A ] 2 5 O ct 1 99 4 Matrix Vieta Theorem

ar X iv : h ep - t h / 92 10 08 3 v 1 1 5 O ct 1 99 2 QUASI - PERIODIC SOLUTIONS FOR MATRIX NONLINEAR SCHRÖDINGER EQUATIONS

The Adler-Kostant-Symes theorem yields isospectral hamiltonian flows on the dual˜g + * of a Lie subalgebrã g + of a loop algebrã g. A general approach relating the method of integration of Krichever, Novikov and Dubrovin to such flows is used to obtain finite-gap solutions of matrix Nonlinear Schrödinger Equations in terms of quotients of θ-functions.

O ct 1 99 8 Another combinatorial determinant

= 1!2! · · ·n!b n(n+1)/2 1 (n = 0, 1, 2, . . . ). In fact, we prove both of these theorems at once, by formulating and proving a common generalization. In both theorems, each row of the matrix is obtained from the previous row by applying a certain transformation of power series: in Theorem 1, the transformation is t 7→ t(1 + a1x + a2x 2 + · · · ), while in Theorem 2, the transformation is t 7→...

m at h . Q A ] 2 5 O ct 1 99 8 Spherical 2 - categories and 4 - manifold invariants

/ 99 10 04 5 v 1 1 1 O ct 1 99 9 Critical visibility for N - particle Greenberger - Horne - Zeilinger correlations to violate local realism

A sequence of Bell inequalities for N-particle systems, which involve three settings of each of the local measuring apparatuses, is derived. For GHZ states, quantum mechanics violates these inequalities by factors exponentially growing with N. The threshold visibilities of the multiparticle sinusoidal interference fringes, for which local realistic theories are ruled out, decrease as (2/3) N .

50 v 1 2 4 O ct 1 99 5 Quantum chaos in A = 46 – 50 atomic nuclei

The spectral statistics of low–lying states of several f p shell nuclei are studied with realistic shell–model calculations. For Ca isotopes, we find significant deviations from the predictions of the random–matrix theory which suggest that some spherical nuclei are not as chaotic in nature as the conventional view assumes.