Multi-soliton dynamics of anti-self-dual gauge fields

We study dynamics of multi-soliton solutions of anti-self-dual Yang-Mills equations for G = GL(2, ℂ) in four-dimensional spaces. The one-soliton solution can be interpreted as a codimension-one soliton in four-dimensional spaces because the principal peak of action density localizes on a three-dimensional hyperplane. We call it the soliton wall. We prove that in the asymptotic region, the n-soliton solution possesses n isolated localized lumps of action density, and interpret it as n intersecting soliton walls. More precisely, each action density lump is essentially the same as a soliton wall because it preserves its shape and “velocity” except for a position shift of principal peak in the scattering process. The position shift results from the nonlinear interactions of the multi-solitons and is called the phase shift. We calculate the phase shift factors explicitly and find that the action densities can be real-valued in three kind of signatures. Finally, we show that the gauge group can be G = SU(2) in the Ultrahyperbolic space 𝕌 (the split signature (+, +, −, −)). This implies that the intersecting soliton walls could be realized in all region in N=2 string theories. It is remarkable that quasideterminants dramatically simplify the calculations and proofs.

Effects of Electric Pulse Modification with Different Technique Parameters on the Liquid Structure of Pure Aluminum

The modification of liquid metal by electric pulse (EP, EPM) is a novel method for grain refinement. In this study, based on the reported structural heredity of EP-modified liquid aluminium, the structure tests of EP-modified liquid aluminium with different technique parameters were conducted by using high temperature X-ray diffractometer. The results show that the quantitative structure changes of EP-modified liquid aluminium have a close relationship with the modifying time and modifying temperature. The decrease of modifying time could result in an obvious weaker principal peak in structure factor curve compared with the optimal EP technique parameters, but a slight increase of coordination number (Ns), correlation radius (rc) and average atom number per cluster (Nat) is still observed under this condition. These facts indicate that the EP-modified liquid aluminum could gain an increasing order degree, and thus have an advantage during the formation of a stable nucleus, eventually leading to a grain-refining solidification structure. On the other hand, the structure factor curve of EP-modified liquid aluminum at the high modifying temperature of 850°C tends to be overlapped with that of the unmodified during the principal peak range. In this case, the competition result between the EP strengthening effect and the destruction of superheating would determine the final structure of EP-modified liquid aluminum.

β Cas: Simultaneous Stromgren Photometry

The observations were carried out in August, 1983 using the 75 cm telescope at Sierra Nevada Observatory in Spain. This telescope is equipped with a six-channel uvbyβ photometer for simultaneous measurements in uvby and the narrow and wide Hβ channels, respectively. The observations were collected during four nights in the four uvby filters and one more night was devoted to measuring β Cas in the n and w bands of the Hβ Crawford system. In all cases, a calibrated neutral filter was used for β Cas. The comparison star was Cl=HR 9085 (V=5.m55, F0III).The analysis of frequencies was carried out using the Discrete Fourier Transform method, as described in López de Coca et al. (1984), to the observational uvby points collected. The periodograms showed a principal peak at v=9.91 cd−l. After prewhitening for this frequency, the resulting periodograms did not show any trace of another peak, suggesting the monoperiodic nature of this star, in agreement with Antonello et al. (1986). Under the assumption of monoperiodicity of β Cas, the classical O-C method can easily be applied. Nine times of maxima were obtained for the four nights by using the method described in Rodriguez et al. (1990), where each light maximum has been derived as an average over the four uvby bands. When the O-C method was applied, the least squares fit converged to a linear ephemeris with the following elements: To=2445568.d5097 (±0.d0007) and Po=0.d10091 (±0.d00002). This linear ephemeris satisfactorily reproduces our data of β Cas.

I.—Excavations at Sparta, 1909: §6.—The Menelaion

A little to the south of Sparta and opposite the hamlet of Psychiko, at the point where the Magoula river runs into the Eurotas, the hills on the left or eastern bank of the latter approach quite close to the river. These hills, which stand high above the plain, have long been identified with those on which, according to Polybius, the Menelaion stood, and as the site of Therapnai. In fact the statements of this author and of Livy make it practically certain that these are the hills in question. They were first explored by Ross in 1833, and he claimed as the shrine of Helen and Menelaos the building that he began to excavate on the principal peak close to the modern chapel of Hagios Elias, and directly above the Eurotas. No other Greek building has yet been found on these heights, so that we may for the present assume with considerable probability that this one, discovered by Ross, is the shrine of Helen and Menelaos mentioned by Herodotus and Pausanias.