Exact solutions and geometric phase factor of time-dependent three-generator quantum systems

There exist a number of typical and interesting systems and/or models, which possess three-generator Lie-algebraic structure, in atomic physics, quantum optics, nuclear physics and laser physics. The well-known fact that all simple 3-generator algebras are either isomorphic to the algebra sl (2, C) or to one of its real forms enables us to treat these time-dependent quantum systems in a unified way. By making use of both the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, the present paper obtains exact solutions of the time-dependent Schr?dinger equations governing various three-generator Lie-algebraic quantum systems. For some quantum systems whose time-dependent Hamiltonians have no quasialgebraic structures, it is shown that the exact solutions can also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator (i.e., the time-independent invariant that commutes with the time-dependent Hamiltonian). The topological property of geometric phase factors and its adiabatic limit in time-dependent systems is briefly discussed. Received 6 July 2002 / Received in final form 21 October 2002 Published online 11 February 2003     

Rational Points on the Sphere

Using only basic tools from the theory of modular forms, the rational points of bounded height on the sphere are counted and shown to be uniformly distributed. The more difficult case of points with a given height is also treated.     

Finite Hurwitz braid group actions for Artin groups

We prove that (with two possible exceptions) the Hurwitz braid group action on the sequence of standard generators of an irreducible Artin group has a finite orbit if and only if the Artin group is of finite type (i.e., the corresponding Coxeter group is finite).     

Natural Operators Lifting Vector Fields to Bundles of Weil Contact Elements

Let A be a Weil algebra. The bijection between all natural operators lifting vector fields from m-manifolds to the bundle functor K ^A of Weil contact elements and the subalgebra of fixed elements SA of the Weil algebra A is determined and the bijection between all natural affinors on K ^A and SA is deduced. Furthermore, the rigidity of the functor K ^A is proved. Requisite results about the structure of SA are obtained by a purely algebraic approach, namely the existence of nontrivial SA is discussed.     

A new algorithm for fixed design regression and denoising

In this paper, we present a new algorithm to estimate a regression function in a fixed design regression model, by piecewise (standard and trigonometric) polynomials computed with an automatic choice of the knots of the subdivision and of the degrees of the polynomials on each sub-interval. First we give the theoretical background underlying the method: the theoretical performances of our penalized least-squares estimator are based on non-asymptotic evaluations of a mean-square type risk. Then we explain how the algorithm is built and possibly accelerated (to face the case when the number of observations is great), how the penalty term is chosen and why it contains some constants requiring an empirical calibration. Lastly, a comparison with some well-known or recent wavelet methods is made: this brings out that our algorithm behaves in a very competitive way in term of denoising and of compression.     

Wave processes in media with hysteretic nonlinearity: Part 2

Nonlinear processes caused by the propagation of low-frequency and high-frequency acoustic pulses in an unbounded medium and the propagation of continuous waves in a ring resonator are theoretically studied on the basis of two hysteretic equations of state for media with imperfect elasticity. The profiles and parameters of pulses, the resonance curve and the Q factor of the resonator, and the ratio of the nonlinear resonance frequency shift to the nonlinear damping decrement are determined. For nonlinear wave processes in such media, the distinctive features that allow one to choose an appropriate hysteretic equation of state for analytically describing the experimental data are revealed.     

Recent developments in algebraic combinatorics

We survey three recent developments in algebraic combinatorics. The first is the theory of cluster algebras and the Laurent phenomenon of Sergey Fomin and Andrei Zelevinsky. The second is the construction of toric Schur functions and their application to computing three-point Gromov-Witten invariants, by Alexander Postnikov. The third development is the construction of intersection cohomology for nonrational fans by Paul Bressler and Valery Lunts and its application by Kalle Karu to the torich-vector of a nonrational polytope. We also briefly discuss the “half hard Lefschetz theorem” of Ed Swartz and its application to matroid complexes. Partially supported by NSF grant #DMS-9988459.     

Bounded imaginary powers of differential operators on manifolds with conical singularities

 We study the minimal and maximal closed extension of a differential operator A on a manifold B with conical singularities, when A acts as an unbounded operator on weighted L _p -spaces over B, 1<p<∞. Under suitable ellipticity assumptions we can define a family of complex powers A ^z , zℂ. We also obtain sufficient information on the resolvent of A to show the boundedness of the purely imaginary powers. Examples concern unique solvability and maximal regularity for the solution of the Cauchy problem for the Laplacian on conical manifolds as well as certain quasilinear diffusion equations. Received: 12 June 2001; in final form: 3 June 2002 / Published online: 1 April 2003 Mathematics Subject Classification (2000): 35J70, 47A10, 35K57     

Reaction of aroyl- and hetaroyltrifluoroacetones with acylhydrazines: Regioselectivity and tautomerism of the condensation products

Acylhydrazines react with 1,3-diketones of the general formula CF₃COCH₂COR (where R is an aryl or hetaryl group) at both carbonyl groups. The reaction at the trifluoroacetyl group is favored by donor substituents in the aromatic ring of the 1,3-diketone or by the 2-furyl and, especially, 2-thienyl group as a hetaryl substituent, as well as by elevated temperature. The condensation products at the carbonyl group contiguous to the aryl or hetaryl ring have the structure of 5-hydroxy-4,5-dihydropyrazoles and do not undergo tautomeric transformations in solution. The condensation products at the trifluoroacetyl group have either 5-hydroxy-4,5-dihydropyrazole or hydrazone structure and give rise to ring-chain tautomeric equilibrium in solution. Electron-withdrawing substituents in the aromatic ring of the 1,3-dicarbonyl fragment and CDCl₃ as solvent favor formation of the cyclic tautomer. The state of the tautomeric equilibrium is weakly sensitive to structural variations in the hydrazine component.     

4-arylamino-2-(2-acetoxyethyl)amino-6-methylpyrimidines: Synthesis,deacetylation, and biological activity

The reaction of 2-(2-acetoxyethyl)amino-4-chloro-6-methylpyrimidine with aromatic amines leads to a series of 4-arylamino-2-(2-acetoxyethyl)amino-6-methylpyrimidines. Deacetylation of these compounds proceeds in both acidic and basic media. Most of the arylaminopyrimides obtained exhibit a pronounced antituberculous effect.