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.

%B Annali di Matematica Pura ed Applicata (1923 -) %8 Jul %G eng %U https://doi.org/10.1007/s10231-019-00887-0 %R 10.1007/s10231-019-00887-0 %0 Report %D 2019 %T On the square distance function from a manifold with boundary %A Giovanni Bellettini %A Alaa Elshorbagy %XWe 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.

%G eng %U http://cvgmt.sns.it/media/doc/paper/4260/manif_with_bound_dist.pdf %0 Journal Article %J arXiv e-prints %D 2019 %T Strong Novikov conjecture for low degree cohomology and exotic group C*-algebras %A Paolo Antonini %A Buss, Alcides %A Engel, Alexander %A Siebenand , Timo %K Mathematics - K-Theory and Homology %K Mathematics - Operator Algebras %XWe 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.

%B arXiv e-prints %P arXiv:1905.07730 %8 May %G eng %0 Journal Article %J Symmetry, Integrability and Geometry. Methods and Applications %D 2018 %T Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane %A Marco Bertola %A José Gustavo Elias Rebelo %A Tamara Grava %XWe 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.

%B Symmetry, Integrability and Geometry. Methods and Applications %I National Academy of Sciences of Ukraine %V 14 %G eng %R 10.3842/SIGMA.2018.091 %0 Report %D 2017 %T Complex Friedrichs systems and applications %A Nenad Antonić %A Krešimir Burazin %A Ivana Crnjac %A Marko Erceg %X 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. %G en %U http://urania.sissa.it/xmlui/handle/1963/35270 %1 35576 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-01-20T12:35:16Z No. of bitstreams: 1 antonic_burazin_crnjac_erceg2016_preprint.pdf: 308035 bytes, checksum: 7c8e2264f51ab9699c6cbe9585fc047a (MD5) %0 Report %D 2017 %T On contact interactions realised as Friedrichs systems %A Marko Erceg %A Alessandro Michelangeli %X 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. %G en %U http://preprints.sissa.it/handle/1963/35298 %1 35604 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-10-10T11:23:22Z No. of bitstreams: 1 SISSA_preprint_48-2017-MATE.pdf: 431531 bytes, checksum: 0368c692a322778cda197c49c320b3a6 (MD5) %0 Report %D 2017 %T Friedrichs systems in a Hilbert space framework: solvability and multiplicity %A Nenad Antonić %A Marko Erceg %A Alessandro Michelangeli %X 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. %G en %U http://preprints.sissa.it/handle/1963/35280 %1 35587 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2017-04-11T07:42:37Z No. of bitstreams: 1 SISSA_preprint_16-2017-MATE.pdf: 323262 bytes, checksum: 17892409d83085ec46707fac64056fc1 (MD5) %0 Journal Article %J Phys. Rev. Lett. %D 2017 %T Universality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation %A Tikan, Alexey %A Billet, Cyril %A Gennady El %A Alexander Tovbis %A Marco Bertola %A Sylvestre, Thibaut %A Gustave, Francois %A Randoux, Stephane %A Genty, Goëry %A Suret, Pierre %A Dudley, John M. %B Phys. Rev. Lett. %I American Physical Society %V 119 %P 033901 %8 Jul %G eng %U https://link.aps.org/doi/10.1103/PhysRevLett.119.033901 %R 10.1103/PhysRevLett.119.033901 %0 Journal Article %J Proc. A. %D 2016 %T Rogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation %A Marco Bertola %A Gennady El %A Alexander Tovbis %B Proc. A. %V 472 %P 20160340, 12 %G eng %U http://dx.doi.org/10.1098/rspa.2016.0340 %R 10.1098/rspa.2016.0340 %0 Journal Article %J Russian Journal of Mathematical Physics. Volume 19, Issue 4, December 2012, Pages 449-460 %D 2012 %T On the critical behavior in nonlinear evolutionary PDEs with small viscocity %A Boris Dubrovin %A Maria Elaeva %X 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. %B Russian Journal of Mathematical Physics. Volume 19, Issue 4, December 2012, Pages 449-460 %I SISSA %G en %U http://hdl.handle.net/1963/6465 %1 6409 %2 Mathematics %4 1 %# MAT/07 FISICA MATEMATICA %$ Submitted by Boris Dubrovin (dubrovin@sissa.it) on 2013-02-08T11:28:04Z\\nNo. of bitstreams: 1\\ndubrovin_elaeva.pdf: 8618184 bytes, checksum: 3c56d1922af7001581ca4ceb4c79cc3b (MD5) %R 10.1134/S106192081204005X %0 Journal Article %J IET systems biology. 2011 Jul; 5(4):269-79 %D 2011 %T Adaptation as a genome-wide autoregulatory principle in the stress response of yeast. %A F Eduati %A B Di Camillo %A G Toffolo %A Claudio Altafini %A Giovanna De Palo %A Mattia Zampieri %X 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. %B IET systems biology. 2011 Jul; 5(4):269-79 %I The Institution of Engineering and Technology %G en %U http://hdl.handle.net/1963/5106 %1 4922 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-11-25T13:05:49Z\\r\\nNo. of bitstreams: 1\\r\\nDePaEdal10.pdf: 2390366 bytes, checksum: f6550b1052efeb53942e47cae9632899 (MD5) %R 10.1049/iet-syb.2009.0050 %0 Journal Article %J Math. Phys. Anal. Geom. %D 2006 %T The PDEs of biorthogonal polynomials arising in the two-matrix model %A Marco Bertola %A B. Eynard %B Math. Phys. Anal. Geom. %V 9 %P 23–52 %G eng %0 Journal Article %J Comm. Math. Phys. %D 2006 %T Semiclassical orthogonal polynomials, matrix models and isomonodromic tau functions %A Marco Bertola %A B. Eynard %A Harnad, J. %B Comm. Math. Phys. %V 263 %P 401–437 %G eng %0 Journal Article %J Lett. Math. Phys. 76 (2006) 187-214 %D 2006 %T Thomae type formulae for singular Z_N curves %A Victor Z. Enolski %A Tamara Grava %X 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. %B Lett. Math. Phys. 76 (2006) 187-214 %G en_US %U http://hdl.handle.net/1963/2125 %1 2118 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-18T08:04:36Z\\nNo. of bitstreams: 1\\n0602017v1.pdf: 315977 bytes, checksum: 2f5c46aa04d9dd6c02200ff627d69a00 (MD5) %R 10.1007/s11005-006-0073-7 %0 Journal Article %J Int. Math. Res. Not. 2004, no. 32, 1619-1683 %D 2004 %T Singular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation %A Victor Z. Enolski %A Tamara Grava %X 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. %B Int. Math. Res. Not. 2004, no. 32, 1619-1683 %G en_US %U http://hdl.handle.net/1963/2540 %1 1579 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-21T10:08:03Z\\nNo. of bitstreams: 1\\n0306050v2.pdf: 587970 bytes, checksum: 3d350364af41be2317176944f1d85868 (MD5) %R 10.1155/S1073792804132625 %0 Journal Article %J Comm. Math. Phys. %D 2003 %T Differential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem %A Marco Bertola %A B. Eynard %A Harnad, J. %B Comm. Math. Phys. %V 243 %P 193–240 %G eng %0 Journal Article %J Teoret. Mat. Fiz. %D 2003 %T The duality of spectral curves that arises in two-matrix models %A Marco Bertola %A B. Eynard %A Kharnad, Dzh. %B Teoret. Mat. Fiz. %V 134 %P 32–45 %G eng %0 Journal Article %J J. Phys. A %D 2003 %T Mixed correlation functions of the two-matrix model %A Marco Bertola %A B. Eynard %B J. Phys. A %V 36 %P 7733–7750 %G eng %0 Journal Article %J Rend. Mat. Appl. 23 (2003) 189-201 %D 2003 %T A note on the integral representation of functionals in the space SBD(O) %A Francois Ebobisse %A Rodica Toader %X 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. %B Rend. Mat. Appl. 23 (2003) 189-201 %I Rendiconti di Matematica %G en_US %U http://hdl.handle.net/1963/3064 %1 1269 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-10-10T08:08:06Z\\nNo. of bitstreams: 1\\n0104264v1.pdf: 157593 bytes, checksum: 8afd09d4c0f34e5fa55e357804395f3d (MD5) %0 Journal Article %J J. Phys. A %D 2003 %T Partition functions for matrix models and isomonodromic tau functions %A Marco Bertola %A B. Eynard %A Harnad, J. %B J. Phys. A %V 36 %P 3067–3083 %G eng %0 Journal Article %J J.Math. Pures Appl. (9) 82 (2003) no.5 , 503 %D 2003 %T A stability result for nonlinear Neumann problems under boundary variations %A Gianni Dal Maso %A Francois Ebobisse %A Marcello Ponsiglione %X 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. %B J.Math. Pures Appl. (9) 82 (2003) no.5 , 503 %I SISSA Library %G en %U http://hdl.handle.net/1963/1618 %1 2500 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:05:23Z (GMT). No. of bitstreams: 1\\nmath.AP0206215.pdf: 329673 bytes, checksum: 40d537fd197fbaec56be0729d3a72ee3 (MD5)\\n Previous issue date: 2002 %R 10.1016/S0021-7824(03)00014-X %0 Journal Article %J Comm. Math. Phys. %D 2002 %T Duality, biorthogonal polynomials and multi-matrix models %A Marco Bertola %A B. Eynard %A Harnad, J. %B Comm. Math. Phys. %V 229 %P 73–120 %G eng %0 Journal Article %J J. Differential Equations 67 (1987), no. 2, 165-184 %D 1987 %T Symmetry breaking in Hamiltonian systems %A Antonio Ambrosetti %A Vittorio Coti Zelati %A Ivar Ekeland %B J. Differential Equations 67 (1987), no. 2, 165-184 %I SISSA Library %G en %U http://hdl.handle.net/1963/409 %1 3558 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T12:32:09Z (GMT). No. of bitstreams: 0\\n Previous issue date: 1985