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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

A functional analysis approach to Arnold diffusion - Massimiliano Berti (Symmetry and perturbation theory (Cala Gonone, 2001), 2001)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Stokes Matrices for Frobenius Manifolds and the 6 Painlevé Equation - Davide Guzzetti (Rokko Lectures in Mathematics, Vol 7 [Issue title: Perspective of Painleve equations], (2000), pages : 101-109, 2000)
These notes are a short review on the theory of Frobenius manifolds and its connection to problems of isomonodromy deformations and to Painlev'e equations.
Keywords: Painlevé equation

Super KP equations and Darboux transformations: another perspective on the Jacobian super KP hierarchy - Gregorio Falqui; Cesare Reina; Alessandro Zampa (J. Geom. Phys. 35 (2000), no. 2-3, 239-272, 2000)

A Uniqueness Condition for Hyperbolic Systems of Conservation Laws - Alberto Bressan; Marta Lewicka (Discrete Contin. Dynam. Systems 6 (2000) 673-682, 2000)
Consider the Cauchy problem for a hyperbolic $n\\\\times n$ system of conservation laws in one space dimension: $$u_t+f(u)_x=0, u(0,x)=\\\\bar u(x).\\\\eqno(CP)$$ Relying on the existence of a continuous semigroup of solutions, we prove that the entropy admissible solution of (CP) is unique within the class of functions $u=u(t,x)$ which have bounded variation along a suitable family of space-like curves.

Value Functions for Bolza Problems with Discontinuous Lagrangians and Hamilton-Jacobi inequalities - Gianni Dal Maso; Helene Frankowska (ESAIM Control Optim. Calc. Var., 5 (2000), n. 5, p. 369-393., 2000)

Well-posedness of the Cauchy problem for n x n systems of conservation laws - Alberto Bressan; Graziano Crasta; Benedetto Piccoli (Mem. Amer. Math. Soc. 146 (2000), no. 694, 134 p., 2000)
Chapter 1 and 2

The anisotropy introduced by the mesh in the finite element approximation of the Mumford-Shah functional - Matteo Negri (Numer. Funct. Anal. Optim. 20 (1999), no. 9-10, 957-982, 1999)
We compute explicitly the anisotropy effect in the H1 term, generated in the approximation of the Mumford-Shah functional by finite element spaces defined on structured triangulations.

Approximation, Stability and control for Conservation Laws - Andrea Marson (SISSA, 1999)

Asymptotic behaviour of nonlinear elliptic higher order equations in perforated domains - Gianni Dal Maso; Igor V. Skrypnik (Journal d\\\'Analyse Mathematique, Volume 79, 1999, Pages: 63-112, 1999)

A bihamiltonian approach to separation of variables in mechanics - Gregorio Falqui; Franco Magri; Marco Pedroni (Proceedings of the workshop on nonlinearity, integrability and all that : twenty years after NEEDS \\\'79, Lecce, Italy, July 1 - 10, 1999 / ed. by M. Boiti. - Singapore : World Scientific, 2000. - p. 258-266, 1999)
This paper is a report on a recent approach to the theory of separability of the Hamilton-Jacobi equations from the viewpoint of bihamiltonian geometry.

Blowup asymptotics for scalar conservation laws with a source - Helge Kristian Jenssen; Carlo Sinestrari (Comm. in Partial Differential Equations 24 (1999) 2237-2261, 1999)

The calibration method for the Mumford-Shah functional - Giovanni Alberti; Guy Bouchitte; Gianni Dal Maso (C. R. Acad. Sci. Paris Ser. I Math. 329 (1999), no. 3, 249-254, 1999)
In this Note we adapt the calibration method to functionals of Mumford-Shah type, and provide a criterion (Theorem 1) to verify that a given function is energy minimizing. Among other applications, we use this criterion to show that certain triple-junction configurations are minimizing (Example 3).

Categorial mirror symmetry for K3 surfaces - Claudio Bartocci; Ugo Bruzzo; Guido Sanguinetti (Comm. Math. Phys. 206 (1999) 265-272, 1999)
We study the structure of a modified Fukaya category ${\\\\frak F}(X)$ associated with a K3 surface $X$, and prove that whenever $X$ is an elliptic K3 surface with a section, the derived category of $\\\\fF(X)$ is equivalent to a subcategory of the derived category ${\\\\bold D}(\\\\hat X)$ of coherent sheaves on the mirror K3 surface $\\\\hat X$.

Correspondence between Minkowski and de Sitter quantum field theory - Marco Bertola; Gorini, Vittorio; Moschella, Ugo; Schaeffer, Richard (Phys. Lett. B, 1999)

Discrete approximation of the Mumford-Shah functional in dimension two - Antonin Chambolle; Gianni Dal Maso (M2AN 33 (1999) 651-672, 1999)

Enhanced gauge symmetries on elliptic K3 - Loriano Bonora; Cesare Reina; Alessandro Zampa (Phys.Lett. B452 (1999) 244-250, 1999)
We show that the geometry of K3 surfaces with singularities of type A-D-E contains enough information to reconstruct a copy of the Lie algebra associated to the given Dynkin diagram. We apply this construction to explain the enhancement of symmetry in F and IIA theories compactified on singular K3\\\'s.

Evans-Vasilesco theorem in Dirichlet spaces - Gianni Dal Maso; Virginia De Cicco (Rendiconti di Matematica e delle sue Applicazioni. vol. 19, Issue 7, (1999), pages : 1-15, 1999)

Extremal faces of the range of a vector measure and a theorem of Lyapunov - Stefano Bianchini (J. Math. Anal. Appl. 231 (1999) 301-318, 1999)

Frobenius manifolds and Virasoro constraints - Boris Dubrovin; Zhang Youjin (Selecta Math. (N.S.) 5 (1999) 423-466, 1999)
For an arbitrary Frobenius manifold a system of Virasoro constraints is constructed. In the semisimple case these constraints are proved to hold true in the genus one approximation. Particularly, the genus $\\\\leq 1$ Virasoro conjecture of T.Eguchi, K.Hori, M.Jinzenji, and C.-S.Xiong and of S.Katz is proved for smooth projective varieties having semisimple quantum cohomology.

Homoclinics and chaotic behaviour for perturbed second order systems - Massimiliano Berti; Bolle, Philippe (Ann. Mat. Pura Appl. (4), 1999)

Hyperbolic Systems of Conservation Laws - Alberto Bressan (Rev. Mat. Complut. 12 (1999) 135-200, 1999)
This is a survey paper, written in the occasion of an invited talk given by the author at the Universidad Complutense in Madrid, October 1998. Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions.

L-1 stability estimates for n x n conservation laws - Alberto Bressan; Tai-Ping Liu; Tong Yang (Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22, 1999)
Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws, each characteristic field being linearly degenerate or genuinely nonlinear. In this paper we explicitly define a functional $\\\\Phi=\\\\Phi(u,v)$, equivalent to the $L^1$ distance, which is `almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon)·(t-s)$ for all $t>s\\\\geq 0$, for every pair of $\\\\epsilon$-approximate solutions $u,v$ with small total variation, generated by a wave-front-tracking algorithm. The small parameter $\\\\epsilon$ here controls the errors in the wave speeds, the maximum size of rarefaction fronts and the total strength of all non-physical waves in $u$ and in $v$. From the above estimate, it follows that front-tracking approximations converge to a unique limit solution, depending Lipschitz continuously on the initial data, in the $L^1$ norm. This provides a new proof of the existence of the standard Riemann semigroup generated by an $n\\\\times n$ system of conservation laws.\\\'\\\'

A Lipschitz selection from the set of minimizers of a nonconvex functional of the gradient - Gianni Dal Maso; Vladimir V. Goncharov; Antonio Ornelas (Nonlinear Analysis, Theory, Methods and Applications. Volume 37, Issue 6, September 1999, Pages 707-717, 1999)
A constructive and improved version of the proof that there exist a continuous map that solves the convexified problem is presented. A Lipschitz continuous map is analyzed such that a map vector minimizes the functional at each vector satisfying Cellina\\\'s condition of existence of minimum. This map is explicitly given by a direct constructive algorithm.

The method of Poisson pairs in the theory of nonlinear PDEs - Gregorio Falqui; Franco Magri; Marco Pedroni (Direct and inverse methods in nonlinear evolution equations : Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, September 5-12, 1999 / Robert Conte ... ; Antonio M. Greco ed. - Berlin : Springer, 2003. - (Lecture Notes in Physics ; 632), 1999)
The aim of these lectures is to show that the methods of classical Hamiltonian mechanics can be profitably used to solve certain classes of nonlinear partial differential equations. The prototype of these equations is the well-known Korteweg-de Vries (KdV) equation.\\nIn these lectures we touch the following subjects:\\ni) the birth and the role of the method of Poisson pairs inside the theory of the KdV equation;\\nii) the theoretical basis of the method of Poisson pairs;\\niii) the Gel\\\'fand-Zakharevich theory of integrable systems on bi-Hamiltonian manifolds;\\niv) the Hamiltonian interpretation of the Sato picture of the KdV flows and of its linearization on an infinite-dimensional Grassmannian manifold.\\nv) the reduction technique(s) and its use to construct classes of solutions;\\nvi) the role of the technique of separation of variables in the study of the reduced systems;\\nvii) some relations intertwining the method of Poisson pairs with the method of Lax pairs.

A multiplicity result for the Yamabe problem on $S\\\\sp n$ - Antonio Ambrosetti; Andrea Malchiodi (J. Funct. Anal. 168 (1999), no. 2, 529-561, 1999)
We prove a multiplicity result for the Yamabe problem on the manifold (S, g), where g is a perturbation of the standard metric g0 of Sn. Solutions are found by variational methods via an abstract perturbation result.

Nonclassical Shocks and the Cauchy Problem for Nonconvex Conservation Laws - Debora Amadori; Paolo Baiti; Philippe G. LeFloch; Benedetto Piccoli (J. Differential Equations 151 (1999) 345-372, 1999)
The Riemann problem for a conservation law with a nonconvex (cubic) flux can be solved in a class of admissible nonclassical solutions that may violate the Oleinik entropy condition but satisfy a single entropy inequality and a kinetic relation. We use such a nonclassical Riemann solver in a front tracking algorithm, and prove that the approximate solutions remain bounded in the total variation norm. The nonclassical shocks induce an increase of the total variation and, therefore, the classical measure of total variation must be modified accordingly. We prove that the front tracking scheme converges strongly to a weak solution satisfying the entropy inequality.

A note on fractional KDV hierarchies. II. The bihamiltonian approach - Paolo Casati; Gregorio Falqui; Marco Pedroni (SISSA Library, 1999)

Oleinik type estimates and uniqueness for n x n conservation laws - Alberto Bressan; Paola Goatin (J. Differential Equations 156 (1999), no. 1, 26--49, 1999)
Let $u_t+f(u)_x=0$ be a strictly hyperbolic $n\\\\times n$ system of conservation laws in one space dimension. Relying on the existence of a semigroup of solutions, we first establish the uniqueness of entropy admissible weak solutions to the Cauchy problem, under a mild assumption on the local oscillation of $u$ in a forward neighborhood of each point in the $t\\\\text{-}x$ plane. In turn, this yields the uniqueness of weak solutions which satisfy a decay estimate on positive waves of genuinely nonlinear families, thus extending a classical result proved by Oleĭnik in the scalar case.

Painlevé transcendents in two-dimensional topological field theory - Boris Dubrovin (The Painlevé property : one century later / Robert Conte ed. - New York : Springer-Verlag, 1999. - (CRM series in mathematical physics). - p. 287-412, 1999)

Perturbation of $Δ u+u^(N+2)/(N-2)=0$, the scalar curvature problem in $R^N$, and related topics - Antonio Ambrosetti; Jesus Garcia Azorero; Ireneo Peral (J. Funct. Anal. 165 (1999) 117-149, 1999)
Some nonlinear elliptic equations on $R^N$ which arise perturbing the problem with the critical Sobolev exponent are studied. In particular, some results dealing with the scalar curvature problem in $R^N$ are given.

Projection singularities of extremals for planar systems - Ugo Boscain; Benedetto Piccoli (SISSA Library, 1999)

Recurrent procedure for the determination of the free energy ε^2 expansion in the topological string theory - Boris Dubrovin; Andrei Ya A Maltsev (arXiv:solv-int/990400, 1999)
We present here the iteration procedure for the determination of free energy ǫ2-expansion using the theory of KdV - type equations. In our approach we use the conservation laws for KdV - type equations depending explicitly on times t1, t2, . . . to find the ǫ2-expansion of u(x, t1, t2, . . .) after the infinite number of shifts of u(x, 0, 0, . . .) ≡ x along t1, t2, . . . in recurrent form. The formulas for the free energy expansion are just obtained then as a result of quite simple integration procedure applied to un(x).

Renormalized solutions of elliptic equations with general measure data - Gianni Dal Maso; Francois Murat; Luigi Orsina; Alain Prignet (Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 4, 741-808, 1999)

On the scalar curvature problem under symmetry - Antonio Ambrosetti; Andrea Malchiodi (SISSA Library, 1999)

Some properties of the solutions of obstacle problems with measure data - Paolo Dall'Aglio; Gianni Dal Maso (Ricerche Matematiche., Supplemento dedicato a Ennio De Giorgi, vol. 48 (1999), page : 99-116, 1999)

Stokes matrices and monodromy of the quantum cohomology of projective spaces - Davide Guzzetti (Comm. Math. Phys. 207 (1999) 341-383, 1999)
n this paper we compute Stokes matrices and monodromy of the quantum cohomology of projective spaces. This problem can be formulated in a \\\"classical\\\" framework, as the problem of computation of Stokes matrices and monodromy of differential equations with regular and irregular singularities. We prove that the Stokes\\\' matrix of the quantum cohomology coincides with the Gram matrix in the theory of derived categories of coherent sheaves. We also study the monodromy group of the quantum cohomology and we show that it is related to hyperbolic triangular groups.

Structural stability and regularity of entropy solutions to hyperbolic systems of conservation laws - Alberto Bressan; Philippe G. LeFloch (Indiana Univ. Math. J. 48 (1999), no. 1, 43--84, 1999)
The paper is concerned with the qualitative structure of entropy solutions to a strictly hyperbolic, genuinely nonlinear system of conservation laws. We first give an accurate description of the local and global wave-front structure of a BV solution, generated by a front tracking algorithm.

Vanishing viscosity solutions of hyperbolic systems on manifolds - Stefano Bianchini; Alberto Bressan (SISSA Library, 1999)

Variational formulation of softening phenomena in fracture mechanics. The one-dimensional case - Andrea Braides; Gianni Dal Maso; Adriana Garroni (Arch. Ration. Mech. Anal. 146 (1999), no. 1, 23--58, 1999)
Starting from experimental evidence, the authors justify a variational model for softening phenomena in fracture of one-dimensional bars where the energy is given by the contribution and interaction of two terms: a typical bulk energy term depending on elastic strain and a discrete part that depends upon the jump discontinuities that occur in fracture. A more formal, rigorous derivation of the model is presented by examining the $\\\\Gamma$-convergence of discrete energy functionals associated to an array of masses and springs. Close attention is paid to the softening and fracture regimes. \\nOnce the continuous model is derived, it is fully analyzed without losing sight of its discrete counterpart. In particular, the associated boundary value problem is studied and a detailed analysis of the stationary points under the presence of a dead load is performed. A final, interesting section on the scale effect on the model is included.

The vector measures whose range is strictly convex - Stefano Bianchini; Carlo Mariconda (J. Math. Anal. Appl. 232 (1999) 1-19, 1999)

Algebraic Solutions to the Painlevé-VI Equation and Reflection Groups - Marta Mazzocco (SISSA, 1998)
Keywords: Painlevé VI equation

Applications of critical point theory to homoclinics and complex dynamics - Ambrosetti, A.; Berti, M. (Discrete Contin. Dynam. Systems, 1998)
Dynamical systems and differential equations, Vol. I (Springfield, MO, 1996)

Asymptotic behavior of nonlinear Dirichlet problems in perforated domains - Gianni Dal Maso; Igor V. Skrypnik (Ann. Mat. Pura Appl. (4) 174 (1998), 13--72, 1998)

Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation - Boris Dubrovin; Zhang Youjin (Comm. Math. Phys. 198 (1998) 311-361, 1998)
We compute the genus one correction to the integrable hierarchy describing coupling to gravity of a 2D topological field theory. The bihamiltonian structure of the hierarchy is given by a classical W-algebra; we compute the central charge of this algebra. We also express the generating function of elliptic Gromov - Witten invariants via tau-function of the isomonodromy deformation problem arising in the theory of WDVV equations of associativity.

Branching points for a class of variational operators - Antonio Ambrosetti (J. Anal. Math. 76 (1998) 321-335, 1998)

On the Cauchy Problem for the Whitham Equations - Tamara Grava (SISSA, 1998)
Keywords: Korteweg de Vries equation

Differential geometry of the space of orbits of a Coxeter group - Boris Dubrovin (J. Differential Geometry Suppl.4 (1998) 181-211, 1998)
Differential-geometric structures on the space of orbits of a finite Coxeter group, determined by Groth\\\\\\\'endieck residues, are calculated. This gives a construction of a 2D topological field theory for an arbitrary Coxeter group.

Diffusion of a particle in presence of N moving point sources - Gianfausto Dell'Antonio; Rodolfo Figari; Alessandro Teta (Annales Poincare Phys.Theor.69:413-424,1998, 1998)

On the Dirichlet problem for vectorial Hamilton-Jacobi equations - Sandro Zagatti (SIAM J. Math. Anal. 29 (1998) 1481-1491, 1998)
We give sufficient conditions for the existence of solutions to the Hamilton--Jacobi equations with Dirichlet boundary condition: $$ \\\\cases{ g(x,{\\\\hbox{\\\\rm det}}Du(x))=0, \\\\ & for a.e. $x\\\\in\\\\Omega,$\\\\cr u(x)=\\\\varphi(x), & for $x\\\\in\\\\partial\\\\Omega,$} $$ obtaining, in addition, an application to the theory of existence of minimizers for a class of nonconvex variational problems.

Error bounds for a deterministic version of the Glimm scheme - Andrea Marson; Alberto Bressan (Arch. Rational Mech. Anal. 142 (1998), no. 2, 155-176, 1998)
Consider the hyperbolic system of conservation laws $u_t F(u)_x=0. Let $u$ be the unique viscosity solution with initial condition $u(0,x)=\\\\bar u(x)$ and let $u^\\\\varepsilon$ be an approximate solution constructed by the Glimm scheme, corresponding to the mesh sizes $\\\\Delta x,\\\\Delta t=O(\\\\Delta x). With a suitable choise of the sampling sequence, we prove the estimate $$ \\\\left\\\\Vert u^\\\\varepsilon(t,\\\\cdot)-u(t,\\\\cdot) \\\\right\\\\Vert_1=o(1)\\\\cdot\\\\sqrt{\\\\Delta x}\\\\vert\\\\ln\\\\Delta x\\\\vert. $$

Extended affine Weyl groups and Frobenius manifolds - Boris Dubrovin; Youjin Zhang (Compositio Mathematica. Volume 111, Issue 2, 1998, Pages 167-219, 1998)

Generation of primordial fluctuations in curved spaces - Schaeffer, Richard; Moschella, Ugo; Marco Bertola; Gorini, Vittorio (Gravit. Cosmol., 1998)

A generic classification of time-optimal planar stabilizing feedbacks - Alberto Bressan; Benedetto Piccoli (SIAM J. Control Optim. 36 (1998) 12-32, 1998)
Consider the problem of stabilization at the origin in minimum time for a planar control system affine with respect to the control. For a family of generic vector fields, a topological equivalence relation on the corresponding time-optimal feedback synthesis was introduced in a previous paper [Dynamics of Continuous, Discrete and Impulsive Systems, 3 (1997), pp. 335--371]. The set of equivalence classes can be put in a one-to-one correspondence with a discrete family of graphs. This provides a classification of the global structure of generic time-optimal stabilizing feedbacks in the plane, analogous to the classification of smooth dynamical systems developed by Peixoto.

Geometric control approach to synthesis theory - Ugo Boscain; Benedetto Piccoli (Rend. Sem. Mat. Univ. Politec. Torino 56 (1998), no. 4, 53-68 (2001), 1998)

Geometry and analytic theory of Frobenius manifolds - Boris Dubrovin (Proceedings of the International Congress of Mathematicians : Berlin 1998, August 18 - 27. II, Invited lectures. - Bielefeld : Universität Bielefeld, Fakultät für Mathematik cop. 1998. - pages : 315-326, 1998)
Main mathematical applications of Frobenius manifolds are\\r\\nin the theory of Gromov - Witten invariants, in singularity theory, in\\r\\ndifferential geometry of the orbit spaces of reflection groups and of their\\r\\nextensions, in the hamiltonian theory of integrable hierarchies. The theory\\r\\nof Frobenius manifolds establishes remarkable relationships between\\r\\nthese, sometimes rather distant, mathematical theories.

Homoclinics and complex dynamics in slowly oscillating systems - Ambrosetti, Antonio; Massimiliano Berti (Discrete Contin. Dynam. Systems, 1998)

Infinite time regular synthesis - Benedetto Piccoli (ESAIM: COCV 3 (1998) 381-405, 1998)
In this paper we provide a new sufficiency theorem for regular syntheses. The concept of regular synthesis is discussed in [12], where a sufficiency theorem for finite time syntheses is proved. There are interesting examples of optimal syntheses that are very regular, but whose trajectories have time domains not necessarily bounded. The regularity assumptions of the main theorem in [12] are verified by every piecewise smooth feedback control generating extremal trajectories that reach the target in finite time with a finite number of switchings. In the case of this paper the situation is even more complicate, since we admit both trajectories with finite and infinite time. We use weak differentiability assumptions on the synthesis and weak continuity assumptions on the associated value function.

Limits of variational problems for Dirichlet forms in varying domains - Gianni Dal Maso; Virginia De Cicco; Lino Notarantonio; Nicoletta A. Tchou (Journal des Mathematiques Pures et Appliquees. Volume 77, Issue 1, January 1998, Pages 89-116, 1998)

Mirror Symmetry on K3 Surfaces as a Hyper-Kähler Rotation - Ugo Bruzzo; Guido Sanguinetti (Lett. Math. Phys. 45 (1998) 295-301, 1998)
We show that under the hypotheses of Strominger, Yau and Zaslow\\\'s paper, a mirror partner of a K3 surface $X$ with a fibration in special Lagrangian tori can be obtained by rotating the complex structure of $X$ within its hyperk\\\\\\\"ahler family of complex structures. The same hypotheses force the B-field to vanish.

Special functions with bounded variation and with weakly differentiable traces on the jump set - Luigi Ambrosio; Andrea Braides; Adriana Garroni (NoDEA Nonlinear Differential Equations Appl. 5 (1998), no. 2, 219--243, 1998)

Uniqueness for discontinuous ODE and conservation laws - Alberto Bressan; Wen Shen (Nonlinear Analysis 34 (1998) 637-652, 1998)
Consider a scalar O.D.E. of the form $\\\\dot x=f(t,x),$ where $f$ is possibly discontinuous w.r.t. both variables $t,x$. Under suitable assumptions, we prove that the corresponding Cauchy problem admits a unique solution, which depends H\\\\\\\"older continuously on the initial data.\\nOur result applies in particular to the case where $f$ can be written in the form $f(t,x)\\\\doteq g\\\\big( u(t,x)\\\\big)$, for some function $g$ and some solution $u$ of a scalar conservation law, say $u_t+F(u)_x=0$. In turn, this yields the uniqueness and continuous dependence of solutions to a class of $2\\\\times 2$ strictly hyperbolic systems, with initial data in $\\\\L^\\\\infty$.

Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium - Massimiliano Berti; Bolle, Philippe (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 1998)

Capacity theory for monotone operators - Gianni Dal Maso; Igor V. Skrypnik (Potential Anal. 7 (1997), no. 4, 765-803, 1997)
If $Au=-div(a(x,Du))$ is a monotone operator defined on the Sobolev space $W^{1,p}(R^n)$, $1< p <+\\\\infty$, with $a(x,0)=0$ for a.e. $x\\\\in R^n$, the capacity $C_A(E,F)$ relative to $A$ can be defined for every pair $(E,F)$ of bounded sets in $R^n$ with $E\\\\subset F$. We prove that $C_A(E,F)$ is increasing and countably subadditive with respect to $E$ and decreasing with respect to $F$. Moreover we investigate the continuity properties of $C_A(E,F)$ with respect to $E$ and $F$.

Comportamento asintotico delle soluzioni di problemi di Dirichlet - Gianni Dal Maso (Bollettino della Unione Matematica Italiana A. Volume 7, Issue SUPPL. 11 part 2, June 1997, Pages 253-277, 1997)
This is the title of the conference which was delivered by prof. Dal Maso in Padova during the XV Congress of the \\\"Unione Mathematica Italiana\\\" (Padova, 11-16 settembre 1995)

On Existence and Continuous Dependence for Systems of Conservation Laws - Paolo Baiti (SISSA, 1997)
Keywords: Conservation laws

Flat pencils of metrics and Frobenius manifolds - Boris Dubrovin (Integrable systems and algebraic geometry : proceedings of the Taniguchi symposium 1997, Kobe, June 30 - July 4, 1997 and Research Institute for Mathematical Sciences, Kyoto University, July 7 - 11, 1997 / eds. M.-H. Saito, Y. Shimizu and K. Ueno. - Sing, 1997)
This paper is based on the author\\\'s talk at 1997 Taniguchi Symposium \\\"Integrable Systems and Algebraic Geometry\\\". We consider an approach to the theory of Frobenius manifolds based on the geometry of flat pencils of contravariant metrics. It is shown that, under certain homogeneity assumptions, these two objects are identical. The flat pencils of contravariant metrics on a manifold $M$ appear naturally in the classification of bihamiltonian structures of hydrodynamics type on the loop space $L(M)$. This elucidates the relations between Frobenius manifolds and integrable hierarchies.

Functionals of the Peierls - Fröhlich Type and the Variational Principle for the Whitham Equations - Boris Dubrovin (Solitons, geometry, and topology : on the crossroad / V. M. Buchstaber, S. P. Novikov editors.- Providence : American Mathematical Society, 1997. - ( American mathematical society translations. Series 2. - vol. 179). - pages : 35-44, 1997)

Heteroclinic solutions for perturbed second order systems - Massimiliano Berti (Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 1997)

Homogeneous tangent vectors and high order necessary conditions for optimal controls - Fabio Ancona (J. Dynam. Control Systems 3 (1997), no. 2, 205--240, 1997)

Kam theorem for generic analytic perturbations of the Guler system - Marta Mazzocco (Z. Angew. Math. Phys. 48 (1997), no. 2, 193-219, 1997)
We apply here KAM theory to the fast rotations of a rigid body with a fixed point, subject to a purely positional potential. The problem is equivalent to a small perturbation of the Euler system. The difficulty is that the unperturbed system is properly degenerate, namely the unperturbed Hamiltonian depends only on two actions. Following the scheme used by Arnol\\\'d for the N-body problem, we use part of the perturbation to remove the degeneracy: precisely, we construct Birkhoff normal form up to a suitable finite order, thus eliminating the two fast angles; the resulting system is nearly integrable and (generically) no more degenerate, so KAM theorem applies. The resulting description of the motion is that, if the initial kinetic energy is sufficiently large, then for most initial data the angular momentum has nearly constant module, and moves slowly in the space, practically following the level curves of the initial potential averaged on the two fast angles; on the same time the body precesses around the instantaneous direction of the angular momentum, essentially as in the Euler-Poinsot motion. We also provide two simple physical examples, where the procedure does apply.

Krichever maps, Faà di Bruno polynomials, and cohomology in KP theory - Gregorio Falqui; Cesare Reina; Alessandro Zampa (Lett. Math. Phys. 42 (1997) 349-361, 1997)
We study the geometrical meaning of the Faa\\\' di Bruno polynomials in the context of KP theory. They provide a basis in a subspace W of the universal Grassmannian associated to the KP hierarchy. When W comes from geometrical data via the Krichever map, the Faa\\\' di Bruno recursion relation turns out to be the cocycle condition for (the Welters hypercohomology group describing) the deformations of the dynamical line bundle on the spectral curve together with the meromorphic sections which give rise to the Krichever map. Starting from this, one sees that the whole KP hierarchy has a similar cohomological meaning.

The semigroup generated by a temple class system with large data - Paolo Baiti; Alberto Bressan (Differential Integral Equations 10 (1997), no. 3, 401-418, 1997)
We consider the Cauchy problem $$u_t + [F(u)]_x=0, u(0,x)=\\\\bar u(x) (*)$$ for a nonlinear $n\\\\times n$ system of conservation laws with coinciding shock and rarefaction curves. Assuming the existence of a coordinates system made of Riemann invariants, we prove the existence of a weak solution of (*) that depends in a lipschitz continuous way on the initial data, in the class of functions with arbitrarily large but bounded total variation.

Shift-differentiability of the flow generated by a conservation law - Alberto Bressan; Graziano Guerra (Discrete Contin. Dynam. Systems 3 (1997), no. 1, 35--58., 1997)
The paper introduces a notion of \\\"shift-differentials\\\" for maps with values in the space BV. These differentials describe first order variations of a given functin $u$, obtained by horizontal shifts of the points of its graph. The flow generated by a scalar conservation law is proved to be generically shift-differentiable, according to the new definition.

Some Problems in the Asymptotic Analysis of Partial Differential Equations in Perforated Domains - Rodica Toader (SISSA, 1997)
Keywords: Dirichlet problems

Some properties of reachable solutions of nonlinear elliptic equations with measure data - Gianni Dal Maso; Annalisa Malusa (Annali della Scuola Normale Superiore di Pisa - Classe di Scienze. Sér. 4, 25 no. 1-2 (1997), p. 375-396, 1997)

Statistics in space dimension two - Gianfausto Dell'Antonio; Rodolfo Figari; Alessandro Teta (Lett. Math. Phys. 40 (1997), no. 3, 235-256, 1997)
We construct as a selfadjoint operator the Schroedinger hamiltonian for a system of $N$ identical particles on a plane, obeying the statistics defined by a representation $\\\\pi_1$ of the braid group. We use quadratic forms and potential theory, and give details only for the free case; standard arguments provide the extension of our approach to the case of potentials which are small in the sense of forms with respect to the laplacian. We also comment on the relation between the analysis given here and other approaches to the problem, and also on the connection with the description of a quantum particle on a plane under the influence of a shielded magnetic field (Aharanov-Bohm effect).

Structural stability for time-optimal planar sytheses - Alberto Bressan; Benedetto Piccoli (Dynam. Contin. Discrete Impuls. Systems 3 (1997), no. 3, 335--371, 1997)

Three-Phase Solutions of the Kadomtsev - Petviashvili Equation - Boris Dubrovin; Ron Flickinger; Harvey Segur (Studies in Applied Mathematics. Year : 1997 ; Volume: 99 ; Issue: 2 ; Pages: 137-203, 1997)
The Kadomtsev]Petviashvili KP. equation is known to admit explicit periodic\\r\\nand quasiperiodic solutions with N independent phases, for any integer\\r\\nN, based on a Riemann theta-function of N variables. For Ns1 and 2,\\r\\nthese solutions have been used successfully in physical applications. This\\r\\narticle addresses mathematical problems that arise in the computation of\\r\\ntheta-functions of three variables and with the corresponding solutions of\\r\\nthe KP equation. We identify a set of parameters and their corresponding\\r\\nranges, such that e¨ery real-valued, smooth KP solution associated with a\\r\\nRiemann theta-function of three variables corresponds to exactly one choice\\r\\nof these parameters in the proper range. Our results are embodied in a\\r\\nprogram that computes these solutions efficiently and that is available to the\\r\\nreader. We also discuss some properties of three-phase solutions.

Viscosity solutions and uniquenessfor systems of inhomogeneous balance laws - Graziano Crasta; Benedetto Piccoli (Discrete Contin. Dynam. Systems 3 (1997), no. 4, 477--5, 1997)

A capacity method for the study of Dirichlet problems for elliptic systems in varying domains - Gianni Dal Maso; Rodica Toader (Rend. Sem. Mat. Univ. Padova 96 (1996), 257--277, 1996)

The semigroup approach to systems of conservation laws - Alberto Bressan (Mat. Contemp. 10 (1996) 21-74, 1996)

Solving Honig generic problem about Volterra integral equations - Giovanni Vidossich (Bull. Polish Acad. Sci. Math. 44 (1996), no. 4, 495--498, 1996)

Some remarks on a variational approach to Arnold's diffusion - Massimiliano Berti (Discrete Contin. Dynam. Systems, 1996)

Capacity and Dirichlet problems in varying domains - Gianni Dal Maso (SISSA Library, 1995)

Classical solutions for a perturbed N-body system - Gianfausto Dell'Antonio (Topological nonlinear analysis, II (Frascati, 1995), 1--86, Progr. Nonlinear Differential Equations Appl., 27, Birkhauser Boston, Boston, MA, 1997, 1995)

An existence result in a problem of the vectorial case of the calculus of variations - Arrigo Cellina; Sandro Zagatti (SIAM, 1995)
SIAM J. Control Optim. 33 (1995) 960-970

Geometry of 2D topological field theories - Boris Dubrovin (Integrable systems and quantum groups : lectures given at the 1st session of the Centro internazionale matematico estivo (C.I.M.E.) held in Montecatini Terme, Italy, June 14-22, 1995 / R. Donagi, B. Dubrovin, E. Frenkel... [et al.] ; editors, M. Francavig, 1995)
These notes are devoted to the theory of “equations of associativity”\\r\\ndescribing geometry of moduli spaces of 2D topological field theories.

Quantum homogeneous spaces at roots of unity - Cesare Reina; Alessandro Zampa (Quantization, Coherent States and Poisson Structures, Proc. XIVth Workshop on Geometric Methods in Physics, Bialowieza, Poland, 9-15 July 1995, eds. A. Strasburger,\\nS.T. Ali, J.-P. Antoine, J.-P. Gazeau , A. Odzijewicz, Polish Scientific Publisher PWN 1, 1995)

Some control problems for the pendulum - Benedetto Piccoli (Proceedings of the 34th IEEE Conference on Decision and Control 4 (1995) 3313-3318, 1995)
The aim of this paper is to illustrate some geometric techniques for the study of nonlinear systems. The pendulum on one hand is good for its simplicity, on the other it presents many of the difficulties one can encounter treating nonlinear systems.

Special functions of bounded deformation - Giovanni Bellettini; Alessandra Coscia; Gianni Dal Maso (SISSA Library, 1995)

Unique solutions of 2x2 conservation laws with large data - Alberto Bressan; Rinaldo M. Colombo (Indiana Univ. Math. J. 44 (1995), no. 3, 677-725, 1995)
For a 2x2 hyperbolic system of conservation laws, we first consider a Riemann problem with arbitrarily large data. A stability assumption is introduced, which yields the existence of a Lipschitz semigroup of solutions, defined on a domain containing all suitably small BV perturbations of the Riemann data. We then establish a uniqueness result for large BV solutions, valid within the same class of functions where a local existence theorem can be proved.

Algebraic-geometrical Darboux coordinates in R-matrix formalism - P. Diener; Boris Dubrovin (SISSA, 1994)

Analysis of Singularity Structures for Quasi-Integrable Hamiltonian Systems - Simonetta Abenda (SISSA, 1994)
Keywords: Hamiltonian systems

Asymptotic Behaviour of Dirichlet Problems in Perforated Domains - Adriana Garroni (SISSA, 1994)
Keywords: Dirichlet problems

Hilbert schemes of points on some K3 surfaces and Gieseker stable boundles - Ugo Bruzzo; Antony Maciocia (MATH PROC CAMBRIDGE 120: 255-261 Part 2, 1994)
By using a Fourier-Mukai transform for sheaves on K3 surfaces we show that for a wide class of K3 surfaces $X$ the punctual Hilbert schemes $\\\\Hilb^n(X)$ can be identified, for all $n\\\\geq 1$, with moduli spaces of Gieseker stable vector bundles on $X$ of rank $1+2n$. We also introduce a new Fourier-Mukai type transform for such surfaces.

Integrable functional equations and algebraic geometry - Boris Dubrovin; A.S. Fokas; P.M. Santini (Duke Mathematical Journal. Volume: 76, Issue: 2, Pages: 645-668, 1994)

Limits of Dirichlet problems in perforated domains: a new formulation - Gianni Dal Maso; Rodica Toader (Rend. Istit. Mat. Univ. Trieste 26 (1994) 339-360, 1994)

A version of Olech\\\'s lemma in a problem of the calculus of variations - Arrigo Cellina; Sandro Zagatti (SIAM J. Control Optim. 32 (1994) 1114-1127, 1994)
This paper studies the solutions of the minimum problem for a functional of the gradient under linear boundary conditions. A necessary and sufficient condition, based on the facial structure of the epigraph of the integrand, is provided for the continuous dependence of the solutions on boundary data.

A Borel-Weil-Bott approach to representations of \rm sl\sb q(2,C) - Davide Franco; Cesare Reina (Lett. Math. Phys. 29 (1993) 215-217, 1993)
We use a quite concrete and simple realization of $\slq$ involving finite difference operators. We interpret them as derivations (in the non-commutative sense) on a suitable graded algebra, which gives rise to the double of the projective line as the non commutative version of the standard homogeneous space.

Dispersion relations for non-linear waves and the Schottky problem - Boris Dubrovin (Important developments in soliton theory / A. S. Fokas, V. E. Zakharov (eds.) - Berlin : Springer-Verlag, 1993. - pages : 86-98, 1993)
An approach to the Schottky problem of specification of periods of holomorphic differentials\\r\\non Riemann surfaces (or, equivalently, specification of Jacobians among all principaly\\r\\npolarized Abelian varieties) based on the theory of Kadomtsev - Petviashvili equation,\\r\\nis discussed.

Geometry and integrability of topological-antitopological fusion - Boris Dubrovin (Communications in Mathematical Physics. Volume 152, Issue 3, March 1993, Pages 539-564, 1993)
Integrability of equations of topological-antitopological fusion (being proposed\\r\\nby Cecotti and Vafa) describing the ground state metric on a given 2D topological\\r\\nfield theory (TFT) model, is proved. For massive TFT models these equations\\r\\nare reduced to a universal form (being independent on the given TFT model) by\\r\\ngauge transformations. For massive perturbations of topological conformal field theory\\r\\nmodels the separatrix solutions of the equations bounded at infinity are found\\r\\nby the isomonodromy deformations method. Also it is shown that the ground state\\r\\nmetric together with some part of the underlined TFT structure can be parametrized\\r\\nby pluriharmonic maps of the coupling space to the symmetric space of real positive\\r\\ndefinite quadratic forms.

Integrable systems and classification of 2D topological field theories - Boris Dubrovin (Integrable systems : the Verdier memorial conference : actes du colloque international de Luminy / Olivier Babelon, Pierre Cartier, Yvette Kosmann-Schwarzbach editors. - Boston [etc.] : Birkhauser, c1993. - p. 313-359, 1993)
In this paper we consider from the point of view of differential geometry and of the\\r\\ntheory of integrable systems the so-called WDVV equations as defining relations of 2-\\r\\ndimensional topological field theory. A complete classification of massive topological conformal\\r\\nfield theories (TCFT) is obtained in terms of monodromy data of an auxillary\\r\\nlinear operator with rational coefficients. Procedure of coupling of a TCFT to topological\\r\\ngravity is described (at tree level) via certain integrable bihamiltonian hierarchies of\\r\\nhydrodynamic type and their τ -functions. A possible role of bihamiltonian formalism in\\r\\ncalculation of high genus corrections is discussed. As a biproduct of this discussion new\\r\\nexamples of infinite dimensional Virasoro-type Lie algebras and their nonlinear analogues\\r\\nare constructed. As an algebro-geometrical applications it is shown that WDVV is just the\\r\\nuniversal system of integrable differential equations (high order analogue of the Painlev´e-\\r\\nVI) specifying periods of Abelian differentials on Riemann surfaces as functions on moduli\\r\\nof these surfaces.

Topological conformal field theory from the point of view of integrable systems - Boris Dubrovin (Integrable quantum field theories / edited by L. Bonora ... \et al.! - New York : Plenum Press, 1993. - page : 283 - 302., 1993)
Recent results on classification of massive topological conformal field theories (TCFT) in terms of monodromy data of auxiliary linear operators with rational coefficients are presented. Procedure of coupling of a TCFT to topological gravity is described (at tree-level approximation) via certain integrable hierarchies of hydrodynamic type and their tau-functions. It is explained how the calculation of the ground state metric on TCFT can be interpreted in terms of harmonic maps. Also a construction of some models via Coxeter groups is described.
NATO ASI series / B ;v. 310

Workshop on point interactions, Trieste, 21-23 December 1992 - Gianfausto Dell'Antonio (SISSA Library, 1993)

Hamiltonian formalism of Whitham-type hierarchies and topological Landau - Ginsburg models - Boris Dubrovin (Communications in Mathematical Physics. Volume 145, Issue 1, March 1992, Pages 195-207, 1992)
We show that the bi-hamiltonian structure of the averaged Gelfand-Dikii\\r\\nhierarchy is involved in the Landau-Ginsburg topological models (for An-Series):\\r\\nthe Casimirs for the first P.B. give the correct coupling parameters for the perturbed\\r\\ntopological minimal model; the correspondence {coupling parameters} ~ {primary\\r\\nfields} is determined by the second P.B. The partition function (at the tree level) and\\r\\nthe chiral algebra for LG models are calculated for any genus g.

Integrable systems in topological field theory - Boris Dubrovin (Nuclear Physics B. Volume 379, Issue 3, 1992, pages : 627-689, 1992)
Integrability of the system of PDE for dependence on coupling parameters of the (tree-level) primary partition function in massive topological field theories, being imposed by the associativity of the perturbed primary chiral algebra, is proved. In the conformal case it is shown that all the topological field theories are classified as solutions of a universal high-order Painlevé-type equation. Another integrable hierarchy (of systems of hydrodynamic type) is shown to describe coupling to gravity of the matter sector of any topological field theory. Different multicritical models with the given structure of primary correlators are identified with particular self-similar solutions of the hierarchy. The partition function of any of the models is calculated as the corresponding tau-function of the hierarchy.

Some Problems in the Calculus of the Variations - Sandro Zagatti (SISSA, 1992)
Keywords: Calculus of variations

Topological "observables" in semiclassical field theories - Margherita Nolasco; Cesare Reina (Phys. Lett. B 297 (1992) 82-88, 1992)
We give a geometrical set up for the semiclassical approximation to euclidean field theories having families of minima (instantons) parametrized by suitable moduli spaces ${\mathcal{M}}$. The standard examples are of course Yang-Mills theory and non-linear $σ$-models. The relevant space here is a family of measure spaces $\tilde{\mathcal{N}} → \mathcal{M}$, with standard fibre a distribution space, given by a suitable extension of the normal bundle to $\mathcal{M}$ in the space of smooth fields. Over $\tilde{\mathcal{N}}$ there is a probability measure $dμ$ given by the twisted product of the (normalized) volume element on $\mathcal{M}$ and the family of gaussian measures with covariance given by the tree propagator $C_φ$ in the background of an instanton $φ \in \mathcal{M}$. The space of "observables", i.e. measurable functions on ($\tilde{\mathcal{N}},\, dμ$), is studied and it is shown to contain a topological sector, corresponding to the intersection theory on $\mathcal{M}$. The expectation value of these topological "observables" does not depend on the covariance; it is therefore exact at all orders in perturbation theory and can moreover be computed in the topological regime by setting the covariance to zero.

A variational method in image segmentation: existence and approximation result - Gianni Dal Maso; Jean-Michel Morel; Sergio Solimini (Acta Math. 168 (1992), no.1-2, p. 89-151, 1992)

A class of absolute retracts of dwarf spheroidal galaxies - Alberto Bressan; Arrigo Cellina; Andrzej Fryszkowski (Proc.Amer.Math.Soc. 112 (1991), no.2, 413, 1991)

Differential geometry of moduli spaces and its applications to soliton equations and to topological conformal field theory - Boris Dubrovin (Preprint n.117, Scuola Normale Superiore, Pisa, November 1991, 31 pp. Published in: Surveys in Differential Geometry , Vol. IV (1999), p. 213 - 238., 1991)
We construct flat Riemannian metrics on moduli spaces of algebraic curves with marked meromorphic function. This gives a new class of exact algebraic-geometry solutions to certain non-linear equations in terms of functions on the moduli spaces. We show that the Riemannian metrics on the moduli spaces coincide with two-point correlators in topological conformal field theory and calculate the partition function for A_n model for arbitrary genus. A universal method for constructing complete families of conservation laws for Whitham-type hierarchies of PDEs is also proposed.

Shape optimization for Dirichlet problems: relaxed formulations and optimally conditions - Giuseppe Buttazzo; Gianni Dal Maso (Appl.Math.Optim. 23 (1991), no.1, p. 17-49., 1991)