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.

1 aBellettini, Giovanni1 aElshorbagy, Alaa1 aPaolini, Maurizio1 aScala, Riccardo uhttps://doi.org/10.1007/s10231-019-00887-000668nas a2200109 4500008004100000245006600041210005900107520027500166100002500441700002100466856007100487 2019 eng d00aOn the square distance function from a manifold with boundary0 asquare distance function from a manifold with boundary3 aWe 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.

1 aBellettini, Giovanni1 aElshorbagy, Alaa uhttp://cvgmt.sns.it/media/doc/paper/4260/manif_with_bound_dist.pdf01020nas a2200181 4500008004100000245008500041210006900126260000800195300002100203520034100224653004000565653003600605100002000641700001800661700002100679700002000700856011800720 2019 eng d00aStrong Novikov conjecture for low degree cohomology and exotic group C*-algebras0 aStrong Novikov conjecture for low degree cohomology and exotic g cMay aarXiv:1905.077303 aWe 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.

10aMathematics - K-Theory and Homology10aMathematics - Operator Algebras1 aAntonini, Paolo1 aBuss, Alcides1 aEngel, Alexander1 aSiebenand, Timo uhttps://www.math.sissa.it/publication/strong-novikov-conjecture-low-degree-cohomology-and-exotic-group-c-algebras01288nas a2200145 4500008004100000245008600041210007000127260004400197490000700241520070800248100001900956700003200975700001801007856011701025 2018 eng d00aPainlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane0 aPainlevé IV Critical Asymptotics for Orthogonal Polynomials in t bNational Academy of Sciences of Ukraine0 v143 aWe 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.

1 aBertola, Marco1 aElias Rebelo, José Gustavo1 aGrava, Tamara uhttps://www.math.sissa.it/publication/painlev%C3%A9-iv-critical-asymptotics-orthogonal-polynomials-complex-plane01282nas a2200133 4500008004100000245004800041210004800089520088200137100002001019700002301039700001801062700001701080856005101097 2017 en d00aComplex Friedrichs systems and applications0 aComplex Friedrichs systems and applications3 aWe 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.1 aAntonić, Nenad1 aBurazin, Krešimir1 aCrnjac, Ivana1 aErceg, Marko uhttp://urania.sissa.it/xmlui/handle/1963/3527000871nas a2200109 4500008004100000245005900041210005600100520051100156100001700667700002900684856004800713 2017 en d00aOn contact interactions realised as Friedrichs systems0 acontact interactions realised as Friedrichs systems3 aWe 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.1 aErceg, Marko1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3529801281nas a2200121 4500008004100000245008200041210006900123520085300192100002001045700001701065700002901082856004801111 2017 en d00aFriedrichs systems in a Hilbert space framework: solvability and multiplicity0 aFriedrichs systems in a Hilbert space framework solvability and 3 aThe 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.1 aAntonić, Nenad1 aErceg, Marko1 aMichelangeli, Alessandro uhttp://preprints.sissa.it/handle/1963/3528000795nas a2200241 4500008004100000245011200041210006900153260003500222300001100257490000800268100001800276700001800294700001600312700002200328700001900350700002300369700002200392700002200414700001800436700001800454700002100472856006000493 2017 eng d00aUniversality of the Peregrine Soliton in the Focusing Dynamics of the Cubic Nonlinear Schrödinger Equation0 aUniversality of the Peregrine Soliton in the Focusing Dynamics o bAmerican Physical SocietycJul a0339010 v1191 aTikan, Alexey1 aBillet, Cyril1 aEl, Gennady1 aTovbis, Alexander1 aBertola, Marco1 aSylvestre, Thibaut1 aGustave, Francois1 aRandoux, Stephane1 aGenty, Goëry1 aSuret, Pierre1 aDudley, John, M. uhttps://link.aps.org/doi/10.1103/PhysRevLett.119.03390100485nas a2200145 4500008004100000022001400041245008800055210006900143300001700212490000800229100001900237700001600256700002200272856004500294 2016 eng d a1364-502100aRogue waves in multiphase solutions of the focusing nonlinear Schrödinger equation0 aRogue waves in multiphase solutions of the focusing nonlinear Sc a20160340, 120 v4721 aBertola, Marco1 aEl, Gennady1 aTovbis, Alexander uhttp://dx.doi.org/10.1098/rspa.2016.034000864nas a2200121 4500008004100000245008100041210006900122260001000191520046700201100002000668700001800688856003600706 2012 en d00aOn the critical behavior in nonlinear evolutionary PDEs with small viscocity0 acritical behavior in nonlinear evolutionary PDEs with small visc bSISSA3 aWe 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.1 aDubrovin, Boris1 aElaeva, Maria uhttp://hdl.handle.net/1963/646501400nas a2200169 4500008004100000245009000041210006900131260005000200520083200250100001401082700001801096700001501114700002201129700002201151700002101173856003601194 2011 en d00aAdaptation as a genome-wide autoregulatory principle in the stress response of yeast.0 aAdaptation as a genomewide autoregulatory principle in the stres bThe Institution of Engineering and Technology3 aThe 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.1 aEduati, F1 aDi Camillo, B1 aToffolo, G1 aAltafini, Claudio1 aDe Palo, Giovanna1 aZampieri, Mattia uhttp://hdl.handle.net/1963/510600479nas a2200133 4500008004100000022001400041245007300055210006800128300001200196490000600208100001900214700001500233856009700248 2006 eng d a1385-017200aThe PDEs of biorthogonal polynomials arising in the two-matrix model0 aPDEs of biorthogonal polynomials arising in the twomatrix model a23–520 v91 aBertola, Marco1 aEynard, B. uhttps://www.math.sissa.it/publication/pdes-biorthogonal-polynomials-arising-two-matrix-model00554nas a2200145 4500008004100000022001400041245008800055210006900143300001400212490000800226100001900234700001500253700001500268856012500283 2006 eng d a0010-361600aSemiclassical orthogonal polynomials, matrix models and isomonodromic tau functions0 aSemiclassical orthogonal polynomials matrix models and isomonodr a401–4370 v2631 aBertola, Marco1 aEynard, B.1 aHarnad, J. uhttps://www.math.sissa.it/publication/semiclassical-orthogonal-polynomials-matrix-models-and-isomonodromic-tau-functions00778nas a2200109 4500008004300000245004900043210004800092520045100140100002300591700001800614856003600632 2006 en_Ud 00aThomae type formulae for singular Z_N curves0 aThomae type formulae for singular ZN curves3 aWe 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.1 aEnolski, Victor Z.1 aGrava, Tamara uhttp://hdl.handle.net/1963/212501974nas a2200109 4500008004300000245009900043210006900142520157600211100002301787700001801810856003601828 2004 en_Ud 00aSingular Z_N curves, Riemann-Hilbert problem and modular solutions of the Schlesinger equation0 aSingular ZN curves RiemannHilbert problem and modular solutions 3 aWe 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.1 aEnolski, Victor Z.1 aGrava, Tamara uhttp://hdl.handle.net/1963/254000596nas a2200145 4500008004100000022001400041245012600055210006900181300001400250490000800264100001900272700001500291700001500306856012900321 2003 eng d a0010-361600aDifferential systems for biorthogonal polynomials appearing in 2-matrix models and the associated Riemann-Hilbert problem0 aDifferential systems for biorthogonal polynomials appearing in 2 a193–2400 v2431 aBertola, Marco1 aEynard, B.1 aHarnad, J. uhttps://www.math.sissa.it/publication/differential-systems-biorthogonal-polynomials-appearing-2-matrix-models-and-associated00495nas a2200145 4500008004100000022001400041245006800055210006300123300001200186490000800198100001900206700001500225700001800240856009100258 2003 eng d a0564-616200aThe duality of spectral curves that arises in two-matrix models0 aduality of spectral curves that arises in twomatrix models a32–450 v1341 aBertola, Marco1 aEynard, B.1 aKharnad, Dzh. uhttps://www.math.sissa.it/publication/duality-spectral-curves-arises-two-matrix-models00444nas a2200133 4500008004100000022001400041245005600055210005500111300001600166490000700182100001900189700001500208856008700223 2003 eng d a0305-447000aMixed correlation functions of the two-matrix model0 aMixed correlation functions of the twomatrix model a7733–77500 v361 aBertola, Marco1 aEynard, B. uhttps://www.math.sissa.it/publication/mixed-correlation-functions-two-matrix-model00630nas a2200121 4500008004300000245007700043210006900120260002900189520021200218100002300430700001900453856003600472 2003 en_Ud 00aA note on the integral representation of functionals in the space SBD(O)0 anote on the integral representation of functionals in the space bRendiconti di Matematica3 aIn 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.1 aEbobisse, Francois1 aToader, Rodica uhttp://hdl.handle.net/1963/306400524nas a2200145 4500008004100000022001400041245007400055210006900129300001600198490000700214100001900221700001500240700001500255856010800270 2003 eng d a0305-447000aPartition functions for matrix models and isomonodromic tau functions0 aPartition functions for matrix models and isomonodromic tau func a3067–30830 v361 aBertola, Marco1 aEynard, B.1 aHarnad, J. uhttps://www.math.sissa.it/publication/partition-functions-matrix-models-and-isomonodromic-tau-functions00672nas a2200133 4500008004100000245008000041210006900121260001800190520022400208100002100432700002300453700002600476856003600502 2003 en d00aA stability result for nonlinear Neumann problems under boundary variations0 astability result for nonlinear Neumann problems under boundary v bSISSA Library3 aIn 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.1 aDal Maso, Gianni1 aEbobisse, Francois1 aPonsiglione, Marcello uhttp://hdl.handle.net/1963/161800492nas a2200145 4500008004100000022001400041245006200055210006000117300001300177490000800190100001900198700001500217700001500232856009900247 2002 eng d a0010-361600aDuality, biorthogonal polynomials and multi-matrix models0 aDuality biorthogonal polynomials and multimatrix models a73–1200 v2291 aBertola, Marco1 aEynard, B.1 aHarnad, J. uhttps://www.math.sissa.it/publication/duality-biorthogonal-polynomials-and-multi-matrix-models00374nas a2200121 4500008004100000245004500041210004500086260001800131100002400149700002600173700001800199856003500217 1987 en d00aSymmetry breaking in Hamiltonian systems0 aSymmetry breaking in Hamiltonian systems bSISSA Library1 aAmbrosetti, Antonio1 aCoti Zelati, Vittorio1 aEkeland, Ivar uhttp://hdl.handle.net/1963/409