An octonion algebra originating in combinatorics

C.H. Yang discovered a polynomial version of the classical Lagrange identity expressing the product of two sums of four squares as another sum of four squares. He used it to give short proofs of some important theorems on composition of delta-codes (now known as T-sequences). We investigate the possible new versions of his polynomial Lagrange identity. Our main result shows that all such identities are equivalent to each other.Comment: 11 pages, A simpler proof of the main theorem, due to Alberto Elduque, is inserted. The paper will appear in the Proc. Amer. Math. So

Tunnelling Effect and Hawking Radiation from a Vaidya Black Hole

In this paper, we extend Parikh' work to the non-stationary black hole. As an example of the non-stationary black hole, we study the tunnelling effect and Hawking radiation from a Vaidya black hole whose Bondi mass is identical to its mass parameter. We view Hawking radiation as a tunnelling process across the event horizon and calculate the tunnelling probability. We find that the result is different from Parikh's work because $\frac{dr_{H}}{dv}$ is the function of Bondi mass m(v)

Comparison of wind turbine tower failure modes under seismic and wind loads

This paper studies the structural responses and failure modes of a 1.5-MW horizontal-axis wind turbine under strong ground motions and wind loading. Ground motions were selected and scaled to match the two design response spectra given by the seismic code, and wind loads were generated considering tropical cyclone scenarios. Nonlinear dynamic time-history analyses were conducted and structural performances under wind loads as well as short- and long-period ground motions compared. The results show that under strong wind loads the collapse of the wind turbine tower is driven by the formation of a plastic hinge at the lower section of the tower. This area is also critical when the tower is subject to most ground motions. However, some short-period earthquakes trigger the collapse of the middle and upper parts of the tower due to the increased contribution of high-order vibration modes. Although long-period ground motions tend to result in greater structural responses, short-period earthquakes may cause brittle failure modes in which the full plastic hinge develops quickly in regions of the tower with only a moderate energy dissipation capacity. Based on these results, recommendations for future turbine designs are proposed

Generation of entangled photons by trapped ions in microcavities under a magnetic field gradient

We propose a potential scheme to generate entangled photons by manipulating trapped ions embedded in two-mode microcavities, respectively, assisted by a magnetic field gradient. By means of the spin-spin coupling due to the magnetic field gradient and the Coulomb repulsion between the ions, we show how to efficiently generate entangled photons by detecting the internal states of the trapped ions. We emphasize that our scheme is advantageous to create complete sets of entangled multi-photon states. The requirement and the experimental feasibility of our proposal are discussed in detail.Comment: 2 Tables, 2 Figures, To appear in Phys. Rev.