A Hopf bundle over a quantum four-sphere from the symplectic group - Giovanni Landi; Chiara Pagani; Cesare Reina (Commun. Math. Phys. 263 (2006) 65-88, 2006)
We construct a quantum version of the SU(2) Hopf bundle $S^7 \\\\to S^4$. The quantum sphere $S^7_q$ arises from the symplectic group $Sp_q(2)$ and a quantum 4-sphere $S^4_q$ is obtained via a suitable self-adjoint idempotent $p$ whose entries generate the algebra $A(S^4_q)$ of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere $S^4$. We compute the fundamental $K$-homology class of $S^4_q$ and pair it with the class of $p$ in the $K$-theory getting the value -1 for the topological charge. There is a right coaction of $SU_q(2)$ on $S^7_q$ such that the algebra $A(S^7_q)$ is a non trivial quantum principal bundle over $A(S^4_q)$ with structure quantum group $A(SU_q(2))$.

Infinite Horizon Noncooperative Differential Games - Alberto Bressan; Fabio Simone Priuli (J. Differential Equations 227 (2006) 230-257, 2006)
For a non-cooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite-horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value\\nfunctions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability.

An instability of the Godunov scheme - Alberto Bressan; Helge Kristian Jenssen; Paolo Baiti (Comm. Pure Appl. Math. 59 (2006) 1604-1638, 2006)
We construct a solution to a $2\\\\times 2$ strictly hyperbolic system of conservation laws, showing that the Godunov scheme \\\\cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or $L^1$ stability estimates can in general be valid for finite difference schemes.

Large Parameter Behavior of Equilibrium Measures - Tamara Grava; Fei-Ran Tian (Commun. Math. Sci. 4 (2006) 551-573, 2006)
We study the equilibrium measure for a logarithmic potential in the presence of an external field V*(x) + tp(x), where t is a parameter, V*(x) is a smooth function and p(x) a monic polynomial. When p(x) is of an odd degree, the equilibrium measure is shown to be supported on a single interval as |t| is sufficiently large. When p(x) is of an even degree, the equilibrium measure is supported on two disjoint intervals as t is negatively large; it is supported on a single interval for convex p(x) as t is positively large and is likely to be supported on multiple disjoint intervals for non-convex p(x).

Local Index Formula on the Equatorial Podles Sphere - Francesco D'Andrea; Ludwik Dabrowski (Lett. Math. Phys. 75 (2006) 235-254, 2006)
We discuss spectral properties of the equatorial Podles sphere. As a preparation we also study the `degenerate\\\' (i.e. $q=0$) case (related to the quantum disk). We consider two different spectral triples: one related to the Fock representation of the Toeplitz algebra and the isopectral one....

Matching Procedure for the Sixth Painlevé Equation (May 2006) - Davide Guzzetti (Journal of Physics A: Mathematical and General, Volume 39, Issue 39, 29 September 2006, Article numberS02, Pages 11973-12031, 2006)
We present a constructive procedure to obtain the critical behavior of\r\nPainleve\' VI transcendents and solve the connection problem. This procedure\r\nyields two and one parameter families of solutions, including trigonometric and\r\nlogarithmic behaviors, and three classes of solutions with Taylor expansion at\r\na critical point.
This paper appeared in May 2006. I put it on the archive now, with more that four years of delay, for completeness sake. The paper is published in J.Phys.A: Math.Gen. 39 (2006), 11973-12031, with some modifications.

N=1 superpotentials from multi-instanton calculus - Francesco Fucito; Jose F. Morales; Rubik Poghossian; Alessandro Tanzini (JHEP01(2006)031, 2006)
In this paper we compute gaugino and scalar condensates in N = 1 supersymmetric gauge\\ntheories with and without massive adjoint matter, using localization formulae over the multi-instanton moduli space. Furthermore we compute the chiral ring relations among the correlators of the N = 1* theory and check this result against the multi-instanton computation finding agreement.

Normal bundles to Laufer rational curves in local Calabi-Yau threefolds - Ugo Bruzzo; Antonio Ricco (Lett. Math. Phys. 76 (2006) 57-63, 2006)
We prove a conjecture by F. Ferrari. Let X be the total space of a nonlinear deformation of a rank 2 holomorphic vector bundle on a smooth rational curve, such that X has trivial canonical bundle and has sections. Then the normal bundle to such sections is computed in terms of the rank of the Hessian of a suitably defined superpotential at its critical points.

On Palais-Smale sequences for H-systems: some examples - Paolo Caldiroli; Roberta Musina (Adv. Differential Equations 11 (2006) 931-960, 2006)
We exhibit a series of examples of Palais-Smale sequences for the Dirichlet problem associated to the mean curvature equation with null boundary condition, and we show that in the case of nonconstant mean curvature functions different kinds of blow up phenomena appear and Palais-Smale sequences may have quite wild behaviour.

The PDEs of biorthogonal polynomials arising in the two-matrix model - Marco Bertola; B. Eynard (Math. Phys. Anal. Geom., 2006)

Periodic solutions of nonlinear wave equations for asymptotically full measure sets of frequencies - P Baldi; Massimiliano Berti (Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni, 2006)
We prove existence and multiplicity of small amplitude periodic solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for asymptotically full measure sets of frequencies, extending the results of [7] to new types of nonlinearities.
cited By (since 1996)5

Q-curvature flow on S^4 - Andrea Malchiodi; Michael Struwe (J. Differential Geom. 73 (2006) 1-44, 2006)

Quantisation of bending flows - Gregorio Falqui; Fabio Musso (Czechoslovak Journal of Physics 56 (2006), n. 10-11, 1143-1148, 2006)
We briefly review the Kapovich-Millson notion of Bending flows as an integrable system on the space of polygons in ${\\\\bf R}^3$, its connection with a specific Gaudin XXX system, as well as the generalisation to $su(r), r>2$. Then we consider the quantisation problem of the set of Hamiltonians pertaining to the problem, quite naturally called Bending Hamiltonians, and prove that their commutativity is preserved at the quantum level.

Quasi-periodic solutions of completely resonant forced wave equations - Massimiliano Berti; Procesi, Michela (Comm. Partial Differential Equations, 2006)

Quasistatic evolution problems for linearly elastic-perfectly plastic materials - Gianni Dal Maso; Antonio DeSimone; Maria Giovanna Mora (Arch. Ration. Mech. Anal. 180 (2006) 237-291, 2006)
The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation. This provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain.

Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials - Antonio Ambrosetti; David Ruiz (Proc. Roy. Soc. Edinburgh Sect. A 136 (2006) 889-907, 2006)
We prove the existence of radial solutions of 1.2) concentrating at a sphere for potentials which might be zero and might decay to zero at\\r\\ninfinity. The proofs use a perturbation technique in a variational setting, through a Lyapunov-Schmidt reduction.

Recent analytical developments in micromagnetics - Antonio DeSimone; Robert V. Kohn; Stefan Müller; Felix Otto (The science of hysteresis / eds. Giorgio Bertotti, Isaak D. Mayergoyz. - Amsterdam: Elsevier, 2006. Vol.2, 269-381., 2006)

Reflection symmetries for multiqubit density operators - Claudio Altafini; Timothy F. Havel (J. Math. Phys. 47 (2006) 032104, 2006)
For multiqubit density operators in a suitable tensorial basis, we show that a number of nonunitary operations used in the detection and synthesis of entanglement are classifiable as reflection symmetries, i.e., orientation changing rotations. While one-qubit reflections correspond to antiunitary symmetries, as is known for example from the partial transposition criterion, reflections on the joint density of two or more qubits are not accounted for by the Wigner Theorem and are well-posed only for sufficiently mixed states. One example of such nonlocal reflections is the unconditional NOT operation on a multiparty density, i.e., an operation yelding another density and such that the sum of the two is the identity operator. This nonphysical operation is admissible only for sufficiently mixed states.

Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions - Marco Bertola; B. Eynard; Harnad, J. (Comm. Math. Phys., 2006)

Semistability vs. nefness for (Higgs) vector bundles - Ugo Bruzzo; Daniel Hernandez Ruiperez (Differential Geom. Appl. 24 (2006) 403-416, 2006)
According to Miyaoka, a vector bundle E on a smooth projective curve is semistable if and only if a certain numerical class in the projectivized bundle PE is nef. We establish a similar criterion for the semistability of Higgs bundles: namely, such a bundle is semistable if and only if for every integer s between 0 and the rank of E, a suitable numerical class in the scheme parametrizing the rank s locally-free Higgs quotients of E is nef. We also extend this result to higher-dimensional complex projective varieties by showing that the nefness of the above mentioned classes is equivalent to the semistability of the Higgs bundle E together with the vanishing of the discriminant of E.

On Separation of Variables for Homogeneous SL(r) Gaudin Systems - Gregorio Falqui; Fabio Musso (Math. Phys. Anal. Geom. 9 (2006), n. 3, 233-262 (2007), 2006)
By means of a recently introduced bihamiltonian structure for the homogeneous Gaudin models, we find a new set of Separation Coordinates for the sl(r) case.

Spectral geometry of k-Minkowski space - Francesco D'Andrea (J. Math. Phys. 47 (2006) 062105, 2006)
After recalling Snyder's idea of using vector fields over a smooth manifold as "coordinates on a noncommutative space", we discuss a two dimensional toy-model whose "dual" noncommutative coordinates form a Lie algebra: this is the well known $κ$-Minkowski space. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of $κ$-Minkowski as linear operators on an Hilbert space study its "spectral properties" and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of M. Dimitrijevic et al. can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.

Stability of planar nonlinear switched systems - Ugo Boscain; Grégoire Charlot; Mario Sigalotti (Discrete Contin. Dyn. Syst. 15 (2006) 415-432, 2006)
We consider the time-dependent nonlinear system ˙ q(t) = u(t)X(q(t)) + (1 − u(t))Y (q(t)), where q ∈ R2, X and Y are two smooth vector fields, globally asymptotically stable at the origin and u : [0,∞) → {0, 1} is an arbitrary measurable function. Analysing the topology of the set where X and Y are parallel, we give some sufficient and some necessary conditions for global asymptotic stability, uniform with respect to u(.). Such conditions can be verified without any integration or construction of a Lyapunov function, and they are robust under small perturbations of the vector fields.

Thomae type formulae for singular Z_N curves - Victor Z. Enolski; Tamara Grava (Lett. Math. Phys. 76 (2006) 187-214, 2006)
We give an elementary and rigorous proof of the Thomae type formula for singular $Z_N$ curves. To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szego kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof inherits principal points of Nakayashiki\\\'s proof [31], obtained for non-singular ZN curves.

Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic field - Ugo Boscain; Paolo Mason (Journal of Mathematical Physics 47, 062101 (2006), 2006)
In this paper we consider the minimum time population transfer problem for the z-component\\nof the spin of a (spin 1/2) particle driven by a magnetic field, controlled along the x axis, with bounded amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds. Let (-E,E) be the two energy levels, and |omega (t)| ≤ M the bound on the field amplitude. For each couple of values E and M, we determine the time optimal synthesis starting from the level -E and we provide the explicit expression of the time optimal trajectories steering the state one to the state two, in terms of a parameter that can be computed solving numerically a suitable equation. For M/E << 1, every time optimal trajectory is bang-bang and in particular the corresponding control is periodic with frequency of the order of the resonance frequency wR = 2E. On the other side, for M/E > 1, the time optimal trajectory steering the state one to the state two is bang-bang with exactly one switching. Fixed E we also prove that for M → ∞ the time needed to reach the state two tends to zero. In the case M/E > 1 there are time optimal trajectories containing a singular arc. Finally we compare these results with some known results of Khaneja, Brockett and Glaser and with those obtained by controlling the magnetic field both on the x and y directions (or with one external field, but in the rotating wave approximation). As byproduct we prove that the qualitative shape of the time optimal synthesis presents different patterns, that cyclically alternate as M/E → 0, giving a partial proof of a conjecture formulated in a previous paper.

On topological M-theory - Giulio Bonelli; Alessandro Tanzini; Maxim Zabzine (Adv. Theor. Math. Phys. 10 (2006) 239-260, 2006)
We construct a gauge fixed action for topological membranes on G2-manifolds such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds.

Topological symmetry of forms, N=1 supersymmetry and S-duality on special manifolds - Laurent Baulieu; Alessandro Tanzini (J. Geom. Phys. 56 (2006) 2379-2401, 2006)
We study the quantization of a holomorphic two-form coupled to a Yang-Mills field on special manifolds in various dimensions, and we show that it yields twisted supersymmetric theories. The construction determines ATQFT\\\'s (Almost Topological Quantum Field Theories), that is, theories with observables that are invariant under changes of metrics belonging to restricted classes. For Kahler manifolds in four dimensions, our topological model is related to N=1 Super Yang-Mills theory. Extended supersymmetries are recovered by considering the coupling with chiral multiplets. We also analyse Calabi-Yau manifolds in six and eight dimensions, and seven dimensional G_2 manifolds of the kind recently discussed by Hitchin. We argue that the two-form field could play an interesting role for the study of the conjectured S-duality in topological string. We finally show that in the case of real forms in six dimensions the partition function of our topological model is related to the square of that of the holomorphic Chern-Simons theory, and we discuss the uplift to seven dimensions and its relation with the recent proposals for the topological M-theory.

Two-matrix model with semiclassical potentials and extended Whitham hierarchy - Marco Bertola (J. Phys. A, 2006)

On universality of critical behaviour in Hamiltonian PDEs - Boris Dubrovin (Geometry, topology, and mathematical physics : S.P. Novikov\\\'s seminar : 2006-2007 / V.M. Buchstaber, I.M. Krichever, editors. - Providence, R.I. : American Mathematical Society, 2008. - pages : 59-109, 2006)
Our main goal is the comparative study of singularities of solutions to\\r\\nthe systems of rst order quasilinear PDEs and their perturbations containing higher\\r\\nderivatives. The study is focused on the subclass of Hamiltonian PDEs with one\\r\\nspatial dimension. For the systems of order one or two we describe the local structure\\r\\nof singularities of a generic solution to the unperturbed system near the point of\\r\\n\\\\gradient catastrophe\\\" in terms of standard objects of the classical singularity theory;\\r\\nwe argue that their perturbed companions must be given by certain special solutions\\r\\nof Painlev e equations and their generalizations.

On variational approach to differential invariants of rank two distributions - Igor Zelenko (Differential Geom. Appl. 24 (2006) 235-259, 2006)
n the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n=5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his \\\"reduction-prolongation\\\" procedure. After Cartan\\\'s work the following questions remained open: first the geometric reason for existence of Cartan\\\'s tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan\\\'s tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in our previous works with A. Agrachev . In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n greater than 4.\\nFor n=5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In our next paper we show that in the case n=5 our fundamental form coincides with Cartan\\\'s tensor.

Variational problems in fracture mechanics - Gianni Dal Maso (2006)
We present some recent existence results for the variational model of crack growth in brittle materials proposed by Francfort and Marigo in 1998. These results, obtained in collaboration with Francfort and Toader, cover the case of arbitrary space dimension with a general quasiconvex bulk energy and with prescribed boundary deformations and applied loads.

WDVV equations and Frobenius manifolds - Boris Dubrovin (Encyclopedia of Mathematical Physics. Vol 1 A : A-C. Oxford: Elsevier, 2006, p. 438-447, 2006)

Asymptotic Morse theory for the equation $\\\\Delta v=2v\\\\sb x\\\\wedge v\\\\sb y$ - Sagun Chanillo; Andrea Malchiodi (Comm. Anal. Geom. 13 (2005) 187-252, 2005)
Given a smooth bounded domain ${\\\\O}\\\\subseteq \\\\R^2$, we consider the equation $\\\\D v = 2 v_x \\\\wedge v_y$ in $\\\\O$, where $v: {\\\\O}\\\\to \\\\R^3$. We prescribe Dirichlet boundary datum, and consider the case in which this datum converges to zero. An asymptotic study of the corresponding Euler functional is performed, analyzing multiple-bubbling phenomena. This allows us to settle a particular case of a question raised by H. Brezis and J.M. Coron.

On the attainable set for Temple class systems with boundary controls - Fabio Ancona; Giuseppe Maria Coclite (SIAM J. Control Optim. 43 (2005) 2166-2190, 2005)
Consider the initial-boundary value problem for a strictly hyperbolic, genuinely nonlinear, Temple class system of conservation laws % $$ u_t+f(u)_x=0, \\\\qquad u(0,x)=\\\\ov u(x), \\\\qquad {{array}{ll} &u(t,a)=\\\\widetilde u_a(t), \\\\noalign{\\\\smallskip} &u(t,b)=\\\\widetilde u_b(t), {array}. \\\\eqno(1) $$ on the domain $\\\\Omega =\\\\{(t,x)\\\\in\\\\R^2 : t\\\\geq 0, a \\\\le x\\\\leq b\\\\}.$ We study the mixed problem (1) from the point of view of control theory, taking the initial data $\\\\bar u$ fixed, and regarding the boundary data $\\\\widetilde u_a, \\\\widetilde u_b$ as control functions that vary in prescribed sets $\\\\U_a, \\\\U_b$, of $\\\\li$ boundary controls. In particular, we consider the family of configurations $$ \\\\A(T) \\\\doteq \\\\big\\\\{u(T,\\\\cdot); ~ u {\\\\rm is a sol. to} (1), \\\\quad \\\\widetilde u_a\\\\in \\\\U_a, \\\\widetilde u_b \\\\in \\\\U_b \\\\big\\\\} $$ that can be attained by the system at a given time $T>0$, and we give a description of the attainable set $\\\\A(T)$ in terms of suitable Oleinik-type conditions. We also establish closure and compactness of the set $\\\\A(T)$ in the $lu$ topology.

On the Blow-up for a Discrete Boltzmann Equation in the Plane - Alberto Bressan; Massimo Fonte (Discrete Contin. Dyn. Syst. 13 (2005) 1-12, 2005)
We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed.

Commuting multiparty quantum observables and local compatibility - Claudio Altafini (Phys. Rev. A 72 (2005) 012112, 2005)
A formula for the commutator of tensor product matrices is used to shows that, for qubits, compatibility of quantum multiparty observables almost never implies local compatibility at each site and to predict when this happens/does not happen in a concise manner. In particular, it is shown that two ``fully nontrivial\\\'\\\' $n$-qubit observables are compatible locally and globally if and only if they are equal up to sign. In addition, the formula gives insight into the construction of new paradoxes of the type of the Kochen-Specker Theorem, which can then be easily rephrased into proposals for new no hidden variable experiments of the type of the ``Bell Theorem without inequalities\\\'\\\'.

Complete systems of invariants for rank 1 curves in Lagrange Grassmannians - Igor Zelenko (Differential geometry and its applications, 367-382, Matfyzpress, Prague, 2005, 2005)
Curves in Lagrange Grassmannians naturally appear when one studies intrinsically \\\"the Jacobi equations for extremals\\\", associated with control systems and geometric structures. In this way one reduces the problem of construction of the curvature-type invariants for these objects to the much more concrete problem of finding of invariants of curves in Lagrange Grassmannians w.r.t. the action of the linear Symplectic group. In the present paper we develop a new approach to differential geometry of so-called rank 1 curves in Lagrange Grassmannian, i.e., the curves with velocities being rank one linear mappings (under the standard identification of the tangent space to a point of the Lagrange Grassmannian with an appropriate space of linear mappings). The curves of this class are associated with \\\"the Jacobi equations for extremals\\\", corresponding to control systems with scalar control and to rank 2 vector distributions. In particular, we construct the tuple of m principal invariants, where m is equal to half of dimension of the ambient linear symplectic space, such that for a given tuple of arbitrary m smooth functions there exists the unique, up to a symplectic transformation, rank 1 curve having this tuple, as the tuple of the principal invariants. This approach extends and essentially simplifies some results of our previous paper (J. Dynamical and Control Systems, 8, 2002, No. 1, 93-140), where only the uniqueness part was proved and in rather cumbersome way. It is based on the construction of the new canonical moving frame with the most simple structural equation.
Proceedings of 9th Conference on Differential Geometry and its Applications, Prague 2004

Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains - Andrea Malchiodi (Geometric and Functional Analysis 15 (6) 1162-1222 (2005), 2005)
We prove new concentration phenomena for the equation −ɛ2 Δu + u = up in a smooth bounded domain R3 and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0.

Conservation laws with time dependent discontinuous coefficients - Giuseppe Maria Coclite; Nils Henrik Risebro (SIAM J. Math. Anal. 36 (2005) 1293-1309, 2005)
We consider scalar conservation laws where the flux function depends discontinuously on both the spatial and temporal location. Our main results are the existence and well-posedness of an entropy solution to the Cauchy problem. The existence is established by showing that a sequence of front tracking approximations is compact in L1, and that the limits are entropy solutions. Then, using the definition of an entropy solution taken form [11], we show that the solution operator is L1 contractive. These results generalize the corresponding results from [16] and [11].

On curvatures and focal points of distributions of dynamical Lagrangian distributions and their reductions by first integrals - Andrei A. Agrachev; Natalia N. Chtcherbakova; Igor Zelenko (J. Dyn. Control Syst. 11 (2005) 297-327, 2005)
Pairs (Hamiltonian system, Lagrangian distribution), called dynamical Lagrangian distributions, appear naturally in Differential Geometry, Calculus of Variations and Rational Mechanics. The basic differential invariants of a dynamical Lagrangian distribution w.r.t. the action of the group of symplectomorphisms of the ambient symplectic manifold are the curvature operator and the curvature form. These invariants can be seen as generalizations of the classical curvature tensor in Riemannian Geometry. In particular, in terms of these invariants one can localize the focal points along extremals of the corresponding variational problems. In the present paper we study the behavior of the curvature operator, the curvature form and the focal points of a dynamical Lagrangian distribution after its reduction by arbitrary first integrals in involution. The interesting phenomenon is that the curvature form of so-called monotone increasing Lagrangian dynamical distributions, which appear naturally in mechanical systems, does not decrease after reduction. It also turns out that the set of focal points to the given point w.r.t. the monotone increasing dynamical Lagrangian distribution and the corresponding set of focal points w.r.t. its reduction by one integral are alternating sets on the corresponding integral curve of the Hamiltonian system of the considered dynamical distributions. Moreover, the first focal point corresponding to the reduced Lagrangian distribution comes before any focal point related to the original dynamical distribution. We illustrate our results on the classical $N$-body problem.

Decay of a bound state under a time-periodic perturbation: a toy case - Michele Correggi; Gianfausto Dell'Antonio (J. Phys. A 38 (2005) 4769-4781, 2005)
We study the time evolution of a three dimensional quantum particle, initially in a bound state, under the action of a time-periodic zero range interaction with ``strength\\\'\\\' (\\\\alpha(t)). Under very weak generic conditions on the Fourier coefficients of (\\\\alpha(t)), we prove complete ionization as (t \\\\to \\\\infty). We prove also that, under the same conditions, all the states of the system are scattering states.

The Dirac operator on SU_q(2) - Ludwik Dabrowski; Giovanni Landi; Andrzej Sitarz; Walter van Suijlekom; Joseph C. Varilly (Commun. Math. Phys. 259 (2005) 729-759, 2005)
We construct a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.
v2: minor changes

Enhanced residual-free bubble method for convection-diffusion problems - Andrea Cangiani; Süli, E. (Internat. J. Numer. Methods Fluids, 2005)
8th ICFD Conference on Numerical Methods for Fluid Dynamics. Part 2

Enhanced RFB method - Andrea Cangiani; Süli, Endre (Numer. Math., 2005)

Explicit Wei–Norman formulae for matrix Lie groups via Putzer\\\'s method - Claudio Altafini (Systems and Control Letters, 54 (11):1121-1130, 2005, 2005)
The Wei–Norman formula locally relates the Magnus solution of a system of linear time-varying ODEs with the solution expressed in terms of products of exponentials by means of a set of nonlinear differential equations in the parameters of the two types of solutions. A closed form expression of such formula is proposed based on the use of Putzer\\\'s method.

A fourth order uniformization theorem on some four manifolds with large total Q-curvature - Zindine Djadli; Andrea Malchiodi (C. R. Acad. Sci. Paris, Ser. I 340 (2005) 341-346., 2005)
Given a four-dimensional manifold (M,g), we study the existence of a conformal metric for which the Q-curvature, associated to a conformally invariant fourth-order operator (the Paneitz operator), is constant. Using a topological argument, we obtain a new result in cases which were still open.

Gel\\\'fand-Zakharevich Systems and Algebraic Integrability: the Volterra Lattice Revisited - Gregorio Falqui; Marco Pedroni (Regul. Chaotic Dyn. 10 (2005) 399-412, 2005)
In this paper we will discuss some features of the bi-Hamiltonian method for solving the Hamilton-Jacobi (H-J) equations by Separation of Variables, and make contact with the theory of Algebraic Complete Integrability and, specifically, with the Veselov-Novikov notion of algebro-geometric (AG) Poisson brackets.

Global solutions of the Hunter-Saxton equation - Alberto Bressan; Adrian Constantin (SIAM J. Math. Anal. 37 (2005) 996-1026, 2005)
We construct a continuous semigroup of weak, dissipative solutions to a nonlinear partial differential equations modeling nematic liquid crystals. A new distance functional, determined by a problem of optimal transportation, yields sharp estimates on the continuity of solutions with respect to the initial data.

Ground states of nonlinear Schroedinger equations with potentials vanishing at infinity - Antonio Ambrosetti; Veronica Felli; Andrea Malchiodi (J. Eur. Math. Soc. 7 (2005) 117-144, 2005)
We deal with a class on nonlinear Schr\\\\\\\"odinger equations \\\\eqref{eq:1} with potentials $V(x)\\\\sim |x|^{-\\\\a}$, $0<\\\\a<2$, and $K(x)\\\\sim |x|^{-\\\\b}$, $\\\\b>0$. Working in weighted Sobolev spaces, the existence of ground states $v_{\\\\e}$ belonging to $W^{1,2}(\\\\Rn)$ is proved under the assumption that $p$ satisfies \\\\eqref{eq:p}. Furthermore, it is shown that $v_{\\\\e}$ are {\\\\em spikes} concentrating at a minimum of ${\\\\cal A}=V^{\\\\theta}K^{-2/(p-1)}$, where $\\\\theta= (p+1)/(p-1)-1/2$.

Hybrid necessary principle - Mauro Garavello; Benedetto Piccoli (SIAM J. Control Optim. 43 (2005) 1867-1887, 2005)
We consider a hybrid control system and general optimal control problems for this system. We suppose that the switching strategy imposes restrictions on control sets and we provide necessary conditions for an optimal hybrid trajectory, stating a hybrid necessary principle (HNP). Our result generalizes various necessary principles available in the literature.
Proceedings of IFAC Conference on Analysis and Design of Hybrid Systems, Saint Malo, France

Ionization for Three Dimensional Time-dependent Point Interactions - Michele Correggi; Gianfausto Dell'Antonio; Rodolfo Figari; Andrea Mantile (Comm. Math. Phys. 257 (2005) 169-192, 2005)
We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the ``strength\\\'\\\' of the interaction (\\\\alpha(t)) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of (\\\\alpha(t)), we prove that there is complete ionization as (t \\\\to \\\\infty), starting from a bound state at time (t = 0). Moreover we prove also that, under the same conditions, all the states of the system are scattering states.

Isomonodromic deformation of resonant rational connections - Marco Bertola; Mo, M. Y. (IMRP Int. Math. Res. Pap., 2005)

The local index formula for SUq(2) - Walter van Suijlekom; Ludwik Dabrowski; Giovanni Landi; Andrzej Sitarz; Joseph C. Varilly (K-Theory 35 (2005) 375-394, 2005)
We discuss the local index formula of Connes-Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula.

Minimal surfaces in pseudohermitian geometry - Jih-Hsin Cheng; JennFang Hwang; Andrea Malchiodi; Paul Yang (Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177., 2005)
We consider surfaces immersed in three-dimensional pseudohermitian manifolds. We define the notion of (p-)mean curvature and of the associated (p-)minimal surfaces, extending some concepts previously given for the (flat) Heisenberg group. We interpret the p-mean curvature not only as the tangential sublaplacian of a defining function, but also as the curvature of a characteristic curve, and as a quantity in terms of calibration geometry. As a differential equation, the p-minimal surface equation is degenerate (hyperbolic and elliptic). To analyze the singular set, we formulate some {\em extension} theorems, which describe how the characteristic curves meet the singular set. This allows us to classify the entire solutions to this equation and to solve a Bernstein-type problem (for graphs over the $xy$-plane) in the Heisenberg group $H_1$. In $H_{1}$, identified with the Euclidean space $R^{3}$, the p-minimal surfaces are classical ruled surfaces with the rulings generated by Legendrian lines. We also prove a uniqueness theorem for the Dirichlet problem under a condition on the size of the singular set in two dimensions, and generalize to higher dimensions without any size control condition. We also show that there are no closed, connected, $C^{2}$ smoothly immersed constant p-mean curvature or p-minimal surfaces of genus greater than one in the standard $S^{3}.$ This fact continues to hold when $S^{3}$ is replaced by a general spherical pseudohermitian 3-manifold.

On the Minimum Problem for Nonconvex Scalar Functionals - Sandro Zagatti (SIAM J. Math. Anal. 37 (2005) 982-995, 2005)
We study the minimum problem for scalar nonconvex functionals defined on Sobolev maps satisfying a Dirichlet boundary condition and refine well-known existence results under standard regularity assumptions.

Modulation of the Camassa-Holm equation and reciprocal transformations - Simonetta Abenda; Tamara Grava (Ann. Inst. Fourier (Grenoble) 55 (2005) 1803-1834, 2005)
We derive the modulation equations or Whitham equations for the Camassa-Holm (CH) equation. We show that the modulation equations are hyperbolic and admit bi-Hamiltonian structure. Furthermore they are connected by a reciprocal transformation to the modulation equations of the first negative flow of the Korteweg de Vries (KdV) equation. The reciprocal transformation is generated by the Casimir of the second Poisson bracket of the KdV averaged flow. We show that the geometry of the bi-Hamiltonian structure of the KdV and CH modulation equations is quite different: indeed the KdV averaged bi-Hamiltonian structure can always be related to a semisimple Frobenius manifold while the CH one cannot.

Multiple clustered layer solutions for semilinear Neumann problems on a ball - Andrea Malchiodi; Wei-Ming Ni; Juncheng Wei (Ann. Inst. H. Poincare Anal. Non Lineaire 22 (2005) 143-163, 2005)

Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy - Ugo Boscain; Thomas Chambrion; Grégoire Charlot (Discrete Contin. Dyn. Syst. Ser. B 5 (2005) 957-990, 2005)
We apply techniques of subriemannian geometry on Lie groups and of optimal synthesis on 2-D manifolds to the population transfer problem in a three-level quantum system driven by two laser pulses, of arbitrary shape and frequency. In the rotating wave approximation, we consider a nonisotropic model i.e. a model in which the two coupling constants of the lasers are different. The aim is to induce transitions from the first to the third level, minimizing 1) the time of the transition (with bounded laser amplitudes),\\n2) the energy of lasers (with fixed final time). After reducing the problem to real variables, for the purpose 1) we develop a theory of time optimal syntheses for distributional problem on 2-D-manifolds, while for the purpose 2) we use techniques of subriemannian geometry on 3-D Lie groups. The complete optimal syntheses are computed.

Nonlinear Schrödinger Equations with vanishing and decaying potentials - Antonio Ambrosetti; Wang Zhi-Qiang (Differential Integral Equations 18 (2005), no. 12, 1321-1332, 2005)

An Optimal Transportation Metric for Solutions of the Camassa-Holm Equation - Alberto Bressan; Massimo Fonte (Methods Appl. Anal. 12 (2005) 191-219, 2005)
In this paper we construct a global, continuous flow of solutions to the Camassa-Holm equation on the entire space H1. Our solutions are conservative, in the sense that the total energy int[(u2 + u2x) dx] remains a.e. constant in time. Our new approach is based on a distance functional J(u, v), defined in terms of an optimal transportation problem, which satisfies d dtJ(u(t), v(t)) ≤ κ · J(u(t), v(t)) for every couple of solutions. Using this new distance functional, we can construct arbitrary solutions as the uniform limit of multi-peakon solutions, and prove a general uniqueness result.

Orbifold Cohomology of ADE-singularities - Fabio Perroni (SISSA, 2005)
Keywords: Orbifolds

Periodic solutions of nonlinear wave equations with non-monotone forcing terms - Massimiliano Berti; Luca Biasco (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 117-124, 2005)

Principal fibrations from noncommutative spheres - Giovanni Landi; Walter van Suijlekom (Comm. Math. Phys. 260 (2005) 203-225, 2005)
We construct noncommutative principal fibrations S_\\\\theta^7 \\\\to S_\\\\theta^4 which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. The algebra inclusion $A(S_\\\\theta^4) \\\\into A(S_\\\\theta^7)$ is an example of a not trivial quantum principal bundle.

Quasi-periodic oscillations for wave equations under periodic forcing - Massimiliano Berti; Procesi, Michela (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 2005)

Quasistatic Crack Growth in Nonlinear Elasticity - Gianni Dal Maso; Gilles A. Francfort; Rodica Toader (Arch. Ration. Mech. Anal. 176 (2005) 165-225, 2005)
In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\\\\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time.

Regularity properties of optimal trajectories of single-input control systems in dimension three - Mario Sigalotti (Journal of Mathematical Sciences 126 (2005) 1561-1573, 2005)
Let q=f(q)+ug(q) be a smooth control system on a three-dimensional manifold. Given a point q 0 of the manifold at which the iterated Lie brackets of f and g satisfy some prescribed independence condition, we analyze the structure of a control function u(t) corresponding to a time-optimal trajectory lying in a neighborhood of q 0. The control turns out to be the concatenation of some bang-bang and some singular arcs. More general optimality criteria than time-optimality are considered. The paper is a step toward to the analysis of generic single-input systems affine in the control in dimension 3. The main techniques used are second-order optimality conditions and, in particular, the index of the second variation of the switching times for bang-bang trajectories.

Self-similar folding patterns and energy scaling in compressed elastic sheets - Sergio Conti; Antonio DeSimone; Stefan Müller (Comput. Methods Appl. Mech. Engrg. 194 (2005) 2534-2549, 2005)
Thin elastic sheets under isotropic compression, such as for example blisters formed by thin films which debonded from the substrate, can exhibit remarkably complex folding patterns. We discuss the scaling of the elastic energy with respect to the film thickness, and show that in certain regimes the optimal energy scaling can be reached\\nby self-similar folding patterns that refine towards the boundary, in agreement with experimental observations. We then extend the analysis\\nto anisotropic compression, and discuss a simplified scalar model which suggests the presence of a transition between a regime where\\nthe deformation is governed by global properties of the domain and another one where the direction of maximal compression dominates and the scale of the folds is mainly determined by the distance to the boundary in the direction of the folds themselves.

A short introduction to optimal control - Ugo Boscain; Benedetto Piccoli (Contrôle non linéaire et applications: Cours donnés à l\\\'école d\\\'été du Cimpa de l\\\'Université de Tlemcen / Sari Tewfit [ed.]. - Paris: Hermann, 2005, 2005)

Solutions of Neumann problems in domains with cracks and applications to fracture mechanics - Gianni Dal Maso (2005)

The spectral geometry of the equatorial Podles sphere - Ludwik Dabrowski; Giovanni Landi; Mario Paschke; Andrzej Sitarz (C. R. Math. 340 (2005) 819-822, 2005)
We propose a slight modification of the properties of a spectral geometry a la Connes, which allows for some of the algebraic relations to be satisfied only modulo compact operators. On the equatorial Podles sphere we construct suq2-equivariant Dirac operator and real structure which satisfy these modified properties.

Stability of solutions of quasilinear parabolic equations - Giuseppe Maria Coclite; Helge Holden (J. Math. Anal. Appl. 308 (2005) 221-239, 2005)
We bound the difference between solutions $u$ and $v$ of $u_t = a\\\\Delta u+\\\\Div_x f+h$ and $v_t = b\\\\Delta v+\\\\Div_x g+k$ with initial data $\\\\phi$ and $ \\\\psi$, respectively, by $\\\\Vert u(t,\\\\cdot)-v(t,\\\\cdot)\\\\Vert_{L^p(E)}\\\\le A_E(t)\\\\Vert \\\\phi-\\\\psi\\\\Vert_{L^\\\\infty(\\\\R^n)}^{2\\\\rho_p}+ B(t)(\\\\Vert a-b\\\\Vert_{\\\\infty}+ \\\\Vert \\\\nabla_x\\\\cdot f-\\\\nabla_x\\\\cdot g\\\\Vert_{\\\\infty}+ \\\\Vert f_u-g_u\\\\Vert_{\\\\infty} + \\\\Vert h-k\\\\Vert_{\\\\infty})^{\\\\rho_p} \\\\abs{E}^{\\\\eta_p}$. Here all functions $a$, $f$, and $h$ are smooth and bounded, and may depend on $u$, $x\\\\in\\\\R^n$, and $t$. The functions $a$ and $h$ may in addition depend on $\\\\nabla u$. Identical assumptions hold for the functions that determine the solutions $v$. Furthermore, $E\\\\subset\\\\R^n$ is assumed to be a bounded set, and $\\\\rho_p$ and $\\\\eta_p$ are fractions that depend on $n$ and $p$. The diffusion coefficients $a$ and $b$ are assumed to be strictly positive and the initial data are smooth.

Stress-dilatancy based modelling of granular materials and extensions to soils with crushable grains - Antonio DeSimone; Claudio Tamagnini (Int. J. Numer. Anal. Met. 29 (2005) 73-101, 2005)
Stress-dilatancy relations have played a crucial role in the understanding of the mechanical behaviour of soils and in the development of realistic constitutive models for their response. Recent investigations on the mechanical behaviour of materials with crushable grains have called into question the validity of classical relations such as those used in critical state soil mechanics.\\nIn this paper, a method to construct thermodynamically consistent (isotropic, three-invariant) elasto-plastic models based on a given stress-dilatancy relation is discussed. Extensions to cover the case of granular materials with crushable grains are also presented, based on the interpretation of some classical model parameters (e.g. the stress ratio at critical state) as internal variables that evolve according to suitable hardening laws.

Time minimal trajectories for two-level quantum systems with drift - Ugo Boscain; Paolo Mason (Decision and Control, 2005 and 2005 European Control Conference. CDC-ECC \\\'05. 44th IEEE Conference on, 2005)
On a two-level quantum system driven by an external field, we consider the population transfer problem from the first to the second level, minimizing the time of transfer, with bounded field amplitude. On the Bloch sphere (i.e. after a suitable Hopf projection), this problem can be attacked with techniques of optimal syntheses on 2-D manifolds.

Time Optimal Synthesis for Left-Invariant Control Systems on SO(3) - Ugo Boscain; Yacine Chitour (SIAM J. Control Optim. 44 (2005) 111-139, 2005)
Consider the control system given by $\\\\dot x=x(f+ug)$, where $x\\\\in SO(3)$, $|u|\\\\leq 1$ and $f,g\\\\in so(3)$ define two perpendicular left-invariant vector fields normalized so that $\\\\|f\\\\|=\\\\cos(\\\\al)$ and $\\\\|g\\\\|=\\\\sin(\\\\al)$, $\\\\al\\\\in ]0,\\\\pi/4[$. In this paper, we provide an upper bound and a lower bound for $N(\\\\alpha)$, the maximum number of switchings for time-optimal trajectories. More precisely, we show that $N_S(\\\\al)\\\\leq N(\\\\al)\\\\leq N_S(\\\\al)+4$, where $N_S(\\\\al)$ is a suitable integer function of $\\\\al$ which for $\\\\al\\\\to 0$ is of order $\\\\pi/(4\\\\alpha).$ The result is obtained by studying the time optimal synthesis of a projected control problem on $R P^2$, where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere $S^2$. It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.

Topological vector symmetry, topological gauge fixing of BRSTQFT and construction of maximal supersymmetry - Laurent Baulieu; Guillaume Bossard; Alessandro Tanzini (JHEP 0508 (2005) 037, 2005)
The scalar and vector topological Yang-Mills symmetries determine a closed and consistent sector of Yang-Mills supersymmetry. We provide a geometrical construction of these symmetries, based on a horizontality condition on reducible manifolds. This yields globally well-defined scalar and vector topological BRST operators. These operators generate a subalgebra of maximally supersymmetric Yang-Mills theory, which is small enough to be closed off-shell with a finite set of auxiliary fields and large enough to determine the Yang-Mills supersymmetric theory. Poincaré supersymmetry is reached in the limit of flat manifolds. The arbitrariness of the gauge functions in BRSTQFTs is thus removed by the requirement of scalar and vector topological symmetry, which also determines the complete supersymmetry transformations in a twisted way. Provided additional Killing vectors exist on the manifold, an equivariant extension of our geometrical framework is provided, and the resulting \\\"equivariant topological field theory\\\" corresponds to the twist of super Yang-Mills theory on omega backgrounds.

Traffic flow on a road network - Giuseppe Maria Coclite; Benedetto Piccoli; Mauro Garavello (SIAM J. Math. Anal. 36 (2005) 1862-1886, 2005)
This paper is concerned with a fluidodynamic model for traffic flow. More precisely, we consider a single conservation law, deduced from conservation of the number of cars,\\ndefined on a road network that is a collection of roads with junctions. The evolution problem is underdetermined at junctions, hence we choose to have some fixed rules for the distribution of traffic plus an optimization criteria for the flux. We prove existence, uniqueness and stability of solutions to the Cauchy problem. Our method is based on wave front tracking approach, see [6], and works also for boundary data and time dependent coefficients of traffic distribution at junctions, so including traffic lights.

Vanishing viscosity solutions of nonlinear hyperbolic systems - Stefano Bianchini; Alberto Bressan (Ann. of Math. 161 (2005) 223-342, 2005)
We consider the Cauchy problem for a strictly hyperbolic, $n\\\\times n$ system in one space dimension: $u_t+A(u)u_x=0$, assuming that the initial data has small total variation.\\nWe show that the solutions of the viscous approximations $u_t+A(u)u_x=\\\\ve u_{xx}$ are defined globally in time and satisfy uniform BV estimates, independent of $\\\\ve$. Moreover, they depend continuously on the initial data in the $\\\\L^1$ distance, with a Lipschitz constant independent of $t,\\\\ve$. Letting $\\\\ve\\\\to 0$, these viscous solutions converge to a unique limit, depending Lipschitz continuously on the initial data. In the conservative case where $A=Df$ is the Jacobian of some flux function $f:\\\\R^n\\\\mapsto\\\\R^n$, the vanishing viscosity limits are precisely the unique entropy weak solutions to the system of conservation laws $u_t+f(u)_x=0$.

Wetting of rough surfaces: a homogenization approach - Antonio DeSimone; Giovanni Alberti (Proc. R. Soc. Lon. Ser. A 461 (2005) 79-97, 2005)
The contact angle of a drop in equilibrium on a solid is strongly affected by the roughness of the surface on which it rests. We study the roughness-induced enhancement of the hydrophobic or hydrophilic properties of a solid surface through homogenization theory. By relying on a variational formulation of the problem, we show that the macroscopic contact angle is associated with the solution of two cell problems, giving the minimal energy per unit macroscopic area for a transition layer between the rough solid surface and a liquid or vapor phase. Our results are valid for both chemically heterogeneous and homogeneous surfaces. In the latter case, a very transparent structure emerges from the variational\\napproach: the classical laws of Wenzel and Cassie-Baxter give bounds for the optimal energy, and configurations of minimal energy are those leading to the smallest macroscopic contact angle in the hydrophobic case, to the largest one in the hydrophilic case.

On almost duality for Frobenius manifolds - Boris Dubrovin (Amer. Math. Soc. Transl. 212 (2004)\\n75-132., 2004)
We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups and Shephard groups, from the Seiberg - Witten duality.

On analytic families of invariant tori for PDEs - Boris Dubrovin (Astérisque. Issue 297, 2004, Pages 35-65, 2004)
We propose to apply a version of the classical Stokes\\r\\nexpansion method to the perturbative construction of invariant tori for\\r\\nPDEs corresponding to solutions quasiperiodic in space and time variables.\\r\\nWe argue that, for integrable PDEs all but finite number of the\\r\\nsmall divisors arising in the perturbative analysis cancel. As an illustrative\\r\\nexample we establish such cancellations for the case of KP equation.\\r\\nIt is proved that, under mild assumptions about decay of the magnitude\\r\\nof the Fourier modes all analytic families of finite-dimensional invariant\\r\\ntori for KP are given by the Krichever construction in terms of thetafunctions\\r\\nof Riemann surfaces. We also present an explicit construction\\r\\nof infinite dimensional real theta-functions and corresponding quasiperiodic\\r\\nsolutions to KP as sums of infinite number of interacting plane\\r\\nwaves.

Asymptotic behaviour and correctors for linear Dirichlet problems with simultaneously varying operators and domains - Gianni Dal Maso; Francois Murat (Ann. Inst. H. Poincaré. Anal. Non Linéaire 21 (2004), (4), p. 445-486., 2004)
We consider a sequence of Dirichlet problems in varying domains (or, more generally, of relaxed Dirichlet problems involving measures in M_0) for second order linear elliptic operators in divergence form with varying matrices of coefficients. When the matrices H-converge to a matrix A^0, we prove that there exist a subsequence and a measure mu^0 in M_0 such that the limit problem is the relaxed Dirichlet problem corresponding to A^0 and mu^0. We also prove a corrector result which provides an explicit approximation of the solutions in the H^1-norm, and which is obtained by multiplying the corrector for the H-converging matrices by some special test function which depends both on the varying matrices and on the varying domains.

Bifurcation of free vibrations for completely resonant wave equations - Massimiliano Berti; Philippe Bolle (Boll. Unione Mat. Ital. Sez. B 7 (2004) 519-528, 2004)
We prove existence of small amplitude, 2 pi/omega -periodic in time solutions of completely resonant nonlinear wave equations with Dirichlet boundary conditions for any frequency omega belonging to a Cantor-like set of positive measure and for a generic set of nonlinearities. The proof relies on a suitable Lyapunov-Schmidt decomposition and a variant of the Nash-Moser Implicit Function Theorem.

Blow-up solutions for the Schrödinger equation in dimension three with a concentrated nonlinearity - Riccardo Adami; Gianfausto Dell'Antonio; Rodolfo Figari; Alessandro Teta (Ann. Inst. H. Poincare Anal. Non Lineaire 21 (2004) 121-137, 2004)
We present some results on the blow-up phenomenon for the Schroedinger equation in dimension three with a nonlinear term supported in a fixed point. We find sufficient conditions for the blow up exploiting the moment of inertia of the solution and the uncertainty principle. In the critical case, we discuss the additional symmetry of the equation and construct a family of explicit blow up solutions.

Calculation of impulsively started incompressible viscous flows - Marra, Andrea; Andrea Mola; Quartapelle, Luigi; Riviello, Luca (Int. J. Numer. Meth. Fluids, 2004)

Coarse-grained models of materials with non-convex free-energy: two case studies - Antonio DeSimone (Computer methods in applied mechanics and engineering , 193 (2004) 5129-5141, 2004)
Bridging across length scales is one of the fundamental challenges in the computational modelling of material systems whose mechanical response is driven by rough energy landscapes. The typical feature of such systems is that of exhibiting fine scale microstructures. Two case studies, namely, nematic elastomers and ferromagnetic shape memory alloys, are presented to illustrate the use of modern techniques from (non-convex) calculus of variations in developing coarse-grained models of microstructure-driven material response.

Coherent control of open quantum dynamical systems - Claudio Altafini (Phys. Rev. A 70 (2004) 062321, 2004)
A systematic analysis of the behavior of the quantum Markovian master equation driven by coherent control fields is proposed. Its irreversible character is formalized using control-theoretic notions and the sets of states that can be reached via cohere nt controls are described. The analysis suggests to which extent (and how) it is possible to counteract the effect of dissipation.

On the convergence rate of vanishing viscosity approximations - Alberto Bressan; Tong Yang (Comm. Pure Appl. Math. 57 (2004) 1075-1109, 2004)
Given a strictly hyperbolic, genuinely nonlinear system of conservation laws, we prove the a priori bound $\\\\big\\\\|u(t,\\\\cdot)-u^\\\\ve(t,\\\\cdot)\\\\big\\\\|_{\\\\L^1}= \\\\O(1)(1+t)\\\\cdot \\\\sqrt\\\\ve|\\\\ln\\\\ve|$ on the distance between an exact BV solution $u$ and a viscous approximation $u^\\\\ve$, letting the viscosity coefficient $\\\\ve\\\\to 0$. In the proof, starting from $u$ we construct an approximation of the viscous solution $u^\\\\ve$ by taking a mollification $u*\\\\phi_{\\\\strut \\\\sqrt\\\\ve}$ and inserting viscous shock profiles at the locations of finitely many large shocks, for each fixed $\\\\ve$. Error estimates are then obtained by introducing new Lyapunov functionals which control shock interactions, interactions between waves of different families and by using sharp decay estimates for positive nonlinear waves.

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; Bolle, Philippe (Nonlinear Anal., 2004)

Periodic orbits close to elliptic tori and applications to the three-body problem - Massimiliano Berti; Biasco, Luca; Valdinoci, Enrico (Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2004)

Periodic solutions of Hamiltonian PDEs - Massimiliano Berti (Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 2004)

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)
Random matrix theory

Periodic solutions of nonlinear wave equations with general nonlinearities - Massimiliano Berti; Bolle, Philippe (Comm. Math. Phys., 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.
60 pages; Latex; 3 figures. The statements of theorems have been\r\n simplified

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; Bolle, Philippe (Discrete Contin. Dyn. Syst., 2002)

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; Bolle, Philippe (Ann. Inst. H. Poincaré C Anal. Non Linéaire, 2002)

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 II: blow up analysis and applications - Zindine Djadli; Andrea Malchiodi; Mohameden Ould Ahmedou (Ann. Sc. Norm. Super. Pisa Cl. Sci., 2002, 1, 387, 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)

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