The elliptic representation of the sixth Painlevé equation. - Davide Guzzetti (Théories asymptotiques et équations de Painlevé : [colloque], Angers, juin 2004 / édité par Éric Delabaere, Michèle Loday-Richaud. - Paris : Société mathématique de France, 2006. - Collection SMF. Séminaires et congrès. - page : 83-101, 2004)
We find a class of solutions of the sixth Painlev´e equation corresponding\r\nto almost all the monodromy data of the associated linear system; actually, all data\r\nbut one point in the space of data. We describe the critical behavior close to the\r\ncritical points by means of the elliptic representation, and we find the relation among\r\nthe parameters at the different critical points (connection problem).
Keywords: Painlevé equation

Energetics and switching of quasi-uniform states in small ferromagnetic particles - François Alouges; Sergio Conti; Antonio DeSimone; Ivo Pokern (M2AN Math. Model. Numer. Anal. 38 (2004) 235-248, 2004)
We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size.

Existence of H-bubbles in a perturbative setting - Paolo Caldiroli; Roberta Musina (Rev. Mat. Iberoamericana 20 (2004) 611-626, 2004)
Given a $C^{1}$ function $H: \\\\mathbb{R}^3 \\\\to \\\\mathbb{R}$, we look for $H$-bubbles, i.e., surfaces in $\\\\mathbb{R}^3$ parametrized by the sphere $\\\\mathbb{S}^2$ with mean curvature $H$ at every regular point. Here we study the case $H(u)=H_{0}(u)+\\\\epsilon H_{1}(u)$ where $H_{0}$ is some \\\"good\\\" curvature (for which there exist $H_{0}$-bubbles with minimal energy, uniformly bounded in $L^{\\\\infty}$), $\\\\epsilon$ is the smallness parameter, and $H_{1}$ is {\\\\em any} $C^{1}$ function.

The Extended Toda Hierarchy - Guido Carlet; Boris Dubrovin; Zhang Youjin (Moscow Math. J. 4 (2004)\\n313-332., 2004)

Fredholm modules for quantum euclidean spheres - Eli Hawkins; Giovanni Landi (J. Geom. Phys. 49 (2004) 272-293, 2004)
The quantum Euclidean spheres, $S_q^{N-1}$, are (noncommutative) homogeneous spaces of quantum orthogonal groups, $\\\\SO_q(N)$. The *-algebra $A(S^{N-1}_q)$ of polynomial functions on each of these is given by generators and relations which can be expressed in terms of a self-adjoint, unipotent matrix. We explicitly construct complete sets of generators for the K-theory (by nontrivial self-adjoint idempotents and unitaries) and the K-homology (by nontrivial Fredholm modules) of the spheres $S_q^{N-1}$. We also construct the corresponding Chern characters in cyclic homology and cohomology and compute the pairing of K-theory with K-homology. On odd spheres (i. e., for N even) we exhibit unbounded Fredholm modules by means of a natural unbounded operator D which, while failing to have compact resolvent, has bounded commutators with all elements in the algebra $A(S^{N-1}_q)$.

A geometric approach to the separability of the Neumann-Rosochatius system - Claudio Bartocci; Gregorio Falqui; Marco Pedroni (Differential Geom. Appl. 21 (2004) 349-360, 2004)
We study the separability of the Neumann-Rosochatius system on the n-dimensional sphere using the geometry of bi-Hamiltonian manifolds. Its well-known separation variables are recovered by means of a separability condition relating the Hamiltonian with a suitable (1,1) tensor field on the sphere. This also allows us to iteratively construct the integrals of motion of the system.

H-bubbles in a perturbative setting: the finite-dimensional reduction\\\'s method - Paolo Caldiroli; Roberta Musina (Duke Math. J. 122 (2004), no. 3, 457--484, 2004)
Given a regular function $H\\\\colon\\\\mathbb{R}^{3}\\\\to\\\\mathbb{R}$, we look for $H$-bubbles, that is, regular surfaces in $\\\\mathbb{R}^{3}$ parametrized on the sphere $\\\\mathbb{S}+^{2}$ with mean curvature $H$ at every point. Here we study the case of $H(u)=H_{0}+\\\\varepsilon H_{1}(u)=:H_{\\\\varepsilon}(u)$, where $H_{0}$ is a nonzero constant, $\\\\varepsilon$ is the smallness parameter, and $H_{1}$ is any $C^{2}$-function. We prove that if $\\\\bar p\\\\in\\\\mathbb{R}^{3}$ is a ``good\\\'\\\' stationary point for the Melnikov-type function $\\\\Gamma(p)=-\\\\int_{|q-p|<|H_{0}|^{-1}}H_{1}(q)\\\\,dq$, then for $|\\\\varepsilon|$ small there exists an $H_{\\\\varepsilon}$-bubble $\\\\omega^{\\\\varepsilon}$ that converges to a sphere of radius $|H_{0}|^{-1}$ centered at $\\\\bar p$, as $\\\\varepsilon\\\\to 0$.

Higher order quasiconvexity reduces to quasiconvexity - Gianni Dal Maso; Irene Fonseca; Giovanni Leoni; Massimiliano Morini (Arch. Ration. Mech. Anal. 171 (2004) 55-81, 2004)
In this paper it is shown that higher order quasiconvex functions suitable in the variational treatment of problems involving second derivatives may be extended to the space of all matrices as classical quasiconvex functions. Precisely, it is proved that a smooth strictly 2-quasiconvex function with p-growth at infinity, p>1, is the restriction to symmetric matrices of a 1-quasiconvex function with the same growth. As a consequence, lower semicontinuity results for second-order variational problems are deduced as corollaries of well-known first order theorems.

On the minimal degree of a common Lyapunov function for planar switched systems - Paolo Mason; Ugo Boscain; Yacine Chitour (43rd IEEE Conference on Decision and Control, 2004, 2786 - 2791 Vol.3, 2004)
In this paper, we consider linear switched systems x(t) = Au(t)x(t), x ε Rn, u ε U, and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a polynomial common Lyapunov function. Then our main result is that the degree of that the common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin.

Multidimensional boundary layers for a singularly perturbed Neumann problem - Andrea Malchiodi; Marcelo Montenegro (Duke Math. J. 124 (2004) 105-143, 2004)

Multiplicity of periodic solutions of nonlinear wave equations - Massimiliano Berti; Philippe Bolle (Nonlinear Anal. 56 (2004) 1011-1046, 2004)

Periodic orbits close to elliptic tori and applications to the three-body problem - Massimiliano Berti; Luca Biasco; Enrico Valdinoci (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004) 87-138, 2004)
We prove, under suitable non-resonance and non-degeneracy ``twist\\\'\\\' conditions, a Birkhoff-Lewis type result showing the existence of infinitely many periodic solutions, with larger and larger minimal period, accumulating onto elliptic invariant tori (of Hamiltonian systems). We prove the applicability of this result to the spatial planetary three-body problem in the small eccentricity-inclination regime. Furthermore, we find other periodic orbits under some restrictions on the period and the masses of the ``planets\\\'\\\'. The proofs are based on averaging theory, KAM theory and variational methods. (Supported by M.U.R.S.T. Variational Methods and Nonlinear Differential Equations.)

Quasi-static evolution in brittle fracture: the case of bounded solutions - Gianni Dal Maso; Gilles A. Francfort; Rodica Toader (Quad. Mat. Dip. Mat. Seconda Univ. Napoli 14 (2004) 245-266, 2004)
The main steps of the proof of the existence result for the quasi-static evolution of cracks in brittle materials, obtained in [7] in the vector case and for a general quasiconvex elastic energy, are presented here under the simplifying assumption that the minimizing sequences involved in the problem are uniformly bounded in $L^\\\\infty$.

Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric - Claudio Altafini (ESAIM: COCV 10 (2004) 526-548, 2004)
For a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the Riemannian exponential map and the Lie group exponential map do not coincide. The consequence is that the reduced equations look more complicated than the original ones. The main scope of this paper is to treat the reduction of second order variational problems (corresponding to geometric splines) on such semidirect products of Lie groups. Due to the semidirect structure, a number of extra terms appears in the reduction, terms that are calculated explicitely. The result is used to compute the necessary conditions of an optimal control problem for a simple mechanical control system having invariant Lagrangian equal to the kinetic energy corresponding to the metric tensor. As an example, the case of a rigid body on the Special Euclidean group is considered in detail.

Representing multiqubit unitary evolutions via Stokes tensors - Claudio Altafini (Phys. Rev. A 70 (2004) 032331, 2004)
For the Stokes tensor parametrization of a multiqubit density operator, we provide an explicit formulation of the corresponding unitary dynamics at the infinitesimal level. The main advantage of this formalism (clearly reminiscent of the ideas of ``coherences\\\'\\\' and ``coupling Hamiltonians\\\'\\\' of spin systems) is that the pattern of correlation between qubits and the pattern of infinitesimal correlation are highlighted simultaneously and can be used constructively for qubit manipulation. For example, it allows to compute explicitly a Rodrigues\\\' formula for the one-parameter orbits of nonlocal Hamiltonians. The result is easily generalizable to orbits of Cartan subalgebras and allows to express the Cartan decomposition of unitary propagators as a linear action directly in terms of the infinitesimal generators.

Resonance of minimizers for n-level quantum systems with an arbitrary cost - Ugo Boscain; Grégoire Charlot (ESAIM COCV 10 (2004) 593-614, 2004)
We consider an optimal control problem describing a laser-induced population transfer on a $n$-level quantum system.\\nFor a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for $n=2$ and $n=3$): instead of looking for minimizers on the sphere $S^{2n-1}\\\\subset\\\\C^n$ one is reduced to look just for minimizers on the sphere $S^{n-1}\\\\subset \\\\R^n$. Moreover, for the reduced problem, we investigate on the question of existence of strict abnormal minimizer.

The role of the spectrum of the Laplace operator on \\\\S2 in the H-bubble problem - Roberta Musina (J. Anal. Math. 94 (2004) 265-291, 2004)

Rotating Singular Perturbations of the Laplacian - Michele Correggi; Gianfausto Dell'Antonio (Ann. Henri Poincare 5 (2004) 773-808, 2004)
We study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for their unitary semigroups. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as (\\\\omega \\\\to \\\\infty).

Semiclassical analysis of constrained quantum systems - Gianfausto Dell'Antonio; Lucattilio Tenuta (J. Phys. A 37 (2004) 5605-5624, 2004)
We study the dynamics of a quantum particle in R^(n+m) constrained by a strong potential force to stay within a distance of order hbar (in suitable units) from a smooth n-dimensional submanifold M. We prove that in the semiclassical limit the evolution of the wave function is approximated in norm, up to terms of order hbar^(1/2), by the evolution of a semiclassical wave packet centred on the trajectory of the corresponding classical constrained system.

Semi-cooperative strategies for differential games - Alberto Bressan; Wen Shen (Internat. J. Game Theory 32 (2004) 561-593, 2004)
The paper is concerned with a non-cooperative differential game for two players. We first consider Nash equilibrium solutions in feedback form. In this case, we show that the Cauchy problem for the value functions is generically ill-posed. Looking at vanishing viscosity approximations, one can construct special solutions in the form of chattering controls, but these also appear to be unstable. In the second part of the paper we propose an alternative \\\"semi-cooperative\\\" pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, we prove that the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.

A sharp decay estimate for positive nonlinear waves - Alberto Bressan; Tong Yang (SIAM J. Math. Anal. 36 (2004) 659-677, 2004)
We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers\\\' equation with impulsive sources.

Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation - Victor Z. Enolski; Tamara Grava (Int. Math. Res. Not. 2004, no. 32, 1619-1683, 2004)
We are solving the classical Riemann-Hilbert problem of rank N>1 on the extended complex plane punctured in 2m+2 points, for NxN quasi-permutation monodromy matrices. Following Korotkin we solve the Riemann-Hilbert problem in terms of the Szego kernel of certain Riemann surfaces branched over the given 2m+2 points. These Riemann surfaces are constructed from a permutation representation of the symmetric group S_N to which the quasi-permutation monodromy representation has been reduced. The permutation representation of our problem generates the cyclic subgroup Z_N. For this reason the corresponding Riemann surfaces of genus N(m-1) have Z_N symmetry. This fact enables us to write the matrix entries of the solution of the NxN Riemann-Hilbert problem as a product of an algebraic function and theta-function quotients. The algebraic function turns out to be related to the Szego kernel with zero characteristics. From the solution of the Riemann- Hilbert problem we automatically obtain a particular solution of the Schlesinger system. The tau-function of the Schlesinger system is computed explicitly. The rank 3 problem with four singular points (0,t,1,\\\\infty) is studied in detail. The corresponding solution of the Riemann-Hilbert problem and the Schlesinger system is given in terms of Jacobi\\\'s theta-function with modulus T=T(t), Im(T)>0. The function T=T(t) is invertible if it belongs to the Siegel upper half space modulo the subgroup \\\\Gamma_0(3) of the modular group. The inverse function t=t(T) generates a solution of a general Halphen system.

Singularity perturbed elliptic equations with symmetry: existence of solutions concetrating on spheres, Part II - Antonio Ambrosetti; Andrea Malchiodi; Wei-Ming Ni (Indiana Univ. Math. J. 53 (2004) 297-392, 2004)

Small BV solutions of hyperbolic noncooperative differential games - Alberto Bressan; Wen Shen (SIAM J. Control Optim. 43 (2004) 194-215, 2004)
The paper is concerned with an n-persons differential game in one space dimension. We state conditions for which the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic. In the positive case, we show that the weak solution of a corresponding system of conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game.

Solitary waves for Maxwell Schrodinger equations - Giuseppe Maria Coclite; Vladimir Georgiev (Electron. J. Differential Equations (2004) 94, 2004)
In this paper we study solitary waves for the coupled system of Schrodinger-Maxwell equations in the three-dimensional space. We prove the existence of a sequence of radial solitary waves for these equations with a fixed L^2 norm. We study the asymptotic behavior and the smoothness of these solutions. We show also that the eigenvalues are negative and the first one is isolated.

Solutions concentrating at curves for some singularly perturbed elliptic problems - Andrea Malchiodi (C. R. Acad. Sci. Paris, Ser. I 338 (2004) 775-780, 2004)

Soluzioni periodiche di PDEs Hamiltoniane - Massimiliano Berti (Bollettino dell\\\'Unione Matematica Italiana Serie 8 7-B (2004), p. 647-661, 2004)

Some remarks on multidimensional systems of conservation laws - Alberto Bressan (Rend. Mat. Acc. Lincei, s. 9, v. 15 (2004) 3-4, pp. 225 - 233, 2004)
This note is concerned with the Cauchy problem for hyperbolic systems of conservation\\nlaws in several space dimensions. We first discuss an example of ill-posedness, for a special system\\nhaving a radial symmetry property. Some conjectures are formulated, on the compactness of the set of\\nflow maps generated by vector fields with bounded variation.

Stability rates for patchy vector fields - Fabio Ancona; Alberto Bressan (ESAIM COCV 10 (2004) 168-200, 2004)
This paper is concerned with the stability of the set of trajectories of a patchy vector field, in the presence of impulsive perturbations. Patchy vector fields are discontinuous, piecewise smooth vector fields that were introduced in Ancona and Bressan (1999) to study feedback stabilization problems. For patchy vector fields in the plane, with polygonal patches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude as the impulsive forcing term.

Superlocalization formulas and supersymmetric Yang-Mills theories - Ugo Bruzzo; Francesco Fucito (Nucl. Phys. B 678 (2004) 638-655, 2004)
By using supermanifold techniques we prove a generalization of the localization formula in equivariant cohomology which is suitable for studying supersymmetric Yang-Mills theories in terms of ADHM data. With these techniques one can compute the reduced partition functions of topological super Yang-Mills theory with 4, 8 or 16 supercharges. More generally, the superlocalization formula can be applied to any topological field theory in any number of dimensions.

Tensor of coherences parametrization of multiqubit density operators for entanglement characterization - Claudio Altafini (Phys. Rev. A 69 (2004) 012311, 2004)
For multiqubit densities, the tensor of coherences (or Stokes tensor) is a real parameterization obtained by the juxtaposition of the affine Bloch vectors of each qubit. While it maintains the tensorial structure of the underlying space, it highlights the pattern of correlations, both classical and quantum, between the subsystems and, due to the affine parameterization, it contains in its components all reduced densities of all orders. The main purpose of our use of this formalism is to deal with entanglement. For example, the detection of bipartite entanglement is straightforward, as it is the synthesis of densities having positive partial transposes between desired qubits. In addition, finding explicit mixtures for families of separable states becomes a feasible issue for few qubit symmetric densities (we compute it for Werner states) and, more important, it provides some insight on the possible origin of entanglement for such densities.

Time-dependent singular interactions - Michele Correggi (SISSA, 2004)
Keywords: Rotating singular interactions

Virasoro Symmetries of the Extended Toda Hierarchy - Boris Dubrovin; Zhang Youjin (Comm. Math.\\nPhys. 250 (2004) 161-193., 2004)
We prove that the extended Toda hierarchy of \\\\cite{CDZ} admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators $L_m$, $m\\\\geq -1$ of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the $CP^1$ Gromov-Witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.

Well-posedness for general 2x2 systems of conservation laws - Fabio Ancona; Andrea Marson (Mem. Amer. Math. Soc. 169 (2004), no. 801, x+170 pp., 2004)

Autonomous integral functionals with discontinous nonconvex integrands: Lipschitz regularity of mimimizers, DuBois-Reymond necessary conditions and Hamilton-Jacobi equations - Gianni Dal Maso; Helene Frankowska (Applied Math.Optim. 48 (2003), no.1, p.39-66, 2003)

Bilinear semiclassical moment functionals and their integral representation - Marco Bertola (J. Approx. Theory, 2003)

The calibration method for the Mumford-Shah functional and free-discontinuity problems - Giovanni Alberti; Guy Bouchitte; Gianni Dal Maso (Calc. Var. Partial Differential Equations 16 (2003) 299-333, 2003)
We present a minimality criterion for the Mumford-Shah functional, and more generally for non convex variational integrals on SBV which couple a surface and a bulk term. This method provides short and easy proofs for several minimality results.

Control Problems for Systems of Conservation Laws - Giuseppe Maria Coclite (SISSA, 2003)
Keywords: Asymptotic Stabilization

Controllability properties for finite dimensional quantum Markovian master equations - Claudio Altafini (J. Math. Phys. 44 (2003) 2357-2372, 2003)
Various notions from geometric control theory are used to characterize the behavior of the Markovian master equation for N-level quantum mechanical systems driven by unitary control and to describe the structure of the sets of reachable states. It is shown that the system can be accessible but neither small-time controllable nor controllable in finite time. In particular, if the generators of quantum dynamical semigroups are unital, then the reachable sets admit easy characterizations as they monotonically grow in time. The two level case is treated in detail.

Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem - Marco Bertola; B. Eynard; Harnad, J. (Comm. Math. Phys., 2003)

Drift in phase space: a new variational mechanism with optimal diffusion time - Massimiliano Berti; Luca Biasco; Philippe Bolle (J. Math. Pures Appl. 82 (2003) 613-664, 2003)
We consider non-isochronous, nearly integrable, a-priori unstable Hamiltonian systems with a (trigonometric polynomial) $O(\\\\mu)$-perturbation which does not preserve the unperturbed tori. We prove the existence of Arnold diffusion with diffusion time $ T_d = O((1/ \\\\mu) \\\\log (1/ \\\\mu))$ by a variational method which does not require the existence of ``transition chains of tori\\\'\\\' provided by KAM theory. We also prove that our estimate of the diffusion time $T_d $ is optimal as a consequence of a general stability result derived from classical perturbation theory.

The duality of spectral curves that arises in two-matrix models - Marco Bertola; B. Eynard; Kharnad, Dzh. (Teoret. Mat. Fiz., 2003)

Effective dynamics for Bloch electrons: Peierls substitution and beyond - Gianluca Panati; Herbert Spohn; Stefan Teufel (Springer, 2003)
We consider an electron moving in a periodic potential and subject to an additional slowly varying external electrostatic potential, $\\\\phi(\\\\epsi x)$, and vector potential $A(\\\\epsi x)$, with $x \\\\in \\\\R^d$ and $\\\\epsi \\\\ll 1$. We prove that associated to an isolated family of Bloch bands there exists an almost invariant subspace of $L^2(\\\\R^d)$ and an effective Hamiltonian governing the evolution inside this subspace to all orders in $\\\\epsi$. To leading order the effective Hamiltonian is given through the Peierls substitution. We explicitly compute the first order correction. From a semiclassical analysis of this effective quantum Hamiltonian we establish the first order correction to the standard semiclassical model of solid state physics.

A finite element approximation of the Griffith\\\'s model in fracture mechanics - Matteo Negri (Numer. Math., 2003, 95, 653, 2003)
The Griffith model for the mechanics of fractures in brittle materials is consider in the weak formulation of SBD spaces. We suggest an approximation, in the sense of Gamma-convergence, by a sequence of discrete functionals defined on finite elements spaces over structured and adaptive triangulations. The quasi-static evolution for boundary value problems is also taken into account and some numerical results are shown.

Free energy of the two-matrix model/dToda tau-function - Marco Bertola (Nuclear Phys. B, 2003)

Gaudin models and bending flows: a geometrical point of view - Gregorio Falqui; Fabio Musso (J. Phys. A: Math. Gen. 36 (2003) 11655-11676, 2003)
In this paper we discuss the bihamiltonian formulation of the (rational XXX) Gaudin models of spin-spin interaction, generalized to the case of sl(r)-valued spins. In particular, we focus on the homogeneous models. We find a pencil of Poisson brackets that recursively define a complete set of integrals of the motion, alternative to the set of integrals associated with the \\\'standard\\\' Lax representation of the Gaudin model. These integrals, in the case of su(2), coincide wih the Hamiltonians of the \\\'bending flows\\\' in the moduli space of polygons in Euclidean space introduced by Kapovich and Millson. We finally address the problem of separability of these flows and explicitly find separation coordinates and separation relations for the r=2 case.

Geometric motion control for a kinematically redundant robotic chain: application to a holonomic mobile manipulator - Claudio Altafini (J. Robotic Syst. 20 (2003) 211-227, 2003)
For kinematically redundant robotic manipulators, the extra degrees of freedom available allows freedom in the generation of the trajectories of the end-effector. In this paper, for this scope, we use techniques for motion control of rigid bodies on Riemannian manifolds (and Lie groups in particular) to design workspace control algorithms for the end-effector of the robotic chain and then to pull them back to joint space, all respecting the different geometric structures of the two underlying model spaces. The trajectory planner makes use of geometric splines. Examples of the different kinds of curves that are obtained via the De Casteljau algorithm in correspondence of different metric structures in SE(3) are reported. The feedback module, instead, consists of a Lyapunov based PD controller defined from a suitable notion of error distance on the Lie group. The motivating application of our work is a holonomic mobile manipulator for which simulation results are described in detail.

Hybrid optimal control: case study of a car with gears - Ciro D'Apice; Mauro Garavello; Rosanna Manzo; Benedetto Piccoli (Int. J. Control 76 (2003) 1272-1284, 2003)
The purpose of this paper is to show the use of some analytical tools for hybrid optimal control. We illustrate both the hybrid maximum principle and the hybrid necessary principle at work on a simple example of a car with gears. The model is sufficiently rich to generate non-trivial optimization problems and the obtained results match with intuition. Finally, computer simulations confirm the theoretical analysis.

An ill posed Cauchy problem for a hyperbolic system in two space dimensions - Alberto Bressan (Università di Padova, 2003)
The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed.

An interior estimate for a nonlinear parabolic equation - Giuseppe Maria Coclite (J.Math.Anal.Appl. 284 (2003) no.1, 49, 2003)

A lemma and a conjecture on the cost of rearrangements - Alberto Bressan (Rend. Sem. Mat. Univ. Padova 110 (2003) 97-102, 2003)
Consider a stack of books, containing both white and black books. Suppose that we want to sort them out, putting the white books on the right, and the black books on the left (fig.~1). This will be done by a finite sequence of elementary transpositions. In other words, if we have a stack of all black books of length $a$ followed by a stack of all white books of length $b$, we are allowed to reverse their order at the cost of $a+b$. We are interested in a lower bound on the total cost of the rearrangement.

On the local structure of optimal trajectories in R3 - Andrei A. Agrachev; Mario Sigalotti (SIAM J. Control Optim. 42 (2003) 513-531, 2003)
We analyze the structure of a control function u(t) corresponding to an optimal trajectory for the system $\\\\dot q =f(q)+u\\\\, g(q)$ in a three-dimensional manifold, near a point where some nondegeneracy conditions are satisfied. The kind of optimality which is studied includes time-optimality. The control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four.

Mixed correlation functions of the two-matrix model - Marco Bertola; B. Eynard (J. Phys. A, 2003)

Motion on submanifolds of noninvariant holonomic constraints for a kinematic control system evolving on a matrix Lie group - Claudio Altafini; Ruggero Frezza (Syst. Control Lett. 50 (2003) 241-250, 2003)
For a control system on a matrix Lie group with one or more configuration constraints that are not left/right invariant, finding the combinations of (kinematic) control inputs satisfying the motion constraints is not a trivial problem. Two methods, one coordinate-dependent and the other coordinate-free are suggested. The first is based on the Wei-Norman formula; the second on the calculation of the annihilator of the coadjoint action of the constraint one-form at each point of the group manifold. The results are applied to a control system on SE(3) with a holonomic inertial constraint involving the noncommutative part in a nontrivial way. The difference in terms of compactness of the result between the two methods is considerable.

Multi-instanton calculus and equivariant cohomology - Ugo Bruzzo; Jose F. Morales; Francesco Fucito; Alessandro Tanzini (J.High Energy Phys. 2003,no.5,054,24 pp., 2003)

Non-linear sigma-models in noncommutative geometry: fields with values in finite spaces - Ludwik Dabrowski; Thomas Krajewski; Giovanni Landi (Mod. Phys. Lett. A 18 (2003) 2371-2379, 2003)
We study sigma-models on noncommutative spaces, notably on noncommutative tori. We construct instanton solutions carrying a nontrivial topological charge q and satisfying a Belavin-Polyakov bound. The moduli space of these instantons is conjectured to consists of an ordinary torus endowed with a complex structure times a projective space $CP^{q-1}$.

A note on singular limits to hyperbolic systems of conservation laws - Stefano Bianchini (Commun. Pure Appl. Ana., 2003, 2, 51-64, 2003)
In this note we consider two different singular limits to hyperbolic system of conservation laws, namely the standard backward schemes for non linear semigroups and the semidiscrete scheme. \\nUnder the assumption that the rarefaction curve of the corresponding hyperbolic system are straight lines, we prove the stability of the solution and the convergence to the perturbed system to the unique solution of the limit system for initial data with small total variation.

A note on the integral representation of functionals in the space SBD(O) - Francois Ebobisse; Rodica Toader (Rend. Mat. Appl. 23 (2003) 189-201, 2003)
In this paper we study the integral representation in the space SBD(O) of special functions with bounded deformation of some L^1-norm lower semicontinuous functionals invariant with respect to rigid motions.

Parameter differentiation and quantum state decomposition for time varying Schrödinger equations - Claudio Altafini (Rep. Math. Phys. 52 (2003) 381-400, 2003)
For the unitary operator, solution of the Schroedinger equation corresponding to a time-varying Hamiltonian, the relation between the Magnus and the product of exponentials expansions can be expressed in terms of a system of first order differential equations in the parameters of the two expansions. A method is proposed to compute such differential equations explicitly and in a closed form.

Partition functions for matrix models and isomonodromic tau functions - Marco Bertola; B. Eynard; Harnad, J. (J. Phys. A, 2003)

Periodic solutions of nonlinear wave equations with general nonlinearities - Massimiliano Berti; Philippe Bolle (Comm.Math.Phys. 243 (2003) no.2, 315, 2003)

Poisson Pencils, Integrability, and Separation of Variables - Gregorio Falqui (SISSA, 2003)
In this paper we will review a recently introduced method for solving the Hamilton-Jacobi equations by the method of Separation of Variables. This method is based on the notion of pencil of Poisson brackets and on the bihamiltonian approach to integrable systems. We will discuss how separability conditions can be intrinsically characterized within such a geometrical set-up, the definition of the separation coordinates being encompassed in the \\\\bih structure itself. We finally discuss these constructions studying in details a particular example, based on a generalization of the classical Toda Lattice.

Positive solutions to a class of quasilinear elliptic equations on R - Antonio Ambrosetti; Wang Zhi-Qiang (Discrete Contin.Dyn.Syst. 9 (2003), no.1, 55-68, 2003)
We discuss the existence of positive solutions of perturbation to a class of quasilinear elliptic equations on R.

Prescribing scalar and boundary mean curvature on the three dimensional half sphere - Zindine Djadli; Andrea Malchiodi; Mohameden Ould Ahmedou (J. Geom. Anal. 13 (2003) 255-289, 2003)
We consider the problem of prescribing the scalar curvature and the boundary mean curvature of the standard half three sphere, by deforming conformally its standard metric. Using blow up analysis techniques and minimax arguments, we prove some existence and compactness results.

Quantum spin coverings and statistics - Ludwik Dabrowski; Cesare Reina (J. Phys. A 36 (2003), no. 13, 3829-3840, 2003)
SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the complex rotation group SO(3,C), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the decomposition of their tensor products and a coquasitriangular structure, with the associated braiding (or statistics). As an example, the case l=3 is discussed in detail.

Second and third order observables of the two-matrix model - Marco Bertola (J. High Energy Phys., 2003)

Separation of variables for Bi-Hamiltonian systems - Gregorio Falqui; Marco Pedroni (Math. Phys. Anal. Geom. 6 (2003) 139-179, 2003)
We address the problem of the separation of variables for the Hamilton-Jacobi equation within the theoretical scheme of bi-Hamiltonian geometry. We use the properties of a special class of bi-Hamiltonian manifolds, called omega-N manifolds, to give intrisic tests of separability (and Staeckel separability) for Hamiltonian systems. The separation variables are naturally associated with the geometrical structures of the omega-N manifold itself. We apply these results to bi-Hamiltonian systems of the Gel\\\'fand-Zakharevich type and we give explicit procedures to find the separated coordinates and the separation relations.

Sequences of Singularly Perturbed Functionals Generating Free-Discontinuity Problems - Massimiliano Morini (SIAM J. Math. Anal. 35 (2003) 759-805, 2003)
We prove that a wide class of singularly perturbed functionals generates as $\\\\Gamma$-limit a functional related to a free-discontinuity problem. Several applications of the result are shown.

Single-Input Control Affine Systems: Local Regularity of Optimal Trajectories and a Geometric Controllability Problem - Mario Sigalotti (SISSA, 2003)

Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, Part I - Antonio Ambrosetti; Andrea Malchiodi; Wei-Ming Ni (Comm. Math. Phys. 235 (2003) no.3, 427-466, 2003)

Some results on the boundary control of systems of conservation laws - Alberto Bressan; Fabio Ancona; Giuseppe Maria Coclite (SIAM J.Control Optim. 41 (2003),no.2, 607, 2003)

Space-adiabatic perturbation theory - Gianluca Panati; Herbert Spohn; Stefan Teufel (Adv. Theor. Math. Phys. 7 (2003) 145-204, 2003)
We study approximate solutions to the Schr\\\\\\\"odinger equation $i\\\\epsi\\\\partial\\\\psi_t(x)/\\\\partial t = H(x,-i\\\\epsi\\\\nabla_x) \\\\psi_t(x)$ with the Hamiltonian given as the Weyl quantization of the symbol $H(q,p)$ taking values in the space of bounded operators on the Hilbert space $\\\\Hi_{\\\\rm f}$ of fast ``internal\\\'\\\' degrees of freedom. By assumption $H(q,p)$ has an isolated energy band. Using a method of Nenciu and Sordoni \\\\cite{NS} we prove that interband transitions are suppressed to any order in $\\\\epsi$. As a consequence, associated to that energy band there exists a subspace of $L^2(\\\\mathbb{R}^d,\\\\Hi _{\\\\rm f})$ almost invariant under the unitary time evolution. We develop a systematic perturbation scheme for the computation of effective Hamiltonians which govern approximately the intraband time evolution. As examples for the general perturbation scheme we discuss the Dirac and Born-Oppenheimer type Hamiltonians and we reconsider also the time-adiabatic theory.

A stability result for nonlinear Neumann problems under boundary variations - Gianni Dal Maso; Francois Ebobisse; Marcello Ponsiglione (J.Math. Pures Appl. (9) 82 (2003) no.5 , 503, 2003)
In this paper we study, in dimension two, the stability of the solutions of some nonlinear elliptic equations with Neumann boundary conditions, under perturbations of the domains in the Hausdorff complementary topology.

Admissible Riemann solvers for genuinely nonlinear P-systems of mixed type - Jean-Marc Mercier; Benedetto Piccoli (J. Differ. Equations, 2002, 180, 395, 2002)

Arnold diffusion: a functional analysis approach - Massimiliano Berti (Pr. Inst. Mat. Nats. Akad. Nauk Ukr. Mat. Zastos., 43, Part 1, 2, Natsīonal. Akad. Nauk Ukraïni, Īnst. Mat., Kiev, 2002, 2002)
We present, in the context of nearly integrable Hamiltonian systems, a functional analysis approach to study the “splitting of the whiskers” and the “shadowing problem” developed in collaboration with P. Bolle in the recent papers [1] and [2] . This method is applied to the problem of Arnold diffusion for nearly integrable partially isochronous systems improving known results.

On the Boundary Control of Systems of Conservation Laws - Alberto Bressan; Giuseppe Maria Coclite (SIAM J. Control Optim. 41 (2002) 607-622, 2002)
The paper is concerned with the boundary controllability of entropy weak solutions to hyperbolic systems of conservation laws. We prove a general result on the asymptotic stabilization of a system near a constant state. On the other hand, we give an example showing that exact controllability in finite time cannot be achieved, in general.

The Calibration Method for Free-Discontinuity Problems on Vector-Valued Maps - Maria Giovanna Mora (J. Convex Anal. 9 (2002) 1-29, 2002)
The calibration method is a classical minimality criterion, which has been recently adapted to functionals with free discontinuities by Alberti, Bouchitté, Dal Maso. In this paper we present a further generalization of this theory to functionals defined on vector-valued maps.

A center manifold technique for tracing viscous waves - Stefano Bianchini; Alberto Bressan (Commun. Pure Appl. Anal. 1 (2002) 161-190, 2002)
In this paper we introduce a new technique for tracing viscous travelling profiles. To illustrate the method, we consider a special 2 x 2 hyperbolic system of conservation laws with viscosity, and show that any solution can be locally decomposed as the sum of 2 viscous travelling profiles. This yields the global existence, stability and uniform BV bounds for every solution with suitably small BV data.

Chaotic dynamics for perturbations of infinite-dimensional Hamiltonian systems - Massimiliano Berti; Carlo Carminati (Nonlinear Anal. 48 (2002) 481-504, 2002)

Coherent state realizations of $\rm su(n+1)$ on the $n$-torus - de Guise, Hubert; Marco Bertola (J. Math. Phys., 2002)

Controllability of quantum mechanical systems by root space decomposition of su(N) - Claudio Altafini (J.Math.Phys. 43(2002), no.5, 2051, 2002)

Curvature theory of boundary phases: the two-dimensional case - Andrea Braides; Andrea Malchiodi (Interfaces Free Bound. 7 (2002) 345-370, 2002)
We describe the behaviour of minimum problems involving non-convex surface integrals in 2D, singularly perturbed by a curvature term. We show that their limit is described by functionals which take into account energies concentrated on vertices of polygons. Non-locality and non-compactness effects are highlighted.

Duality, biorthogonal polynomials and multi-matrix models - Marco Bertola; B. Eynard; Harnad, J. (Comm. Math. Phys., 2002)

The Elliptic Representation of the General Painlevé 6 Equation - Davide Guzzetti (Communications on Pure and Applied Mathematics, Volume 55, Issue 10, October 2002, Pages 1280-1363, 2002)
We study the analytic properties and the critical behavior of the elliptic\r\nrepresentation of solutions of the Painlev\\\'e 6 equation. We solve the\r\nconnection problem for elliptic representation in the generic case and in a\r\nnon-generic case equivalent to WDVV equations of associativity.

The Elliptic Representation of the Painleve 6 Equation - Davide Guzzetti (Deformation of differential equations and asymptotic analysis / Yoshishige Haraoka. - Kyōto : Kyoto University, Research Institute for Mathematical Sciences, 2002. - RIMS kokyuroku, volume 1296 . - page: 112-123, 2002)
We review our results on the elliptic representation of the sixth Painleve’ equation
Keywords: Painleve equations

Existence of minimal H-bubbles - Paolo Caldiroli; Roberta Musina (Commun. Contemp. Math. 4 (2002) 177-209, 2002)

Fast Arnold diffusion in systems with three time scales - Massimiliano Berti; Philippe Bolle (Discrete Contin. Dyn. Syst. 8 (2002) 795-811, 2002)
We consider the problem of Arnold Diffusion for nearly integrable partially isochronous Hamiltonian systems with three time scales. By means of a careful shadowing analysis, based on a variational technique, we prove that, along special directions, Arnold diffusion takes place with fast (polynomial) speed, even though the \\\"splitting determinant\\\" is exponentially small.

Flow Stability of Patchy Vector Fields and Robust Feedback Stabilization - Fabio Ancona; Alberto Bressan (SIAM J. Control Optim. 41 (2002) 1455-1476, 2002)
The paper is concerned with patchy vector fields, a class of discontinuous, piecewise smooth vector fields that were introduced in AB to study feedback stabilization problems. We prove the stability of the corresponding solution set w.r.t. a wide class of impulsive perturbations. These results yield the robusteness of patchy feedback controls in the presence of measurement errors and external disturbances.

Following a path of varying curvature as an output regulation problem - Claudio Altafini (IEEE Trans. Automatic Control 47 (2002) 1551-1556, 2002)
Given a path of nonconstant curvature, local asymptotic stability can be proven for the general n trailer whenever the curvature can be considered as the output of an exogenous dynamical system. The controllers that provide convergence to zero of the tracking error chosen for the path-following problem are composed of a prefeedback that input-output linearizes the system, plus a linear controller.

A functional analysis approach to Arnold diffusion - Massimiliano Berti; Philippe Bolle (Ann. Inst. H. Poincare Anal. Non Lineaire 19 (2002) 395-450, 2002)
We discuss in the context of nearly integrable Hamiltonian systems a functional analysis approach to the \\\"splitting of separatrices\\\" and to the \\\"shadowing problem\\\". As an application we apply our method to the problem of Arnold Diffusion for nearly integrable partially isochronous systems improving known results.

On the generation of sequential unitary gates from continuous time Schrodinger equations driven by external fields - Claudio Altafini (Quantum Inf.Process. 1 (2002),no.3,207, 2002)

Geometry of Jacobi Curves I - Andrei A. Agrachev; Igor Zelenko (J. Dynam. Control Systems 8 (2002) 93-140, 2002)
Jacobi curves are deep generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions.\\nIn the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of \\\"flat curves\\\".

Geometry of Jacobi curves II - Andrei A. Agrachev; Igor Zelenko (J. Dynam. Control Systems 8 (2002), no. 2, 167--215, 2002)

Global calibrations for the non-homogeneous Mumford-Shah functional - Massimiliano Morini (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002) 603-648, 2002)
Using a calibration method we prove that, if $\\\\Gamma\\\\subset \\\\Omega$ is a closed regular hypersurface and if the function $g$ is discontinuous along $\\\\Gamma$ and regular outside, then the function $u_{\\\\beta}$ which solves $$ \\\\begin{cases} \\\\Delta u_{\\\\beta}=\\\\beta(u_{\\\\beta}-g)& \\\\text{in $\\\\Omega\\\\setminus\\\\Gamma$} \\\\partial_{\\\\nu} u_{\\\\beta}=0 & \\\\text{on $\\\\partial\\\\Omega\\\\cup\\\\Gamma$} \\\\end{cases} $$ is in turn discontinuous along $\\\\Gamma$ and it is the unique absolute minimizer of the non-homogeneous Mumford-Shah functional $$ \\\\int_{\\\\Omega\\\\setminus S_u}|\\\\nabla u|^2 dx +{\\\\cal H}^{n-1}(S_u)+\\\\beta\\\\int_{\\\\Omega\\\\setminus S_u}(u-g)^2 dx, $$ over $SBV(\\\\Omega)$, for $\\\\beta$ large enough. Applications of the result to the study of the gradient flow by the method of minimizing movements are shown.

Instanton algebras and quantum 4-spheres - Ludwik Dabrowski; Giovanni Landi (Differential Geom. Appl. 16 (2002) 277-284, 2002)
We study some generalized instanton algebras which are required to describe `instantonic complex rank 2 bundles\\\'. The spaces on which the bundles are defined are not prescribed from the beginning but rather are obtained from some natural requirements on the instantons. They turn out to be quantum 4-spheres $S^4_q$, with $q\\\\in\\\\IC$, and the instantons are described by self-adjoint idempotents e. We shall also clarify some issues related to the vanishing of the first Chern-Connes class $ch_1(e)$ and on the use of the second Chern-Connes class $ch_2(e)$ as a volume form.

Jacobi groups, Jacobi forms and their applications - Marco Bertola (Isomonodromic deformations and applications in physics (Montréal, QC, 2000), 2002)

On the K+P problem for a three-level quantum system: optimality implies resonance - Ugo Boscain; Thomas Chambrion; Jean-Paul Gauthier (J.Dynam. Control Systems 8 (2002),no.4, 547, 2002)

Linearized elasticity as gamma-limit of finite elasticity - Gianni Dal Maso; Matteo Negri; Danilo Percivale (Set-Valued Anal. 10 (2002), p.165-183, 2002)

Local calibrations for minimizers of the Mumford-Shah functional with a triple junction - Maria Giovanna Mora (Commun. Contemp. Math. 4 (2002) 297-326, 2002)
We prove that, if u is a function satisfying all Euler conditions for the Mumford-Shah functional and the discontinuity set of u is given by three line segments meeting at the origin with equal angles, then there exists a neighbourhood U of the origin such that u is a minimizer of the Mumford-Shah functional on U with respect to its own boundary conditions on the boundary of U. The proof is obtained by using the calibration method.

On a Lyapunov functional relating shortening curves and viscous conservation laws - Stefano Bianchini; Alberto Bressan (Nonlinear Anal. 51 (2002) 649-662, 2002)
We study a nonlinear functional which controls the area swept by a curve moving in the plane in the direction of curvature. In turn, this yields a priori estimates on solutions to a class of parabolic equations and of scalar viscous conservation laws. A further application provides an estimate on the \\\"change of shape\\\" of a BV solution to a scalar conservation law.

A model for the quasi-static growth of a brittle fracture: existence and approximation results - Gianni Dal Maso; Rodica Toader (Math. Models Methods Appl. Sci., 12 (2002), no. 12, 1773, 2002)
We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.

A model for the quasi-static growth of brittle fractures based on local minimization - Gianni Dal Maso; Rodica Toader (Math.Models Methods Appl. Sci., 12 (2002) , p.1773-1800., 2002)
We study a variant of the variational model for the quasi-static growth of brittle fractures proposed by Francfort and Marigo.9 The main feature of our model is that, in the discrete-time formulation, in each step we do not consider absolute minimizers of the energy, but, in a sense, we look for local minimizers which are sufficiently close to the approximate solution obtained in the previous step. This is done by introducing in the variational problem an additional term which penalizes the L2-distance between the approximate solutions at two consecutive times. We study the continuous-time version of this model, obtained by passing to the limit as the time step tends to zero, and show that it satisfies (for almost every time) some minimality conditions which are slightly different from those considered in Refs. 9 and 8, but are still enough to prove (under suitable regularity assumptions on the crack path) that the classical Griffith\\\'s criterion holds at the crack tips. We also prove that, if no initial crack is present and if the data of the problem are sufficiently smooth, no crack will develop in this model, provided the penalization term is large enough.

A Model for the Quasi-Static Growth of Brittle Fractures: Existence and Approximation Results - Gianni Dal Maso; Rodica Toader (Arch. Ration. Mech. Anal. 162 (2002) 101-135, 2002)
We give a precise mathematical formulation of a variational model for the irreversible quasi-static evolution of brittle fractures proposed by G.A. Francfort and J.-J. Marigo, and based on Griffith\\\'s theory of crack growth. In the two-dimensional case we prove an existence result for the quasi-static evolution and show that the total energy is an absolutely continuous function of time, although we can not exclude that the bulk energy and the surface energy may present some jump discontinuities. This existence result is proved by a time discretization process, where at each step a global energy minimization is performed, with the constraint that the new crack contains all cracks formed at the previous time steps. This procedure provides an effective way to approximate the continuous time evolution.

A multiplicity result for the Schrodinger-Maxwell equations with negative potential - Giuseppe Maria Coclite (Ann. Pol. Math. 79 (2002) 21-30, 2002)
We prove the existence of a sequence of radial solutions with negative energy of the Schrödinger-Maxwell equations under the action of a negative potential.

Multiplicity results for the Yamabe problem on Sn - Antonio Ambrosetti (Proceedings of the National Academy of Sciences of the United States of America. 2002 Nov; 99(24):15252-6, 2002)
We discuss some results related to the existence of multiple solutions for the Yamabe problem.

An optimal fast-diffusion variational method for non isochronous system - Luca Biasco; Massimiliano Berti; Philippe Bolle (SISSA Library, 2002)

Optimal stability and instability results for a class of nearly integrable Hamiltonian systems - Massimiliano Berti; Luca Biasco; Philippe Bolle (Atti.Accad.Naz.Lincei Cl.Sci.Fis.Mat.Natur.Rend.Lincei (9) Mat.Appl.13(2002),no.2,77-84, 2002)

The passage from nonconvex discrete systems to variational problems in Sobolev spaces: the one-dimensional case - Andrea Braides; Maria Stella Gelli; Mario Sigalotti (Proc. Steklov Inst. Math. 236 (2002) 395-414, 2002)

On a Poisson reduction for Gel\\\'fand-Zakharevich manifolds - Gregorio Falqui; Marco Pedroni (Rep.Math.Phys.50 (2002), no.3, 395, 2002)

Prescribing a fourth oder conformal invariant on the standard sphere - Part I: a perturbation result - Zindine Djadli; Mohameden Ould Ahmedou; Andrea Malchiodi (Commun. Contemp. Math., 2002, 4, 375, 2002)

Prescribing a fourth oder conformal invariant on the standard sphere - Part II: blow up analysis and applications - Zindine Djadli; Andrea Malchiodi; Mohameden Ould Ahmedou (Ann. Sc. Norm. Super. Pisa Cl. Sci., 2002, 1, 387, 2002)

Quantum mechanics and stochastic mechanics for compatible observables at different times - Michele Correggi; Giovanni Morchio (Ann.Physics 296 (2002), no.2, 371, 2002)

On the reachability of quantized control systems - Antonio Bicchi; Alessia Marigo; Benedetto Piccoli (IEEE Trans. Automat. Contr. 47 (2002) 546-563, 2002)
In this paper, we study control systems whose input sets are quantized, i.e., finite or regularly distributed on a mesh. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report results on the reachable set of linear quantized systems, and on a particular but interesting class of nonlinear systems, i.e., nonholonomic chained-form systems. For such systems, we provide a complete characterization of the reachable set, and, in case the set is discrete, a computable method to completely and succinctly describe its structure. Implications and open problems in the analysis and synthesis of quantized control systems are addressed.

The reachable set of a linear endogenous switching system - Claudio Altafini (Systems Control Lett. 47 (2002) 343-353, 2002)
In this work, switching systems are named endogenous when their switching pattern is controllable. Linear endogenous switching systems can be considered as a particular class of bilinear control systems. The key idea is that both types of systems are equivalent to polysystems, i.e. to systems whose flow is piecewise smooth. The reachable set of a linear endogenous switching system can be studied consequently. The main result is that, in general, it has the structure of a semigroup, even when the Lie algebra rank condition is satisfied since the logic inputs cannot reverse the direction of the flow. The adaptation of existing controllability criteria for bilinear systems is straightforward.

Relatively stable bundles over elliptic fibrations - Claudio Bartocci; Ugo Bruzzo; Daniel Hernandez Ruiperez; Jose M. Munoz Porras (Math. Nachr. 238 (2002) 23-36, 2002)
We consider a relative Fourier-Mukai transform defined on elliptic fibrations over an arbitrary normal base scheme. This is used to construct relative Atiyah sheaves and generalize Atiyah\\\'s and Tu\\\'s results about semistable sheaves over elliptic curves to the case of elliptic fibrations. Moreover we show that this transform preserves relative (semi)stability of sheaves of positive relative degree.

The scalar curvature problem on $S\\\\sp n$: an approach via Morse theory - Andrea Malchiodi (Calc. Var. Partial Differential Equations 14 (2002), no. 4, 429-445, 2002)

Singular elliptic problems with critical growth - Paolo Caldiroli; Andrea Malchiodi (Comm. Partial Differential Equations 27 (2002), no. 5-6, 847-876, 2002)

Solutions concentrating on spheres to symmetric singularly perturbed problems - Antonio Ambrosetti; Andrea Malchiodi; Wei-Ming Ni (C.R.Math.Acad.Sci. Paris 335 (2002),no.2,145-150, 2002)
We discuss some existence results concerning problems (NLS) and (N), proving the existence of radial solutions concentrating on a sphere.

Space-adiabatic Decoupling in Quantum Dynamics - Gianluca Panati (SISSA, 2002)

Space-adiabatic perturbation theory in quantum dynamics - Gianluca Panati; Herbert Spohn; Stefan Teufel (Physical review letters. 2002 Jun; 88(25 Pt 1):250405, 2002)
A systematic perturbation scheme is developed for approximate solutions to the time-dependent Schrödinger equation with a space-adiabatic Hamiltonian. For a particular isolated energy band, the basic approach is to separate kinematics from dynamics. The kinematics is defined through a subspace of the full Hilbert space for which transitions to other band subspaces are suppressed to all orders, and the dynamics operates in that subspace in terms of an effective intraband Hamiltonian. As novel applications, we discuss the Born-Oppenheimer theory to second order and derive for the first time the nonperturbative definition of the g factor of the electron within nonrelativistic quantum electrodynamics.

Stability of planar switched systems: the linear single input case - Ugo Boscain (SIAM J. Control Optim. 41 (2002), no. 1, 89-112, 2002)
We study the stability of the origin for the dynamical system $\\\\dot x(t)=u(t)Ax(t)+(1-u(t))Bx(t),$ where A and B are two 2 × 2 real matrices with eigenvalues having strictly negative real part, $x\\\\in {\\\\mbox{{\\\\bf R}}}^2$, and $u(.):[0,\\\\infty[\\\\to[0,1]$ is a completely random measurable function. More precisely, we find a (coordinates invariant) necessary and sufficient condition on A and B for the origin to be asymptotically stable for each function u(.). The result is obtained without looking for a common Lyapunov function but studying the locus in which the two vector fields Ax and Bx are collinear. There are only three relevant parameters: the first depends only on the eigenvalues of A, the second depends only on the eigenvalues of B, and the third contains the interrelation among the two systems, and it is the cross ratio of the four eigenvectors of A and B in the projective line CP1. In the space of these parameters, the shape and the convexity of the region in which there is stability are studied.

On the Stability of the Standard Riemann Semigroup - Stefano Bianchini; Rinaldo M. Colombo (P. Am. Math. Soc., 2002, 130, 1961, 2002)

On the Yamabe problem and the scalar curvature problems under boundary conditions - Antonio Ambrosetti; Li YanYan; Andrea Malchiodi (Math. Ann., 2002, 322, 667, 2002)

Adiabatic limits of closed orbits for some Newtonian systems in R-n - Andrea Malchiodi (Asymptotic Anal., 2001, 25, 149-181, 2001)
We deal with a Newtonian system like x\\\'\\\' + V\\\'(x) = 0. We suppose that V: \\\\R^n \\\\to \\\\R possesses an (n-1)-dimensional compact manifold M of critical points, and we prove the existence of arbitrarity slow periodic orbits. When the period tends to infinity these orbits, rescaled in time, converge to some closed geodesics on M.

Bihamiltonian geometry and separation of variables for Toda lattices - Gregorio Falqui; Franco Magri; Marco Pedroni (J. Nonlinear Math. Phys. 8 (2001), suppl., 118-127, 2001)

A case study in vanishing viscosity - Stefano Bianchini; Alberto Bressan (Discrete Cont. Dyn. Syst. 7 (2001) 449-476, 2001)

Complex Lagrangian embeddings of moduli spaces of vector bundles - Ugo Bruzzo; Fabio Pioli (Differential Geom. Appl. 14 (2001) 151-156, 2001)
By means of a Fourier-Mukai transform we embed moduli spaces of stable bundles on an algebraic curve C as isotropic subvarieties of moduli spaces of mu-stable bundles on the Jacobian variety J(C). When g(C)=2 this provides new examples of special Lagrangian submanifolds.

Controllability for discrete systems with a finite control set - Yacine Chitour; Benedetto Piccoli (Math. Control Signals Systems 14 (2001) 173-193, 2001)

On the Critical Behavior, the Connection Problem and the Elliptic Representation of a Painlevé VI Equation - Davide Guzzetti (Mathematical Physics, Analysis and Geometry 4: 293–377, 2001, 2001)
In this paper we find a class of solutions of the sixth Painlevé equation appearing in\r\nthe theory of WDVV equations. This class covers almost all the monodromy data associated to\r\nthe equation, except one point in the space of the data. We describe the critical behavior close to\r\nthe critical points in terms of two parameters and we find the relation among the parameters at\r\nthe different critical points (connection problem). We also study the critical behavior of Painlevé\r\ntranscendents in the elliptic representation.
Keywords: Painleve Equations, Isomonodromy deformations

Dieletric breakdown: optimal bounds - Adriana Garroni; Vincenzo Nesi; Marcello Ponsiglione (Proc. of the Royal Society of London Series A-Mathematical Physical and Engineering Sciences 457 (2001): p. 2317-2335, OCT. 8, 2001, 2001)

Dirac operator on the standard Podles quantum sphere - Ludwik Dabrowski; Andrzej Sitarz (Noncommutative geometry and quantum groups (Warsaw 2001),49,Banach Center Publ., 61, Polish Acad.Sci., Warsaw,2003, 2001)

Existence and nonexistence results for a class of nonlinear, singular Sturm-Liouville equations - Paolo Caldiroli; Roberta Musina (Adv. Differential Equations 6 (2001), no. 3, 303-326, 2001)

Extremal synthesis for generic planar systems - Ugo Boscain; Benedetto Piccoli (J. Dynam. Control Systems, 2001, 7, 209, 2001)

Finite Difference Approximation of Free Discontinuity Problems - Massimo Gobbino; Maria Giovanna Mora (Proc. Royal Soc. Edinb. Ser. A 131 (2001), no. 3, 567-595, 2001)

A Fourier transform for sheaves on real tori. I. The equivalence Sky(T)~ Loc (T) - Ugo Bruzzo; Giovanni Marelli; Fabio Pioli (J. Geom. Phys. 39 (2001), no. 2, 174--182, 2001)

Free-discontinuity problems: calibration and approximation of solutions - Massimiliano Morini (SISSA, 2001)
Keywords: Calibration of solutions

Gamma-limit of periodic obstacles - Gianni Dal Maso; Paola Trebeschi (Acta Appl. Math., 2001, 65, 207-215, 2001)
We compute the Gamma-limit of a sequence obstacle functionals in the case of periodic obstacles.

A Glimm type functional for a special Jin-Xin relaxation model - Stefano Bianchini (Ann. Inst. H. Poincare\\\' Anal. Non Lineaire 18 (2001), no. 1, 19-42, 2001)

Global continuous Riemann solver for nonlinear elasticity - Jean-Marc Mercier; Benedetto Piccoli (Arch. Ration. Mech. An., 2001, 156, 89, 2001)

Instantons on the Quantum 4-Spheres S^4_q - Ludwik Dabrowski; Giovanni Landi; Tetsuya Masuda (Comm. Math. Phys. 221 (2001) 161-168, 2001)
We introduce noncommutative algebras $A_q$ of quantum 4-spheres $S^4_q$, with $q\\\\in\\\\IR$, defined via a suspension of the quantum group $SU_q(2)$, and a quantum instanton bundle described by a selfadjoint idempotent $e\\\\in \\\\Mat_4(A_q)$, $e^2=e=e^*$. Contrary to what happens for the classical case or for the noncommutative instanton constructed in Connes-Landi, the first Chern-Connes class $ch_1(e)$ does not vanish thus signaling a dimension drop. The second Chern-Connes class $ch_2(e)$ does not vanish as well and the couple $(ch_1(e), ch_2(e))$ defines a cycle in the $(b,B)$ bicomplex of cyclic homology.

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds - Davide Guzzetti (Mathematical Physics, Analysis and Geometry 4: 245–291, 2001, 2001)
We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we\r\nexplicitly compute a parametric form of the solutions of theWDVV equations in terms of Painlevé VI\r\ntranscendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric\r\nform to explicitly construct polynomial and algebraic solutions and to derive the generating\r\nfunction of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective\r\nspace. The procedure is a relevant application of the theory of isomonodromic deformations.
Keywords: Frobenius Manifolds, Painleve Equations, Isomonodromy deformations

Lax representation and Poisson geometry of the Kowalevski top - Gregorio Falqui (J. Phys. A 34 (2001) 2077-2085, 2001)
We discuss the Poisson structure underlying the two-field Kowalevski gyrostat and the Kowalevski top. We start from their Lax structure and construct a suitable pencil of Poisson brackets which endows these systems with the structure of bi-Hamiltonian completely integrable systems. We study the Casimir functions of such pencils, and show how it is possible to frame the Kowalevski systems within the so-called Gel\\\'fand-Zakharevich bi-Hamiltonian setting for integrable systems.

Lie triple systems and warped products - Marco Bertola; Gouthier, D. (Rend. Mat. Appl. (7), 2001)

Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set - Maria Giovanna Mora; Massimiliano Morini (Ann. I. H. Poincare - An., 2001, 18, 403, 2001)

A monotonicity approach to nonlinear Dirichlet problems in perforated domains - Gianni Dal Maso; Igor V. Skrypnik (Adv. Math. Sci. Appl. 11 (2001) 721-751, 2001)
We study the asymptotic behaviour of solutions to Dirichlet problems in perforated domains for nonlinear elliptic equations associated with monotone operators. The main difference with respect to the previous papers on this subject is that no uniformity is assumed in the monotonicity condition. Under a very general hypothesis on the holes of the domains, we construct a limit equation, which is satisfied by the weak limits of the solutions. The additional term in the limit problem depends only on the local behaviour of the holes, which can be expressed in terms of suitable nonlinear capacities associated with the monotone operator.

Morse properties for the minimum time function on 2-D manifolds - Ugo Boscain; Benedetto Piccoli (J. Dynam. Control Systems 7 (2001), no. 3, 385--423, 2001)

On the Multi-Instanton Measure for Super Yang-Mills Theories - Ugo Bruzzo; Francesco Fucito; Alessandro Tanzini; Gabriele Travaglini (Nuclear Phys. B 611 (2001), no. 1-3, 205--226., 2001)

Multiple positive solutions of some elliptic equations in \\\\bold R\\\\sp N - Andrea Malchiodi (Nonlinear Anal. 43 (2001) 159-172, 2001)

Multiplicity results for some nonlinear Schrodinger equations with potentials - Antonio Ambrosetti; Andrea Malchiodi; Simone Secchi (Arch. Ration. Mech. An., 2001, 159, 253, 2001)

Non-compactness and multiplicity results for the Yamabe problem on Sn - Massimiliano Berti; Andrea Malchiodi (J. Funct. Anal. 180 (2001) 210-241, 2001)

A note on the super Krichever map - Gregorio Falqui; Cesare Reina; Alessandro Zampa (J. Geom. Phys. 37 (2001), no. 1-2, 169-181, 2001)
We consider the geometrical aspects of the Krichever map in the context of Jacobian Super KP hierarchy. We use the representation of the hierarchy based\\non the Fa`a di Bruno recursion relations, considered as the cocycle condition for the natural double complex associated with the deformations of super Krichever data. Our approach is based on the construction of the universal super divisor (of degree g), and a local universal family of geometric data which give the map into the Super Grassmannian.

Numerical Methods for Free-Discontinuity Problems Based on Approximations by Γ-Convergence - Matteo Negri (SISSA, 2001)
Keywords: Mumford-Shah functional

Numerical minimization of the Mumford-Shah functional - Matteo Negri; Maurizio Paolini (Calcolo, 2001, 38, 67, 2001)

Picard and Chazy solutions to the Painlevé VI equation - Marta Mazzocco (Math. Ann. 321 (2001) 157-195, 2001)
I study the solutions of a particular family of Painlevé VI equations with the parameters $β=γ=0, δ=1/2$ and $2α=(2μ-1)^2$, for $2μ\in\mathbb{Z}$. I show that the case of half-integer $μ$ is integrable and that the solutions are of two types: the so-called Picard solutions and the so-called Chazy solutions. I give explicit formulae for them and completely determine their asymptotic behaviour near the singular points $0,1,∞$ and their nonlinear monodromy. I study the structure of analytic continuation of the solutions to the PVI$μ$ equation for any $μ$ such that $2μ\in\mathbb{Z}$. As an application, I classify all the algebraic solutions. For $μ$ half-integer, I show that they are in one to one correspondence with regular polygons or star-polygons in the plane. For $μ$ integer, I show that all algebraic solutions belong to a one-parameter family of rational solutions.

S^2 type parametric surfaces with prescribed mean curvature and minimal energy - Paolo Caldiroli; Roberta Musina (Nonlinear equations : methods, models and applications (Bergamo, 2001) / Daniela Lupo, Carlo D. Pagani, Bernhard Ruf, editors. - Basel : Birkhäuser, 2003. - (Progress in nonlinear differential equations and their applications; 54). - p. 61-77, 2001)

On the spreading of characteristics for non-convex conservation laws - Helge Kristian Jenssen; Carlo Sinestrari (Proc. Roy. Soc. Edinburgh Sect. A 131 (2001) 909-925, 2001)
We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the flux function has one point of inflection. It is well known that in the convex case the characteristic speed satisfies a one-sided Lipschitz estimate. Using Dafermos\\\' theory of generalized characteristics, we show that the characteristic speed in the non-convex case satisfies an Hölder estimate. In addition, we give a one-sided Lipschitz estimate with an error term given by the decrease of the total variation of the solution.

Stability of L-infinity solutions for hyperbolic systems with coinciding shocks and rarefactions - Stefano Bianchini (Siam J. Math. Anal., 2001, 33, 959, 2001)
We consider a hyperbolic system of conservation laws u_t + f(u)_x = 0 and u(0,\\\\cdot) = u_0, where each characteristic field is either linearly degenerate or genuinely nonlinear. Under the assumption of coinciding shock and rarefaction curves and the existence of a set of Riemann coordinates $w$, we prove that there exists a semigroup of solutions $u(t) = \\\\mathcal{S}_t u_0$, defined on initial data $u_0 \\\\in L^\\\\infty$. The semigroup $\\\\mathcal{S}$ is continuous w.r.t. time and the initial data $u_0$ in the $L^1_{\\\\text{loc}}$ topology. Moreover $\\\\mathcal{S}$ is unique and its trajectories are obtained as limits of wave front tracking approximations.

Stationary states for a two-dimensional singular Schrodinger equation - Paolo Caldiroli; Roberta Musina (Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 4 (2001), no. 3, 609-633., 2001)

On the subanalyticity of Carnot-Caratheodory distances - Andrei A. Agrachev; Jean-Paul Gauthier (Ann. I. H. Poincare - An., 2001, 18, 359, 2001)

On the symmetric scalar curvature problem on S\\\\sp n - Antonio Ambrosetti; Andrea Malchiodi (J. Differential Equations 170 (2001) 228-245, 2001)
We discuss some existence results dealing with the scalar curvature problem on S\\\\sp n in the presence of various symmetries.

Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems - Paolo Baiti; Philippe G. LeFloch; Benedetto Piccoli (J. Differential Equations 172 (2001) 59-82, 2001)

Uniqueness of solutions to Hamilton-Jacobi equations arising in the Calculus of Variations - Gianni Dal Maso; Helene Frankowska (Optimal control and partial differential equations : in honour of professor Alain Bensoussan\\\'s 60th birthday / edited by José Luis Menaldi, Edmundo Rofman, and Agnès Sulem.,Amsterdam : IOS Press, 2001, p. 335-345, 2001)
We prove the uniqueness of the viscosity solution to the Hamilton-Jacobi equation associated with a Bolza problem of the Calculus of Variations, assuming that the Lagrangian is autonomous, continuous, superlinear, and satisfies the usual convexity hypothesis. Under the same assumptions we prove also the uniqueness, in a class of lower semicontinuous functions, of a slightly different notion of solution, where classical derivatives are replaced only by subdifferentials. These results follow from a new comparison theorem for lower semicontinuous viscosity supersolutions of the Hamilton-Jacobi equation, that is proved in the general case of lower semicontinuous Lagrangians.

Warped products with special Riemannian curvature - Marco Bertola; Gouthier, Daniele (Bol. Soc. Brasil. Mat. (N.S.), 2001)

3D superconformal theories from Sasakian seven-manifolds: new nontrivial evidences for AdS_4/CFT_3 - Davide Fabbri; Pietro Fré; Leonardo Gualtieri; Cesare Reina; Alessandro Tomasiello; Alberto Zaffaroni; Alessandro Zampa (Nucl.Phys. B577 (2000) 547-608, 2000)

Abnormal extremals for minimum time on the plane - Ugo Boscain; Benedetto Piccoli (SISSA Library, 2000)

Arnold's Diffusion in nearly integrable isochronous Hamiltonian systems - Massimiliano Berti; Philippe Bolle (SISSA Library, 2000)

A(SLq(2)) at roots of unity is a free module over A(SL(2)) - Ludwik Dabrowski; Cesare Reina; Alessandro Zampa (Lett. Math. Phys., 2000, 52, 339, 2000)

A bi-Hamiltonian theory for stationary KDV flows and their separability - Gregorio Falqui; Franco Magri; Marco Pedroni; Jorge P. Zubelli (Regul. Chaotic Dyn. 5 (2000), no. 1, 33-52, 2000)

BV estimates for multicomponent chromatography with relaxation - Alberto Bressan; Wen Shen (Discrete Contin. Dynam. Systems 6 (2000) 21-38, 2000)
We consider the Cauchy problem for a system of $2n$ balance laws which arises from the modelling of multi-component chromatography: $$\\\\left\\\\{ \\\\eqalign{u_t+u_x&=-{1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr v_t&={1\\\\over\\\\ve}\\\\big( F(u)-v\\\\big),\\\\cr}\\\\right. \\\\eqno(1)$$ This model describes a liquid flowing with unit speed over a solid bed. Several chemical substances are partly dissolved in the liquid, partly deposited on the solid bed. Their concentrations are represented respectively by the vectors $u=(u_1,\\\\ldots,u_n)$ and $v=(v_1,\\\\ldots,v_n)$. We show that, if the initial data have small total variation, then the solution of (1) remains with small variation for all times $t\\\\geq 0$. Moreover, using the $\\\\L^1$ distance, this solution depends Lipschitz continuously on the initial data, with a Lipschitz constant uniform w.r.t.~$\\\\ve$. Finally we prove that as $\\\\ve\\\\to 0$, the solutions of (1) converge to a limit described by the system $$\\\\big(u+F(u)\\\\big)_t+u_x=0,\\\\qquad\\\\qquad v=F(u).\\\\eqno(2)$$ The proof of the uniform BV estimates relies on the application of probabilistic techniques. It is shown that the components of the gradients $v_x,u_x$ can be interpreted as densities of random particles travelling with speed 0 or 1. The amount of coupling between different components is estimated in terms of the expected number of crossing of these random particles. This provides a first example where BV estimates are proved for general solutions to a class of $2n\\\\times 2n$ systems with relaxation.

BV solutions for a class of viscous hyperbolic systems - Stefano Bianchini; Alberto Bressan (Indiana Univ. Math. J. 49 (2000) 1673-1714, 2000)

The Calibration Method for Free Discontinuity Problems - Gianni Dal Maso (European Congress of Mathematics. Volume I : Barcelona, July 10-14, 2000 / Carles Casacuberta ... [et al.], editors. , Boston : Birkhauser, 2001, p. 317-326., 2000)
The calibration method is used to identify some minimizers of the Mumford-Shah functional. The method is then extended to more general free discontinuity problems.

On the convergence of Godunov scheme for nonlinear hyperbolic systems - Alberto Bressan; Helge Kristian Jenssen (Chinese Ann. Math. B, 2000, 21, 269, 2000)

Decomposing quantum fields on branes - Marco Bertola; Bros, Jacques; Gorini, Vittorio; Moschella, Ugo; Schaeffer, Richard (Nuclear Phys. B, 2000)

Diffusion time and splitting of separatrices for nearly integrable - Massimiliano Berti; Philippe Bolle (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 2000, 11, 235, 2000)

An elementary approach to the polynomial $\\\\tau$-functions of the KP Hierarchy - Gregorio Falqui; Franco Magri; Marco Pedroni; Jorge P. Zubelli (Theor. Math. Phys. 122 (2000) 17-28, 2000)
We give an elementary construction of the solutions of the KP hierarchy associated with polynomial τ-functions starting with a geometric approach to soliton equations based on the concept of a bi-Hamiltonian system. As a consequence, we establish a Wronskian formula for the polynomial τ-functions of the KP hierarchy. This formula, known in the literature, is obtained very directly.

Elliptic variational problems in $ R\\\\sp N$ with critical growth - Antonio Ambrosetti; Jesus Garcia Azorero; Ireneo Peral (J. Differential Equations 168 (2000), no. 1, 10--32, 2000)

Existence and multiplicity results for some nonlinear elliptic equations: a survey. - Antonio Ambrosetti; Jesus Garcia Azorero; Ireneo Peral (Rend. Mat. Appl., 2000, 20, 167, 2000)

Existence and multiplicity results for some problems in Riemannian geometry - Andrea Malchiodi (SISSA, 2000)
Keywords: Yamabe problem

Frobenius manifold structure on orbit space of Jacobi groups. I - Marco Bertola (Differential Geom. Appl., 2000)

Frobenius manifold structure on orbit space of Jacobi groups. II - Marco Bertola (Differential Geom. Appl., 2000)

Functionals depending on curvatures with constraints - Maria Giovanna Mora; Massimiliano Morini (Rend. Sem. Mat. Univ. Padova 104 (2000), 173--199, 2000)

A general construction of conformal field theories from scalar anti-de Sitter quantum field theories - Marco Bertola; Bros, Jacques; Moschella, Ugo; Schaeffer, Richard (Nuclear Phys. B, 2000)

High-order Averaging and Stability of Time-Varying Systems - Andrey Sarychev (SISSA Library, 2000)

Inverse problem for Semisimple Frobenius Manifolds Monodromy Data and the Painlevé VI Equation - Davide Guzzetti (SISSA Library, 2000)

Local calibrations for minimizers of the Mumford-Shah functional with rectilinear discontinuity sets - Gianni Dal Maso; Maria Giovanna Mora; Massimiliano Morini (J. Math. Pures Appl. 79, 2 (2000) 141-162, 2000)

Minimization of functionals of the gradient by Baire's theorem - Sandro Zagatti (SIAM J. Control Optim. 38 (2000) 384-399, 2000)
We give sufficient conditions for the existence of solutions of the minimum problem $$ {\mathcal{P}}_{u_0}: \qquad \hbox{Minimize}\quad \int_Ω g(Du(x))dx, \quad u\in u_0 + W_0^{1,p}(Ω,{\mathbb{R}}), $$ based on the structure of the epigraph of the lower convex envelope of g, which is assumed be lower semicontinuous and to grow at infinity faster than the power p with p larger than the dimension of the space. No convexity conditions are required on g, and no assumptions are made on the boundary datum $u_0\in W_0^{1,p}(Ω,\mathbb{R})$.

Monodromy of certain Painlevé-VI transcendents and reflection groups - Boris Dubrovin; Marta Mazzocco (Invent. Math. 141 (2000) 55-147, 2000)
We study the global analytic properties of the solutions of a particular family of Painleve\\\' VI equations with the parameters $\\\\beta=\\\\gamma=0$, $\\\\delta={1\\\\over2}$ and $\\\\alpha$ arbitrary. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. This result is used to classify all the algebraic solutions of our Painleve\\\' VI equation.

A note on the scalar curvature problem in the presence of symmetries - Antonio Ambrosetti; Li YanYan; Andrea Malchiodi (Ricerche Mat. 49 (2000), suppl., 169-176, 2000)

Principal invariants of Jacobi curves - Andrei A. Agrachev; Igor Zelenko (Nonlinear control in the Year 2000 / Alberto Isidori, Francoise Lamnabhi-Lagarrigue, Witold Respondek (eds.) - Springer : Berlin, 2001. - (Lecture notes in control and information sciences ; 258). - ISBN 1-85233-363-4 (v. 1). - p. 9-22., 2000)
Jacobi curves are far going generalizations of the spaces of \\\"Jacobi fields\\\" along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. Differential geometry of these curves provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. In the present paper we mainly discuss two principal invariants: the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmanian providing the curve with a natural projective structure, and a fundamental form, which is a 4-oder differential on the curve.

Quantized control systems and discrete nonholonomy - Alessia Marigo; Benedetto Piccoli; Antonio Bicchi (Lagrangian and Hamiltonian Methods for Nonlinear Control : a proc. volume from the IFAC Workshop. Princeton, New Jersey, 16-18 March 2000 / ed. by N.E. Leonard, R. Ortega. - Oxford : Pergamon, 2000, 2000)

Reachability Analysis for a Class of Quantized Control Systems - Alessia Marigo; Benedetto Piccoli; Antonio Bicchi (Proc. 39th IEEE Int. Conf. on Decision and Control 4 (2000) 3963-3968, 2000)
We study control systems whose input sets are quantized. We specifically focus on problems relating to the structure of the reachable set of such systems, which may turn out to be either dense or discrete. We report on some results on the reachable set of linear quantized systems, and study in detail an interesting class of nonlinear systems, forming the discrete counterpart of driftless nonholonomic continuous systems. For such systems, we provide a complete characterization of the reachable set, and, in the case the set is discrete, a computable method to describe its lattice structure.

Reduction of bi-Hamiltonian systems and separation of variables: an example from the Boussinesq hierarchy - Gregorio Falqui; Franco Magri; G. Tondo (Theor. Math. Phys. 122 (2000) 176-192, 2000)
We discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel\\\'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.

Regular Synthesis and Sufficiency Conditions for Optimality - Benedetto Piccoli; Hector J. Sussmann (SIAM J. Control Optim. 39 (2000) 359-410, 2000)
We propose a definition of \\\"regular synthesis\\\" that is more general than those suggested by other authors such as Boltyanskii and Brunovsky, and an even more general notion of \\\"regular presynthesis.\\\" We give a complete proof of the corresponding sufficiency theorem, a slightly weaker version of which had been stated in an earlier article, with only a rough outline of the proof. We illustrate the strength of our result by showing that the optimal synthesis for the famous Fuller problem satisfies our hypotheses. We also compare our concept of synthesis with the simpler notion of a \\\"family of solutions of the closed-loop equation arising from an optimal feedback law,\\\" and show by means of examples why the latter is inadequate, and why the difficulty cannot be resolved byusing other concepts of solution--such as Filippov solutions, or the limits of sample-and-hold solutions recently proposed as feedback solutions by Clarke, Ledyaev, Subbotin and Sontag -for equations with a non-Lipschitz and possibly discontinuous right-hand side.

A Remark on One-Dimensional Many-Body Problems with Point Interactions - Sergio Albeverio; Ludwik Dabrowski; Shao-Ming Fei (Int. J. Mod. Phys. B 14 (2000) 721-727, 2000)
The integrability of one dimensional quantum mechanical many-body problems with general contact interactions is extensively studied. It is shown that besides the pure (repulsive or attractive) $\\\\delta$-function interaction there is another singular point interactions which gives rise to a new one-parameter family of integrable quantum mechanical many-body systems. The bound states and scattering matrices are calculated for both bosonic and fermionic statistics.

Scalar curvature under boundary conditions - Antonio Ambrosetti; Li YanYan; Andrea Malchiodi (Cr. Acad. Sci. I-Math, 2000, 330, 1013, 2000)

The semigroup generated by a Temple class system with non-convex flux function - Stefano Bianchini (Differential Integral Equations 13 (2000) 1529-1550, 2000)
We consider the Cauchy problem for a nonlinear n × n system of conservation laws of Temple class, i.e. with coinciding shock and rarefaction curves and with a coordinate system made of Riemann invariants. Without any assumption on the convexity of the flux function, we prove the existence of a semigroup made of weak solutions of the equations and depending Lipschitz continuously on the initial data with bounded total variation.

On the shift differentiability of the flow generated by a hyperbolic system of conservation laws - Stefano Bianchini (Discrete Contin. Dynam. Systems 6 (2000), no. 2, 329-350, 2000)

Some Properties of Non-linear sigma-Models in Noncommutative Geometry - Ludwik Dabrowski; Thomas Krajewski; Giovanni Landi (Int. J. Mod. Phys. B 14 (2000) 2367-2382, 2000)
We introduce non-linear $\\\\sigma$-models in the framework of noncommutative geometry with special emphasis on models defined on the noncommutative torus. We choose as target spaces the two point space and the circle and illustrate some characteristic features of the corresponding $\\\\sigma$-models. In particular we construct a $\\\\sigma$-model instanton with topological charge equal to 1. We also define and investigate some properties of a noncommutative analogue of the Wess-Zumino-Witten model.

Stability of L^infty Solutions of Temple Class Systems - Alberto Bressan; Paola Goatin (Differential Integral Equations 13 (2000) 1503-1528, 2000)
Let $u_t+f(u)_x=0$ be a strictly hyperbolic, genuinely nonlinear system of conservation laws of Temple class. In this paper, a continuous semigroup of solutions is constructed on a domain of $L^∞$ functions, with possibly unbounded variation. Trajectories depend Lipschitz continuously on the initial data, in the $L^1$ distance. Moreover, we show that a weak solution of the Cauchy problem coincides with the corresponding semigroup trajectory if and only if it satisfies an entropy condition of Oleinik type, concerning the decay of positive waves.

On a Steffen\\\'s result about parametric surfaces with prescribed mean curvature - Roberta Musina; Paolo Caldiroli (SISSA Library, 2000)