@article {mine-02-02-011, title = {MicroMotility: State of the art, recent accomplishments and perspectives on the mathematical modeling of bio-motility at microscopic scales}, journal = {Mathematics in Engineering}, volume = {2}, number = {mine-02-02-01}, year = {2020}, pages = {230}, type = {Perspective}, abstract = {

Mathematical modeling and quantitative study of biological motility (in particular, of motility at microscopic scales) is producing new biophysical insight and is offering opportunities for new discoveries at the level of both fundamental science and technology. These range from the explanation of how complex behavior at the level of a single organism emerges from body architecture, to the understanding of collective phenomena in groups of organisms and tissues, and of how these forms of swarm intelligence can be controlled and harnessed in engineering applications, to the elucidation of processes of fundamental biological relevance at the cellular and sub-cellular level. In this paper, some of the most exciting new developments in the fields of locomotion of unicellular organisms, of soft adhesive locomotion across scales, of the study of pore translocation properties of knotted DNA, of the development of synthetic active solid sheets, of the mechanics of the unjamming transition in dense cell collectives, of the mechanics of cell sheet folding in volvocalean algae, and of the self-propulsion of topological defects in active matter are discussed. For each of these topics, we provide a brief state of the art, an example of recent achievements, and some directions for future research.

}, keywords = {active matter, adhesive locomotion, cell motility, cell sheet folding, knotted DNA, topological defects, unicellular swimmers, unjamming transition}, issn = {2640-3501}, doi = {10.3934/mine.2020011}, url = {http://dx.doi.org/10.3934/mine.2020011}, author = {Daniele Agostinelli and Roberto Cerbino and Del Alamo, Juan C and Antonio DeSimone and Stephanie H{\"o}hn and Cristian Micheletti and Giovanni Noselli and Eran Sharon and Julia Yeomans} } @article {Dell{\textquoteright}Antonio2018, title = {Lp-Boundedness of Wave Operators for the Three-Dimensional Multi-Centre Point Interaction}, journal = {Annales Henri Poincar{\'e}}, volume = {19}, number = {1}, year = {2018}, month = {Jan}, pages = {283{\textendash}322}, abstract = {

We prove that, for arbitrary centres and strengths, the wave operators for three-dimensional Schr{\"o}dinger operators with multi-centre local point interactions are bounded in Lp(R3)for 1\<p\<3 and unbounded otherwise.

}, issn = {1424-0661}, doi = {10.1007/s00023-017-0628-4}, url = {https://doi.org/10.1007/s00023-017-0628-4}, author = {Gianfausto Dell{\textquoteright}Antonio and Alessandro Michelangeli and Raffaele Scandone and Kenji Yajima} } @article {MR3505204, title = {Correlation functions of the KdV hierarchy and applications to intersection numbers over $\overline\CalM_g,n$}, journal = {Phys. D}, volume = {327}, year = {2016}, pages = {30{\textendash}57}, issn = {0167-2789}, doi = {10.1016/j.physd.2016.04.008}, url = {http://dx.doi.org/10.1016/j.physd.2016.04.008}, author = {Marco Bertola and Boris Dubrovin and Di Yang} } @article {BertoDiDub, title = {Simple Lie Algebras and Topological ODEs}, journal = {Int. Math. Res. Not.}, volume = {2016}, number = {rnw285}, year = {2016}, author = {Marco Bertola and Boris Dubrovin and Di Yang} } @article {2015, title = {The deal.II Library, Version 8.2}, journal = {Archive of Numerical Software, vol. 3, n. 100, (2015), pages : 1-8}, year = {2015}, abstract = {This paper provides an overview of the new features of the finite element library deal.II version 8.2}, doi = {10.11588/ans.2015.100.18031}, url = {http://urania.sissa.it/xmlui/handle/1963/34464}, author = {Wolfgang Bangerth and Timo Heister and Luca Heltai and G. Kanschat and Martin Kronbichler and Matthias Maier and Bruno Turcksin and T. D. Young} } @article {MR3342766, title = {The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy}, journal = {J. Phys. A}, volume = {48}, number = {19}, year = {2015}, pages = {195205, 20}, issn = {1751-8113}, doi = {10.1088/1751-8113/48/19/195205}, url = {http://dx.doi.org/10.1088/1751-8113/48/19/195205}, author = {Marco Bertola and Di Yang} } @article {2014, title = {Global Structure of Admissible BV Solutions to Piecewise Genuinely Nonlinear, Strictly Hyperbolic Conservation Laws in One Space Dimension}, number = {Communications in Partial Differential Equations;Volume 39; issue 2; pp. 244-273;}, year = {2014}, publisher = {Taylor \& Francis}, abstract = {

The paper describes the qualitative structure of an admissible BV solution to a strictly hyperbolic system of conservation laws whose characteristic families are piecewise genuinely nonlinear. More precisely, we prove that there are a countable set of points Θ and a countable family of Lipschitz curves T{script} such that outside T{script} ∪ Θ the solution is continuous, and for all points in T{script}{set minus}Θ the solution has left and right limit. This extends the corresponding structural result in [7] for genuinely nonlinear systems. An application of this result is the stability of the wave structure of solution w.r.t. -convergence. The proof is based on the introduction of subdiscontinuities of a shock, whose behavior is qualitatively analogous to the discontinuities of the solution to genuinely nonlinear systems.

}, doi = {10.1080/03605302.2013.775153}, url = {http://urania.sissa.it/xmlui/handle/1963/34694}, author = {Stefano Bianchini and Lei Yu} } @article {2014, title = {Structure of entropy solutions to general scalar conservation laws in one space dimension}, journal = {Journal of Mathematical Analysis and Applications}, volume = {428}, number = {SISSA;11/2014/MATE}, year = {2014}, month = {08/2015}, pages = {356-386}, publisher = {SISSA}, chapter = {356}, doi = {https://doi.org/10.1016/j.jmaa.2015.03.006}, url = {https://www.sciencedirect.com/science/article/pii/S0022247X15002218}, author = {Stefano Bianchini and Lei Yu} } @article {2013, title = {The deal.II Library, Version 8.1}, number = {arXiv:1312.2266;}, year = {2013}, note = {5 pages}, institution = {SISSA}, abstract = {This paper provides an overview of the new features of the finite element library deal.II version 8.0.}, url = {http://hdl.handle.net/1963/7236}, author = {Wolfgang Bangerth and Timo Heister and Luca Heltai and G. Kanschat and Martin Kronbichler and Matthias Maier and Bruno Turcksin and T. D. Young} } @article {2013, title = {N=2 gauge theories on toric singularities, blow-up formulae and W-algebrae}, number = {arXiv:1208.0790;}, year = {2013}, note = {33 pages, 1 figure; v2: discussions on U(1) gauge theory and\r\n Frenkel-Kac construction have been added in section 5.1, and typos corrected;\r\n v3: published version; v4: typos corrected}, institution = {SISSA}, abstract = {We compute the Nekrasov partition function of gauge theories on the\r\n(resolved) toric singularities C^2/\\Gamma in terms of blow-up formulae. We\r\ndiscuss the expansion of the partition function in the \\epsilon_1,\\epsilon_2\r\n\\to 0 limit along with its modular properties and how to derive them from the\r\nM-theory perspective. On the two-dimensional conformal field theory side, our\r\nresults can be interpreted in terms of representations of the direct sum of\r\nHeisenberg plus W_N-algebrae with suitable central charges, which can be\r\ncomputed from the fan of the resolved toric variety.We provide a check of this\r\ncorrespondence by computing the central charge of the two-dimensional theory\r\nfrom the anomaly polynomial of M5-brane theory. Upon using the AGT\r\ncorrespondence our results provide a candidate for the conformal blocks and\r\nthree-point functions of a class of the two-dimensional CFTs which includes\r\nparafermionic theories.}, url = {http://hdl.handle.net/1963/6577}, author = {Giulio Bonelli and Kazunobu Maruyoshi and Alessandro Tanzini and Futoshi Yagi} } @mastersthesis {2013, title = {The structure and regularity of admissible BV solutions to hyperbolic conservation laws in one space dimension}, year = {2013}, school = {SISSA}, abstract = {This thesis is devoted to the study of the qualitative properties of admissible BV solutions to the strictly hyperbolic conservation laws in one space dimension by using wave-front tracking approximation. This thesis consists of two parts: {\textbullet} SBV-like regularity of vanishing viscosity BV solutions to strict hyperbolic systems of conservation laws. {\textbullet} Global structure of admissible BV solutions to strict hyperbolic conservation laws.}, author = {Lei Yu} } @article {2012, title = {A Codazzi-like equation and the singular set for C1 smooth surfaces in the Heisenberg group.}, journal = {Journal fur die Reine und Angewandte Mathematik, Issue 671, October 2012, Pages 131-198}, year = {2012}, publisher = {SISSA}, abstract = {In this paper, we study the structure of the singular set for a C 1 smooth surface in the 3-dimensional Heisenberg group ℍ 1. We discover a Codazzi-like equation for the p-area element along the characteristic curves on the surface. Information obtained from this ordinary differential equation helps us to analyze the local configuration of the singular set and the characteristic curves. In particular, we can estimate the size and obtain the regularity of the singular set. We understand the global structure of the singular set through a Hopf-type index theorem. We also justify the Codazzi-like equation by proving a fundamental theorem for local surfaces in ℍ 1}, doi = {10.1515/CRELLE.2011.159}, url = {http://hdl.handle.net/1963/6556}, author = {Andrea Malchiodi and Paul Yang and Jih-Hsin Cheng and JennFang Hwang} } @article {bianchini2012sbv, title = {SBV-like regularity for general hyperbolic systems of conservation laws in one space dimension}, journal = {Rend. Istit. Mat. Univ. Trieste}, volume = {44}, year = {2012}, pages = {439{\textendash}472}, author = {Stefano Bianchini and Lei Yu} } @article {2011, title = {Axial symmetry of some steady state solutions to nonlinear Schr{\"o}dinger equations}, journal = {Proc. Amer. Math. Soc. 139 (2011), 1023-1032}, number = {SISSA;75/2010/M}, year = {2011}, publisher = {American Mathematical Society}, abstract = {In this note, we show the axial symmetry of steady state solutions of nonlinear Schrodinger equations when the exponent of the nonlinearity is between the critical Sobolev exponent of n dimensional space and n - 1 dimensional space.}, keywords = {Nonlinear Schr{\"o}dinger equation}, doi = {10.1090/S0002-9939-2010-10638-X}, url = {http://hdl.handle.net/1963/4100}, author = {Changfeng Gui and Andrea Malchiodi and Haoyuan Xu and Paul Yang} } @article {2011, title = {D-branes, surface operators, and ADHM quiver representations}, number = {SISSA;01/2011/FM}, year = {2011}, note = {45 pages, v2: minor corrections}, institution = {SISSA}, abstract = {A supersymmetric quantum mechanical model is constructed for BPS states bound to surface operators in five dimensional SU(r) gauge theories using D-brane engineering. This model represents the effective action of a certain D2-brane configuration, and is naturally obtained by dimensional reduction of a quiver $(0,2)$ gauged linear sigma model. In a special stability chamber, the resulting moduli space of quiver representations is shown to be smooth and isomorphic to a moduli space of framed quotients on the projective plane. A precise conjecture relating a K-theoretic partition function of this moduli space to refined open string invariants of toric lagrangian branes is formulated for conifold and local P^1 x P^1 geometries.}, url = {http://hdl.handle.net/1963/4133}, author = {Ugo Bruzzo and Duiliu-Emanuel Diaconescu and M. Yardim and G. Pan and Yi Zhang and Chuang Wu-yen} } @article {2011, title = {Generalized matrix models and AGT correspondence at all genera}, journal = {JHEP, Volume 2011, Issue 7, 2011, Article number055}, number = {arXiv:1011.5417;}, year = {2011}, note = {This version is published in : Journal of High Energy Physics, Volume 2011, Issue 7, 2011, Article number055}, publisher = {SISSA}, abstract = {We study generalized matrix models corresponding to n-point Virasoro\r\nconformal blocks on Riemann surfaces with arbitrary genus g. Upon AGT\r\ncorrespondence, these describe four dimensional N=2 SU(2)^{n+3g-3} gauge\r\ntheories with generalized quiver diagrams. We obtain the generalized matrix\r\nmodels from the perturbative evaluation of the Liouville correlation functions\r\nand verify the consistency of the description with respect to degenerations of\r\nthe Riemann surface. Moreover, we derive the Seiberg-Witten curve for the N=2\r\ngauge theory as the spectral curve of the generalized matrix model, thus\r\nproviding a check of AGT correspondence at all genera.}, doi = {10.1007/JHEP07(2011)055}, url = {http://hdl.handle.net/1963/6568}, author = {Giulio Bonelli and Kazunobu Maruyoshi and Alessandro Tanzini and Futoshi Yagib} } @article {2008, title = {Frobenius Manifolds and Central Invariants for the Drinfeld - Sokolov Bihamiltonian Structures}, journal = {Adv. Math. 219 (2008) 780-837}, number = {arXiv.org;0710.3115}, year = {2008}, abstract = {The Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations.}, doi = {10.1016/j.aim.2008.06.009}, url = {http://hdl.handle.net/1963/2523}, author = {Boris Dubrovin and Liu Si-Qi and Zhang Youjin} } @article {2007, title = {Asymptotic variational wave equations}, journal = {Arch. Ration. Mech. Anal. 183 (2007) 163-185}, number = {arXiv.org;math/0502124v1}, year = {2007}, abstract = {We investigate the equation $(u_t + (f(u))_x)_x = f\\\'\\\'(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave equation $u_{tt} - c(u) (c(u)u_x)_x =0$ which models some liquid crystals with a natural sinusoidal $c$. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view.\\nWe establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function $f$ has Lipschitz continuous second-order derivative. In the case where $f$ is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when $f$ is not convex, we show that the dissipative solutions do not depend continuously on the initial data.}, doi = {10.1007/s00205-006-0014-8}, url = {http://hdl.handle.net/1963/2182}, author = {Alberto Bressan and Zhang Ping and Zheng Yuxi} } @article {2006, title = {Conservative Solutions to a Nonlinear Variational Wave Equation}, journal = {Comm. Math. Phys. 266 (2006) 471-497}, number = {arXiv.org;math/0502058v1}, year = {2006}, abstract = {We establish the existence of a conservative weak solution to the Cauchy problem for the nonlinear variational wave equation $u_{tt} - c(u)(c(u)u_x)_x=0$, for initial data of finite energy. Here $c(\\\\cdot)$ is any smooth function with uniformly positive bounded values.}, doi = {10.1007/s00220-006-0047-8}, url = {http://hdl.handle.net/1963/2184}, author = {Alberto Bressan and Zheng Yuxi} } @article {2006, title = {Extended affine Weyl groups and Frobenius manifolds -- II}, number = {SISSA;90/2005/FM}, year = {2006}, abstract = {For the root system of type $B_l$ and $C_l$, we generalize the result of \\\\cite{DZ1998} by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of \\\\cite{DZ1998}.}, url = {http://hdl.handle.net/1963/1787}, author = {Boris Dubrovin and Zhang Youjin and Zuo Dafeng} } @article {2006, title = {On Hamiltonian perturbations of hyperbolic systems of conservation laws I: quasitriviality of bihamiltonian perturbations}, journal = {Comm. Pure Appl. Math. 59 (2006) 559-615}, number = {arXiv.org;math/0410027}, year = {2006}, abstract = {We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the so-called quasi-Miura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following \\\\cite{LZ1}, the invariants of such bihamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.}, doi = {10.1002/cpa.20111}, url = {http://hdl.handle.net/1963/2535}, author = {Boris Dubrovin and Liu Si-Qi and Zhang Youjin} } @article {2005, title = {Minimal surfaces in pseudohermitian geometry}, journal = {Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (5), 4 (2005) 129-177.}, number = {arXiv:math/0401136;}, year = {2005}, publisher = {Scuola Normale Superiore}, abstract = {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.}, doi = {10.2422/2036-2145.2005.1.05}, url = {http://hdl.handle.net/1963/4579}, author = {Jih-Hsin Cheng and JennFang Hwang and Andrea Malchiodi and Paul Yang} } @article {2004, title = {On the convergence rate of vanishing viscosity approximations}, journal = {Comm. Pure Appl. Math. 57 (2004) 1075-1109}, number = {SISSA;60/2003/M}, year = {2004}, publisher = {Wiley}, abstract = {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.}, doi = {10.1002/cpa.20030}, url = {http://hdl.handle.net/1963/2915}, author = {Alberto Bressan and Tong Yang} } @article {2004, title = {The Extended Toda Hierarchy}, journal = {Moscow Math. J. 4 (2004)\\n313-332.}, number = {arXiv.org;nlin/0306060}, year = {2004}, url = {http://hdl.handle.net/1963/2542}, author = {Guido Carlet and Boris Dubrovin and Zhang Youjin} } @article {2004, title = {A sharp decay estimate for positive nonlinear waves}, journal = {SIAM J. Math. Anal. 36 (2004) 659-677}, number = {SISSA;59/2003/M}, year = {2004}, publisher = {SIAM}, abstract = {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.}, doi = {10.1137/S0036141003427774}, url = {http://hdl.handle.net/1963/2916}, author = {Alberto Bressan and Tong Yang} } @article {2004, title = {Virasoro Symmetries of the Extended Toda Hierarchy}, journal = {Comm. Math.\\nPhys. 250 (2004) 161-193.}, number = {arXiv.org;math/0308152}, year = {2004}, abstract = {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.}, doi = {10.1007/s00220-004-1084-9}, url = {http://hdl.handle.net/1963/2544}, author = {Boris Dubrovin and Zhang Youjin} } @article {2002, title = {On the Yamabe problem and the scalar curvature problems under boundary conditions}, journal = {Math. Ann., 2002, 322, 667}, number = {SISSA;52/00/M}, year = {2002}, publisher = {SISSA Library}, doi = {10.1007/s002080100267}, url = {http://hdl.handle.net/1963/1510}, author = {Antonio Ambrosetti and Li YanYan and Andrea Malchiodi} } @article {2000, title = {A note on the scalar curvature problem in the presence of symmetries}, journal = {Ricerche Mat. 49 (2000), suppl., 169-176}, number = {SISSA;150/99/M}, year = {2000}, publisher = {SISSA Library}, url = {http://hdl.handle.net/1963/1365}, author = {Antonio Ambrosetti and Li YanYan and Andrea Malchiodi} } @article {2000, title = {Scalar curvature under boundary conditions}, journal = {Cr. Acad. Sci. I-Math, 2000, 330, 1013}, number = {SISSA;48/00/M}, year = {2000}, publisher = {SISSA Library}, doi = {10.1016/S0764-4442(00)00312-8}, url = {http://hdl.handle.net/1963/1506}, author = {Antonio Ambrosetti and Li YanYan and Andrea Malchiodi} } @article {1999, title = {Frobenius manifolds and Virasoro constraints}, journal = {Selecta Math. (N.S.) 5 (1999) 423-466}, number = {SISSA;84/98/FM}, year = {1999}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s000290050053}, url = {http://hdl.handle.net/1963/2883}, author = {Boris Dubrovin and Zhang Youjin} } @article {1999, title = {L-1 stability estimates for n x n conservation laws}, journal = {Arch. Ration. Mech. Anal. 149 (1999), no. 1, 1--22}, number = {SISSA;80/98/M}, year = {1999}, publisher = {Springer}, abstract = {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 {\textquoteleft}almost decreasing\\\', i.e., $\\\\Phi(u(t),v(t))-\\\\Phi(u(s),v(s))\\\\leq\\\\break O (\\\\epsilon){\textperiodcentered}(t-s)$ 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.\\\'\\\'}, doi = {10.1007/s002050050165}, url = {http://hdl.handle.net/1963/3373}, author = {Alberto Bressan and Tai-Ping Liu and Tong Yang} } @article {1998, title = {Bihamiltonian Hierarchies in 2D Topological Field Theory At One-Loop Approximation}, journal = {Comm. Math. Phys. 198 (1998) 311-361}, number = {SISSA;152/97/FM}, year = {1998}, publisher = {Springer}, abstract = {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.}, doi = {10.1007/s002200050480}, url = {http://hdl.handle.net/1963/3696}, author = {Boris Dubrovin and Zhang Youjin} }