Binary Nonlinearization of Lax pairs of Kaup-Newell Soliton Hierarchy

Kaup-Newell soliton hierarchy is derived from a kind of Lax pairs different from the original ones. Binary nonlinearization procedure corresponding to the Bargmann symmetry constraint is carried out for those Lax pairs. The proposed Lax pairs together with adjoint Lax pairs are constrained as a hierarchy of commutative, finite dimensional integrable Hamiltonian systems in the Liouville sense, which also provides us with new examples of finite dimensional integrable Hamiltonian systems. A sort of involutive solutions to the Kaup-Newell hierarchy are exhibited through the obtained finite dimensional integrable systems and the general involutive system engendered by binary nonlinearization is reduced to a specific involutive system generated by mono-nonlinearization.Comment: 15 pages, plain+ams tex, to be published in Il Nuovo Cimento

The Quasar Pair Q 1634+267 A, B and the Binary QSO vs. Dark Lens Hypotheses

Deep HST/NICMOS H (F160W) band observations of the z=1.96 quasar pair Q 1634+267A,B reveal no signs of a lens galaxy to a 1 sigma threshold of approximately 22.5 mag. The minimum luminosity for a normal lens galaxy would be a 6L_* galaxy at z > 0.5, which is 650 times greater than our detection threshold. Our observation constrains the infrared mass-to-light ratio of any putative, early-type, lens galaxy to (M/L)_H > 690h_65 (1200h_65) for Omega_0=0.1 (1.0) and H_0=65h_65 km/s/Mpc. We would expect to detect a galaxy somewhere in the field because of the very strong Mg II absorption lines at z=1.1262 in the Q 1634+267 A spectrum, but the HST H-band, I-band (F785LP) and V-band (F555W) images require that any associated galaxy be very under-luminous less than 0.1 L^*_H (1.0 L^*_I) if it lies within less than 40 h^{-1} (100 h^{-1}) kpc from Q 1634+267 A,B. While the large image separation (3.86 arcsec) and the lack of a lens galaxy strongly favor interpreting Q 1634+267A,B as a binary quasar system, the spectral similarity remains a puzzle. We estimate that at most 0.06% of randomly selected quasar pairs would have spectra as similar to each other as the spectra of Q 1634+267 A and B. Moreover, spectral similarities observed for the 14 quasar pairs are significantly greater than would be expected for an equivalent sample of randomly selected field quasars. Depending on how strictly we define similarity, we estimate that only 0.01--3% of randomly drawn samples of 14 quasar pairs would have as many similar pairs as the observational sample.Comment: 24 pages, including 4 figures, LaTex, ApJ accepted, comments from the editor included, minor editorial change

On the existence of complete disjoint NP-pairs

Disjoint NP-pairs are an interesting model of computation with important applications in cryptography and proof complexity. The question whether there exists a complete disjoint NP-pair was posed by Razborov in 1994 and is one of the most important problems in the field. In this paper we prove that there exists a many-one hard disjoint NP-pair which is computed with access to a very weak oracle (a tally NP-oracle). In addition, we exhibit candidates for complete NP-pairs and apply our results to a recent line of research on the construction of hard tautologies from pseudorandom generators