagrangian, which is originally given by the determinant of a $4 \times 4$ matrix composed of the metric tensor $g$ and the field strength tensor $F$, using the determinant of a $(4 \cdot 2^n) \times (4 \cdot 2^n)$ matrix $H_{4 \cdot 2^{n}}$. If the elements of $H_{4 \cdot 2^{n}}$ are given by the linear combination of $g$ and $F$, it is found, based on the representation matrix for the multiplication operator of the Cayley-Dickson algebras, that $H_{4 \cdot 2^{n}}$ is distinguished by a single parameter, where distinguished matrices are not similar matrices. We also give a reasonable condition to fix the paramete