In this paper, we estimate from above the area of the graph of a singular map u taking a disk to three vectors, the vertices of a triangle, and jumping along three $\mathcal{C}^2$-embedded curves that meet transversely at only one point of the disk. We show that the singular part of the relaxed area can be estimated from above by the solution of a Plateau-type problem involving three entangled nonparametric area-minimizing surfaces. The idea is to ``fill the hole'' in the graph of the singular map with a sequence of approximating smooth two-codimensional surfaces of graph-type, by imagining three minimal surfaces, placed vertically over the jump of u, coupled together via a triple point in the target triangle. Such a construction depends on the choice of a target triple point, and on a connection passing through it, which dictate the boundary condition for the three minimal surfaces. We show that the singular part of the relaxed area of u cannot be larger than what we obtain by minimizing over all possible target triple points and all corresponding connections.

UR - https://doi.org/10.1007/s10231-019-00887-0 ER - TY - RPRT T1 - On the square distance function from a manifold with boundary Y1 - 2019 A1 - Giovanni Bellettini A1 - Alaa Elshorbagy AB -We characterize arbitrary codimensional smooth manifolds $\mathcal{M}$ with boundary embedded in $\mathbb{R}^n$ using the square distance function and the signed distance function from $\mathcal{M}$ and from its boundary. The results are localized in an open set.

UR - http://cvgmt.sns.it/media/doc/paper/4260/manif_with_bound_dist.pdf ER - TY - JOUR T1 - Strong Novikov conjecture for low degree cohomology and exotic group C*-algebras JF - arXiv e-prints Y1 - 2019 A1 - Paolo Antonini A1 - Buss, Alcides A1 - Engel, Alexander A1 - Siebenand , Timo KW - Mathematics - K-Theory and Homology KW - Mathematics - Operator Algebras AB -We strengthen a result of Hanke–Schick about the strong Novikov conjecture for low degree cohomology by showing that their non-vanishing result for the maximal group $C^*$-algebra even holds for the reduced group $C^*$-algebra. To achieve this we provide a Fell absorption principle for certain exotic crossed product functors.

ER - TY - JOUR T1 - Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane JF - Symmetry, Integrability and Geometry. Methods and Applications Y1 - 2018 A1 - Marco Bertola A1 - José Gustavo Elias Rebelo A1 - Tamara Grava AB -We study the asymptotic behaviour of orthogonal polynomials in the complex plane that are associated to a certain normal matrix model. The model depends on a parameter and the asymptotic distribution of the eigenvalues undergoes a transition for a special value of the parameter, where it develops a corner-type singularity. In the double scaling limit near the transition we determine the asymptotic behaviour of the orthogonal polynomials in terms of a solution of the Painlev´e IV equation. We determine the Fredholm determinant associated to such solution and we compute it numerically on the real line, showing also that the corresponding Painlev´e transcendent is pole-free on a semiaxis.

PB - National Academy of Sciences of Ukraine VL - 14 ER - TY - RPRT T1 - Complex Friedrichs systems and applications Y1 - 2017 A1 - Nenad Antonić A1 - Krešimir Burazin A1 - Ivana Crnjac A1 - Marko Erceg AB - We provide a suitable extension of the theory of abstract Friedrichs systems from real Hilbert spaces to the complex Hilbert space setting, which allows for applications to partial differential equations with complex coeffcients. We also provide examples where the involved Hilbert space is not the space of square integrable functions, as it was the case in previous works, but rather its closed subspace or the space Hs(Rd;Cr), for real s. This setting appears to be suitable for particular systems of partial differential equations, such as the Dirac system, the Dirac-Klein-Gordon system, the Dirac-Maxwell system, and the time-harmonic Maxwell system, which are all addressed in the paper. Moreover, for the time-harmonic Maxwell system we also applied a suitable version of the two-field theory with partial coercivity assumption which is developed in the paper. UR - http://urania.sissa.it/xmlui/handle/1963/35270 U1 - 35576 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - On contact interactions realised as Friedrichs systems Y1 - 2017 A1 - Marko Erceg A1 - Alessandro Michelangeli AB - We realise the Hamiltonians of contact interactions in quantum mechanics within the framework of abstract Friedrichs systems. In particular, we show that the construction of the self-adjoint (or even only closed) operators of contact interaction supported at a fixed point can be associated with the construction of the bijective realisations of a suitable pair of abstract Friedrich operators. In this respect, the Hamiltonians of contact interaction provide novel examples of abstract Friedrich systems. UR - http://preprints.sissa.it/handle/1963/35298 U1 - 35604 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Friedrichs systems in a Hilbert space framework: solvability and multiplicity Y1 - 2017 A1 - Nenad Antonić A1 - Marko Erceg A1 - Alessandro Michelangeli AB - The Friedrichs (1958) theory of positive symmetric systems of first order partial differential equations encompasses many standard equations of mathematical physics, irrespective of their type. This theory was recast in an abstract Hilbert space setting by Ern, Guermond and Caplain (2007), and by Antonić and Burazin (2010). In this work we make a further step, presenting a purely operator-theoretic description of abstract Friedrichs systems, and proving that any pair of abstract Friedrichs operators admits bijective extensions with a signed boundary map. Moreover, we provide suffcient and necessary conditions for existence of infinitely many such pairs of spaces, and by the universal operator extension theory (Grubb, 1968) we get a complete identification of all such pairs, which we illustrate on two concrete one-dimensional examples. UR - http://preprints.sissa.it/handle/1963/35280 U1 - 35587 U2 - Mathematics U4 - 1 ER - TY - JOUR T1 - Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation JF - Phys. Rev. Lett. Y1 - 2017 A1 - Tikan, Alexey A1 - Billet, Cyril A1 - Gennady El A1 - Alexander Tovbis A1 - Marco Bertola A1 - Sylvestre, Thibaut A1 - Gustave, Francois A1 - Randoux, Stephane A1 - Genty, Goëry A1 - Suret, Pierre A1 - Dudley, John M. PB - American Physical Society VL - 119 UR - https://link.aps.org/doi/10.1103/PhysRevLett.119.033901 ER - TY - JOUR T1 - Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation JF - Proc. A. Y1 - 2016 A1 - Marco Bertola A1 - Gennady El A1 - Alexander Tovbis VL - 472 UR - http://dx.doi.org/10.1098/rspa.2016.0340 ER - TY - JOUR T1 - On the critical behavior in nonlinear evolutionary PDEs with small viscocity JF - Russian Journal of Mathematical Physics. Volume 19, Issue 4, December 2012, Pages 449-460 Y1 - 2012 A1 - Boris Dubrovin A1 - Maria Elaeva AB - We address the problem of general dissipative regularization of the quasilinear transport equation. We argue that the local behavior of solutions to the regularized equation near the point of gradient catastrophe for the transport equation is described by the logarithmic derivative of the Pearcey function, a statement generalizing the result of A.M.Il\\\'in \\\\cite{ilin}. We provide some analytic arguments supporting such conjecture and test it numerically. PB - SISSA UR - http://hdl.handle.net/1963/6465 U1 - 6409 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Adaptation as a genome-wide autoregulatory principle in the stress response of yeast. JF - IET systems biology. 2011 Jul; 5(4):269-79 Y1 - 2011 A1 - F Eduati A1 - B Di Camillo A1 - G Toffolo A1 - Claudio Altafini A1 - Giovanna De Palo A1 - Mattia Zampieri AB - The gene expression response of yeast to various types of stresses/perturbations shows a common functional and dynamical pattern for the vast majority of genes, characterised by a quick transient peak (affecting primarily short genes) followed by a return to the pre-stimulus level. Kinetically, this process of adaptation following the transient excursion can be modelled using a genome-wide autoregulatory mechanism by means of which yeast aims at maintaining a preferential concentration in its mRNA levels. The resulting feedback system explains well the different time constants observable in the transient response, while being in agreement with all the known experimental dynamical features. For example, it suggests that a very rapid transient can be induced also by a slowly varying concentration of the gene products. PB - The Institution of Engineering and Technology UR - http://hdl.handle.net/1963/5106 U1 - 4922 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - The PDEs of biorthogonal polynomials arising in the two-matrix model JF - Math. Phys. Anal. Geom. Y1 - 2006 A1 - Marco Bertola A1 - B. Eynard VL - 9 ER - TY - JOUR T1 - Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions JF - Comm. Math. Phys. Y1 - 2006 A1 - Marco Bertola A1 - B. Eynard A1 - Harnad, J. VL - 263 ER - TY - JOUR T1 - Thomae type formulae for singular Z_N curves JF - Lett. Math. Phys. 76 (2006) 187-214 Y1 - 2006 A1 - Victor Z. Enolski A1 - Tamara Grava AB - We give an elementary and rigorous proof of the Thomae type formula for singular $Z_N$ curves. To derive the Thomae formula we use the traditional variational method which goes back to Riemann, Thomae and Fuchs. An important step of the proof is the use of the Szego kernel computed explicitly in algebraic form for non-singular 1/N-periods. The proof inherits principal points of Nakayashiki\\\'s proof [31], obtained for non-singular ZN curves. UR - http://hdl.handle.net/1963/2125 U1 - 2118 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation JF - Int. Math. Res. Not. 2004, no. 32, 1619-1683 Y1 - 2004 A1 - Victor Z. Enolski A1 - Tamara Grava AB - We are solving the classical Riemann-Hilbert problem of rank N>1 on the extended complex plane punctured in 2m+2 points, for NxN quasi-permutation monodromy matrices. Following Korotkin we solve the Riemann-Hilbert problem in terms of the Szego kernel of certain Riemann surfaces branched over the given 2m+2 points. These Riemann surfaces are constructed from a permutation representation of the symmetric group S_N to which the quasi-permutation monodromy representation has been reduced. The permutation representation of our problem generates the cyclic subgroup Z_N. For this reason the corresponding Riemann surfaces of genus N(m-1) have Z_N symmetry. This fact enables us to write the matrix entries of the solution of the NxN Riemann-Hilbert problem as a product of an algebraic function and theta-function quotients. The algebraic function turns out to be related to the Szego kernel with zero characteristics. From the solution of the Riemann- Hilbert problem we automatically obtain a particular solution of the Schlesinger system. The tau-function of the Schlesinger system is computed explicitly. The rank 3 problem with four singular points (0,t,1,\\\\infty) is studied in detail. The corresponding solution of the Riemann-Hilbert problem and the Schlesinger system is given in terms of Jacobi\\\'s theta-function with modulus T=T(t), Im(T)>0. The function T=T(t) is invertible if it belongs to the Siegel upper half space modulo the subgroup \\\\Gamma_0(3) of the modular group. The inverse function t=t(T) generates a solution of a general Halphen system. UR - http://hdl.handle.net/1963/2540 U1 - 1579 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem JF - Comm. Math. Phys. Y1 - 2003 A1 - Marco Bertola A1 - B. Eynard A1 - Harnad, J. VL - 243 ER - TY - JOUR T1 - The duality of spectral curves that arises in two-matrix models JF - Teoret. Mat. Fiz. Y1 - 2003 A1 - Marco Bertola A1 - B. Eynard A1 - Kharnad, Dzh. VL - 134 ER - TY - JOUR T1 - Mixed correlation functions of the two-matrix model JF - J. Phys. A Y1 - 2003 A1 - Marco Bertola A1 - B. Eynard VL - 36 ER - TY - JOUR T1 - A note on the integral representation of functionals in the space SBD(O) JF - Rend. Mat. Appl. 23 (2003) 189-201 Y1 - 2003 A1 - Francois Ebobisse A1 - Rodica Toader AB - In this paper we study the integral representation in the space SBD(O) of special functions with bounded deformation of some L^1-norm lower semicontinuous functionals invariant with respect to rigid motions. PB - Rendiconti di Matematica UR - http://hdl.handle.net/1963/3064 U1 - 1269 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Partition functions for matrix models and isomonodromic tau functions JF - J. Phys. A Y1 - 2003 A1 - Marco Bertola A1 - B. Eynard A1 - Harnad, J. VL - 36 N1 - Random matrix theory ER - TY - JOUR T1 - A stability result for nonlinear Neumann problems under boundary variations JF - J.Math. Pures Appl. (9) 82 (2003) no.5 , 503 Y1 - 2003 A1 - Gianni Dal Maso A1 - Francois Ebobisse A1 - Marcello Ponsiglione AB - In this paper we study, in dimension two, the stability of the solutions of some nonlinear elliptic equations with Neumann boundary conditions, under perturbations of the domains in the Hausdorff complementary topology. PB - SISSA Library UR - http://hdl.handle.net/1963/1618 U1 - 2500 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Duality, biorthogonal polynomials and multi-matrix models JF - Comm. Math. Phys. Y1 - 2002 A1 - Marco Bertola A1 - B. Eynard A1 - Harnad, J. VL - 229 ER - TY - JOUR T1 - Symmetry breaking in Hamiltonian systems JF - J. Differential Equations 67 (1987), no. 2, 165-184 Y1 - 1987 A1 - Antonio Ambrosetti A1 - Vittorio Coti Zelati A1 - Ivar Ekeland PB - SISSA Library UR - http://hdl.handle.net/1963/409 U1 - 3558 U2 - Mathematics U3 - Functional Analysis and Applications ER -