Liouville's equation and the lattice sinh-Gordon model

The equivalence of the Liouville lattice model and the massless version of the sinh-Gordon lattice model is established. Thus Liouville's equation is included in the class of systems to which the quantum method of the inverse problem is applicable. A Hamiltonian of the quantum lattice model is constructed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 161, pp. 24–44, 1987.     

A remark on Liouville's formula on small time scales

We present a new proof of the Liouville formula for a d-dimensional linear dynamic system xΔ=A(t)x on a time scale T, where T is in a sense small. Our proof demonstrates that Liouville's formula on small time scales is a direct consequence of its well-known counterpart for ordinary differential equations.     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated to a metric g and we prove dispersion and Strichartz estimates for the solution of this equation in terms of the geometry of the trajectories associated to g. In particular, we obtain global Strichartz estimates in time for metrics where dispersion estimate is false even locally in time. We also study the analogy between Strichartz estimates obtained for the Liouville equation and the Schrödinger equation with variable coefficients. To cite this article: D. Salort, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Editorial: Hierarchical and bilevel programming

Approximately twenty years ago the modern interest for hierarchical programming was initiated by J. Bracken and J.M. McGill [9], [10]. The activities in the field have ever grown lively, both in terms of theoretical developments and terms of the diversity of the applications. The collection of seven papers in this issue covers a diverse number of topics and provides a good picture of recent research activities in the field of bilevel and hierarchical programming. The papers can be roughly divided into three categories; Linear bilevel programming is addressed in the first two papers by Gendreau et al and Moshirvaziri et al; The following three papers by Nicholls, Loridan & Morgan, and Kalashnikov & Kalashnikova are concerned with nonlinear bilevel programming; and, finally, Wen & Lin and Nagase & Aiyoshi address hierarchical decision making issues relating to both biobjective and bilevel programming.     

Sturm–Liouville operators on time scales

We establish the connection between Sturm–Liouville equations on time scales and Sturm–Liouville equations with measure-valued coefficients. Based on this connection, we generalize several results for Sturm–Liouville equations on time scales, which have been obtained by various authors in the past.     

Liouville and geodesic Ricci solitons

On a tangent bundle endowed with a pseudo-Riemannian metric of complete lift type two classes of Ricci solitons are obtained: a 1-parameter family of shrinking Liouville Ricci solitons if the base manifold is Ricci flat and a steady geodesic Ricci soliton if the base manifold is flat. A nonexistence result of geodesic Ricci solitons for the tangent bundle of a non-flat space form is also provided. To cite this article: M. Crasmareanu, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Stability of Huang's update for the conjugate gradient method

The conditions under which Huang's conjugate gradient method generates descent directions are given and discussed. Bounds for the condition number of the inverse Hessian matrix are estimated for the case of a symmetric update.The author is much indebted to Professor C. G. Broyden of the Essex University Computing Center, for valuable advice and criticism; he is also grateful to Drs. J. Greenstadt and A. R. Gourlay for having sent copies of their unpublished papers.     

Quantum group structure and local fields in the algebraic approach to 2D gravity

This review contains a summary of the work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables — the Liouville exponentials and the Liouville field itself — and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.Supported by DFG.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 104, No. 1, pp. 158–191, July, 1995.     

Exact master equations describing reduced dynamics of the Wigner function

Master equations of different types describe the evolution (reduced dynamics) of a subsystem of a larger system generated by the dynamic of the latter system. Since, in some cases, the (exact) master equations are relatively complicated, there exist numerous approximations for such equations, which are also called master equations. In the paper, we develop an exact master equation describing the reduced dynamics of the Wigner function for quantum systems obtained by a quantization of a Hamiltonian system with a quadratic Hamilton function. First, we consider an exact master equation for first integrals of ordinary differential equations in infinite-dimensional locally convex spaces. After this, we apply the results obtained to develop an exact master equation corresponding to a Liouville-type equation (which is the equation for first integrals of the (system of) Hamilton equation(s)); the latter master equation is called the master Liouville equation; it is a linear first-order differential equation with respect to a function of real variables taking values in a space of functions on the phase space. If the Hamilton equation generating the Liouville equation is linear, then the vector fields that define the first-order linear differential operators in the master Liouville equations are also linear, which in turn implies that for a Gaussian reference state the Fourier transform of a solution of the master Liouville equation also satisfies a linear differential equation. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 203–219, 2005.     

Actions hamiltoniennes de tores et jacobiennes généraliséesTorus actions and generalized Jacobians

I describe the fibration in Liouville tori of an integrable system on the coadjoint orbits of SU(n) in terms of the (relative) generalized Jacobian of a family of singular curves. The integrable system contains the moment mapping of the maximal torus—these periodic Hamiltonians are responsible for the noncompact part of the Jacobian. To cite this article: M. Audin, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 37–42     

A “Sup + C Inf” inequality for Liouville‐type equations with singular potentials

We generalize a Harnack‐type inequality (I. Shafrir, C. R. Acad. Sci. Paris, 315 (1992), 159–164), for solutions of Liouville equations to the case where the weight function may admit zeroes or singularities of power‐type |x|^2α, with α ∈ (?1, 1). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Fibrations 2-isotropes en tores

In this Note one proves a kind of Arnold–Liouville theorem for a 2-isotropic foliation, notion introduced in a previous Note (Turiel, 2008) for any closed r-form, and one studies some properties of the singular integer nets associated to it. To cite this article: F.-J. Turiel, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

Courants algébriques et courants de Liouville avec conditions sur les tranches

In this Note, we study closed positive currents with algebraic or Liouville slices (parallel or concurrent). We improve some results due to Blel, Mimouni and Raby in the case of parallel slices and due to Gruman and Amamou–Ben Farah in the case of concurrent slices. To cite this article: N. Ghiloufi, K. Dabbek, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On some conformally invariant fully nonlinear equationsSur certaines équations completement nonlinéaires invariantes par transformation conforme

We outline proofs of our results in [7] on Liouville type theorems, Harnack type inequalities, and existence and compactness of solutions to some conformally invariant fully nonlinear elliptic equations of second order on locally conformally flat Riemannian manifolds. Details will appear in [7]. To cite this article: A. Li, Y.Y. Li, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 305–310.     

The NP-completeness column: An ongoing guide

This is the ninth edition of a quarterly column which provides continuing coverage of new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book “Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., New York, 1979 (hereinafter referred to as “[G&J]”; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C-355, AT&T Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). If the results are unpublished, please state explicitly that you would like them to be mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this Journal.     

Three-point function in the minimal Liouville gravity

We revisit the problem of the structure constants of the operator product expansions in the minimal models of conformal field theory, rederiving these previously known constants and presenting them in a form particularly useful in Liouville gravity applications. We discuss the analytic relation between our expression and the structure constant in the Liouville field theory and also give the three- and two-point correlation numbers on the sphere in the minimal Liouville gravity in the general form.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 218–234, February, 2005.     

On the boundary value problem for the Sturm–Liouville equation with the discontinuous coefficient

We consider a boundary value problem for the Sturm–Liouville equation with piecewise‐constant leading coefficient. We prove that some integral representations for the solutions of the considered equation can be obtained by using classical transformation operators for the Sturm–Liouville operator at the end points of a finite interval. We also investigate the spectral characteristics of the boundary value problem, prove the completeness and expansion theorem. Copyright © 2012 John Wiley & Sons, Ltd.     

The NP-completeness column: An ongoing guide

This is the seventh edition of a quarterly column the purpose of which is to provide continuing coverage of new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book “Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco 1979 (hereinafter referred to as “[G&J]”; previous columns will be referred to by their dates). A background equivalent to that given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C-355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results to be mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this Journal.