We consider an n×n linear system of ODEs with an irregular singularity of Poincar\'e rank 1 at z=∞, holomorphically depending on parameter t within a polydisc in Cn centred at t=0. The eigenvalues of the leading matrix at z=∞ coalesce along a locus Δ contained in the polydisc, passing through t=0. Namely, z=∞ is a resonant irregular singularity for t∈Δ. We analyse the case when the leading matrix remains diagonalisable at Δ. We discuss the existence of fundamental matrix solutions, their asymptotics, Stokes phenomenon and monodromy data as t varies in the polydisc, and their limits for t tending to points of Δ. When the deformation is isomonodromic away from Δ, it is well known that a fundamental matrix solution has singularities at Δ. When the system also has a Fuchsian singularity at z=0, we show under minimal vanishing conditions on the residue matrix at z=0 that isomonodromic deformations can be extended to the whole polydisc, including Δ, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisc. These data can be computed just by considering the system at fixed t=0. Conversely, if the t-dependent system is isomonodromic in a small domain contained in the polydisc not intersecting Δ, if the entries of the Stokes matrices with indices corresponding to coalescing eigenvalues vanish, then we show that Δ is not a branching locus for the fundamental matrix solutions. The importance of these results for the analytic theory of Frobenius Manifolds is explained. An application to Painlev\'e equations is discussed.

PB - Duke University Press VL - 168 UR - https://doi.org/10.1215/00127094-2018-0059 ER - TY - JOUR T1 - Iterative map-making with two-level preconditioning for polarized cosmic microwave background data sets. A worked example for ground-based experiments JF - ASTRONOMY & ASTROPHYSICS Y1 - 2018 A1 - Puglisi, Giuseppe A1 - Poletti, Davide A1 - Fabbian, Giulio A1 - Baccigalupi, Carlo A1 - Luca Heltai A1 - Stompor, Radek VL - 618 UR - https://arxiv.org/abs/1801.08937 ER - TY - JOUR T1 - The injectivity radius of Lie manifolds JF - ArXiv e-prints Y1 - 2017 A1 - Paolo Antonini A1 - G. De Philippis A1 - Nicola Gigli KW - (58J40) KW - 53C21 KW - Mathematics - Differential Geometry AB -We prove in a direct, geometric way that for any compatible Riemannian metric on a Lie manifold the injectivity radius is positive

UR - https://arxiv.org/pdf/1707.07595.pdf ER - TY - JOUR T1 - Integrability of dominated decompositions on three-dimensional manifolds JF - Ergodic Theory and Dynamical Systems Y1 - 2017 A1 - Stefano Luzzatto A1 - Sina Türeli A1 - Khadim Mbacke War AB -

We investigate the integrability of two-dimensional invariant distributions (tangent sub-bundles) which arise naturally in the context of dynamical systems on 3-manifolds. In particular, we prove unique integrability of dynamically dominated and volume-dominated Lipschitz continuous invariant decompositions as well as distributions with some other regularity conditions.

Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an "Almeida-Molino" integrable lift when the base manifold is simply connected. On the other hand, we notice that the classical de Rham isomorphism provides a universal integrable algebroid. Using it we construct a "de Rham" integrable lift for any given transitive Abelian Lie algebroid.

UR - https://arxiv.org/pdf/1707.04855.pdf ER - TY - THES T1 - Instanton counting on compact manifolds Y1 - 2016 A1 - Massimiliano Ronzani KW - Supersymmetry AB - In this thesis we analyze supersymmetric gauge theories on compact manifolds and their relation with representation theory of infinite Lie algebras associated to conformal field theories, and with the computation of geometric invariants and superconformal indices. The thesis contains the work done by the candidate during the doctorate programme at SISSA under the supervision of A. Tanzini and G. Bonelli. • in Chapter 2, we consider N = 2 supersymmetric gauge theories on four manifolds admitting an isometry. Generalized Killing spinor equations are derived from the consistency of supersymmetry algebrae and solved in the case of four manifolds admitting a U(1) isometry. This is used to explicitly compute the supersymmetric path integral on S2 × S2 via equivariant localization. The building blocks of the resulting partition function are shown to contain the three point functions and the conformal blocks of Liouville Gravity. • in Chapter 3, we provide a contour integral formula for the exact partition function of N = 2 supersymmetric U(N) gauge theories on compact toric four-manifolds by means of supersymmetric localisation. We perform the explicit evaluation of the contour integral for U(2) N = 2∗ theory on P2 for all instanton numbers. In the zero mass case, corresponding to the N = 4 supersymmetric gauge theory, we obtain the generating function of the Euler characteristics of instanton moduli spaces in terms of mock-modular forms. In the decoupling limit of infinite mass we find that the generating function of local and surface observables computes equivariant Donaldson invariants, thus proving in this case a long-standing conjecture by N. Nekrasov. In the case of vanishing first Chern class the resulting equivariant Donaldson polynomials are new. • in Chapter 4, we explore N = (1, 0) superconformal six-dimensional theories arising from M5 branes probing a transverse Ak singularity. Upon circle compactification to five dimensions, we describe this system with a dual pq-web of five-branes and propose the spectrum of basic five-dimensional in- stanton operators driving global symmetry enhancement. For a single M5 brane, we find that the exact partition function of the 5d quiver gauge theory matches the 6d (1, 0) index, which we compute by letter counting. We finally show which relations among vertex correlators of qW algebrae are implied by the S-duality of the pq-web. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/35219 U1 - 35521 U2 - Mathematics U4 - 1 U5 - MAT/07 ER - TY - JOUR T1 - Integrability of C1 invariant splittings JF - Dynamical Systems Y1 - 2016 A1 - Stefano Luzzatto A1 - Sina Türeli A1 - Khadim Mbacke War AB -We derive some new conditions for integrability of dynamically defined C1 invariant splittings, formulated in terms of the singular values of the iterates of the derivative of the diffeomorphism which defines the splitting.

PB - Taylor & Francis VL - 31 UR - https://doi.org/10.1080/14689367.2015.1057480 ER - TY - THES T1 - Integrability of continuous bundles and applications to dynamical systems Y1 - 2016 A1 - Khadim Mbacke War AB - In this dissertation we study the problem of integrability of bundles with low regularities. PB - SISSA U1 - 35529 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - RPRT T1 - Isogeometric analysis-based reduced order modelling for incompressible linear viscous flows in parametrized shapes Y1 - 2016 A1 - Filippo Salmoiraghi A1 - Francesco Ballarin A1 - Luca Heltai A1 - Gianluigi Rozza AB - In this work we provide a combination of isogeometric analysis with reduced order modelling techniques, based on proper orthogonal decomposition, to guarantee computational reduction for the numerical model, and with free-form deformation, for versatile geometrical parametrization. We apply it to computational fluid dynamics problems considering a Stokes flow model. The proposed reduced order model combines efficient shape deformation and accurate and stable velocity and pressure approximation for incompressible viscous flows, computed with a reduced order method. Efficient offine-online computational decomposition is guaranteed in view of repetitive calculations for parametric design and optimization problems. Numerical test cases show the efficiency and accuracy of the proposed reduced order model. PB - Springer, AMOS Advanced Modelling and Simulation in Engineering Sciences UR - http://urania.sissa.it/xmlui/handle/1963/35199 U1 - 35493 U2 - Mathematics U4 - 1 U5 - MAT/08 ER - TY - THES T1 - Integrability of Continuous Tangent Sub-bundles Y1 - 2015 A1 - Sina Türeli KW - Dynamical Systems, Global Analysis, Frobenius Theorem, Integrability AB - In this thesis, the main aim is to study the integrability properties of continuous tangent sub-bundles, especially those that arise in the study of dynamical systems. After the introduction and examples part we start by studying integrability of such sub-bundles under different regularity and dynamical assumptions. Then we formulate a continuous version of the classical Frobenius theorem and state some applications to such bundles, to ODE and PDE. Finally we close of by stating some ongoing work related to interactions between integrability, sub-Riemannian geometry and contact geometry. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/34630 U1 - 34833 U2 - Mathematics U5 - MAT/05 ER - TY - THES T1 - Interaction functionals, Glimm approximations and Lagrangian structure of BV solutions for Hyperbolic Systems of Conservations Laws Y1 - 2015 A1 - Stefano Modena KW - Hyperbolic conservation laws AB - This thesis is a contribution to the mathematical theory of Hyperbolic Conservation Laws. Three are the main results which we collect in this work. The first and the second result (denoted in the thesis by Theorem A and Theorem B respectively) deal with the following problem. The most comprehensive result about existence, uniqueness and stability of the solution to the Cauchy problem \begin{equation}\tag{$\mathcal C$} \label{E:abstract} \begin{cases} u_t + F(u)_x = 0, \\u(0, x) = \bar u(x), \end{cases} \end{equation} where $F: \R^N \to \R^N$ is strictly hyperbolic, $u = u(t,x) \in \R^N$, $t \geq 0$, $x \in \R$, $\TV(\bar u) \ll 1$, can be found in [Bianchini, Bressan 2005], where the well-posedness of \eqref{E:abstract} is proved by means of vanishing viscosity approximations. After the paper [Bianchini, Bressan 2005], however, it seemed worthwhile to develop a \emph{purely hyperbolic} theory (based, as in the genuinely nonlinear case, on Glimm or wavefront tracking approximations, and not on vanishing viscosity parabolic approximations) to prove existence, uniqueness and stability results. The reason of this interest can be mainly found in the fact that hyperbolic approximate solutions are much easier to study and to visualize than parabolic ones. Theorems A and B in this thesis are a contribution to this line of research. In particular, Theorem A proves an estimate on the change of the speed of the wavefronts present in a Glimm approximate solution when two of them interact; Theorem B proves the convergence of the Glimm approximate solutions to the weak admissible solution of \eqref{E:abstract} and provides also an estimate on the rate of convergence. Both theorems are proved in the most general setting when no assumption on $F$ is made except the strict hyperbolicity. The third result of the thesis, denoted by Theorem C, deals with the Lagrangian structure of the solution to \eqref{E:abstract}. The notion of Lagrangian flow is a well-established concept in the theory of the transport equation and in the study of some particular system of conservation laws, like the Euler equation. However, as far as we know, the general system of conservations laws \eqref{E:abstract} has never been studied from a Lagrangian point of view. This is exactly the subject of Theorem C, where a Lagrangian representation for the solution to the system \eqref{E:abstract} is explicitly constructed. The main reasons which led us to look for a Lagrangian representation of the solution of \eqref{E:abstract} are two: on one side, this Lagrangian representation provides the continuous counterpart in the exact solution of \eqref{E:abstract} to the well established theory of wavefront approximations; on the other side, it can lead to a deeper understanding of the behavior of the solutions in the general setting, when the characteristic field are not genuinely nonlinear or linearly degenerate. PB - SISSA UR - http://urania.sissa.it/xmlui/handle/1963/34542 U1 - 34739 U2 - Mathematics U4 - 1 U5 - MAT/05 ER - TY - JOUR T1 - An improvement on geometrical parameterizations by transfinite maps JF - Comptes Rendus Mathematique Y1 - 2014 A1 - Jäggli, C. A1 - Laura Iapichino A1 - Gianluigi Rozza AB - We present a method to generate a non-affine transfinite map from a given reference domain to a family of deformed domains. The map is a generalization of the Gordon-Hall transfinite interpolation approach. It is defined globally over the reference domain. Once we have computed some functions over the reference domain, the map can be generated by knowing the parametric expressions of the boundaries of the deformed domain. Being able to define a suitable map from a reference domain to a desired deformation is useful for the management of parameterized geometries. VL - 352 ER - TY - JOUR T1 - Infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy Y1 - 2014 A1 - Chaozhong Wu A1 - Dafeng Zuo AB - Following the approach of Carlet et al. (2011) [9], we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type A defined by Dubrovin and Zhang. PB - Elsevier UR - http://urania.sissa.it/xmlui/handle/1963/35026 U1 - 35264 U2 - Mathematics U4 - 1 ER - TY - RPRT T1 - Integrability of Dirac reduced bi-Hamiltonian equations Y1 - 2014 A1 - Alberto De Sole A1 - Victor G. Kac A1 - Daniele Valeri AB - First, we give a brief review of the theory of the Lenard-Magri scheme for a non-local bi-Poisson structure and of the theory of Dirac reduction. These theories are used in the remainder of the paper to prove integrability of three hierarchies of bi-Hamiltonian PDE's, obtained by Dirac reduction from some generalized Drinfeld-Sokolov hierarchies. PB - SISSA UR - http://hdl.handle.net/1963/7247 N1 - 15 pages U1 - 7286 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - RPRT T1 - An irreducible symplectic orbifold of dimension 6 with a Lagrangian Prym fibration Y1 - 2014 A1 - Tommaso Matteini KW - Irreducible symplectic variety, Lagrangian fibration, Prym variety, automorphism of symplectic varieties AB - A new example of an irreducible symplectic variety of dimension 6, with only finite quotient singularities, is described as a relative compactified Prymian of a family of genus 4 curves with involution. It is associated to a K3 surface which is a double cover of a cubic surface. It has a natural Lagrangian fibration in abelian 3-folds with polarization type (1,1,2). It does not admit any symplectic resolution. U1 - 7360 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - JOUR T1 - On an isomonodromy deformation equation without the Painlevé property Y1 - 2014 A1 - Boris Dubrovin A1 - Andrey Kapaev AB - We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1) it governs the isomonodromy deformations of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2) it does not possesses the Painlev\'e property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data. PB - Maik Nauka-Interperiodica Publishing UR - http://hdl.handle.net/1963/6466 N1 - 34 pages, 8 figures, references added U1 - 6410 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - An improved geometric inequality via vanishing moments, with applications to singular Liouville equations JF - Communications in Mathematical Physics 322, nr.2 (2013): 415-452 Y1 - 2013 A1 - Mauro Bardelloni A1 - Andrea Malchiodi PB - SISSA UR - http://hdl.handle.net/1963/6561 U1 - 6486 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Inversion formulae for the $\romancosh$-weighted Hilbert transform JF - Proc. Amer. Math. Soc. Y1 - 2013 A1 - Marco Bertola A1 - Katsevich, A. A1 - Alexander Tovbis VL - 141 UR - http://dx.doi.org/10.1090/S0002-9939-2013-11642-4 ER - TY - RPRT T1 - Introduction to Riemannian and sub-Riemannian geometry Y1 - 2012 A1 - Andrei A. Agrachev A1 - Davide Barilari A1 - Ugo Boscain PB - SISSA UR - http://hdl.handle.net/1963/5877 U1 - 5747 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems JF - Matematische Annalen 349 (2011) 75-115 Y1 - 2011 A1 - Guido Carlet A1 - Boris Dubrovin A1 - Luca Philippe Mertens AB - We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold. PB - Springer UR - http://hdl.handle.net/1963/3584 U1 - 716 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Infinitely many positive solutions for a Schrödinger–Poisson system JF - Nonlinear Analysis: Theory, Methods & Applications Y1 - 2011 A1 - Pietro d’Avenia A1 - Alessio Pomponio A1 - Giusi Vaira KW - Non-autonomous Schrödinger–Poisson system KW - Perturbation method AB -We are interested in the existence of infinitely many positive solutions of the Schrödinger–Poisson system −Δu+u+V(|x|)ϕu=|u|p−1u,x∈R3,−Δϕ=V(|x|)u2,x∈R3, where V(|x|) is a positive bounded function, 1<p<5 and V(r

VL - 74 UR - http://www.sciencedirect.com/science/article/pii/S0362546X11003518 ER - TY - RPRT T1 - Instantons on ALE spaces and Super Liouville Conformal Field Theories Y1 - 2011 A1 - Giulio Bonelli A1 - Kazunobu Maruyoshi A1 - Alessandro Tanzini AB - We provide evidence that the conformal blocks of N=1 super Liouville\\r\\nconformal field theory are described in terms of the SU(2) Nekrasov partition\\r\\nfunction on the ALE space O_{P^1}(-2). PB - SISSA UR - http://hdl.handle.net/1963/4262 N1 - 10 pages U1 - 3987 U2 - Physics U3 - Elementary Particle Theory U4 - -1 ER - TY - JOUR T1 - An Integro-Extremization Approach for Non Coercive and Evolution Hamilton-Jacobi Equations JF - Journal of Convex Analysis 18 (2011) 1141-1170 Y1 - 2011 A1 - Sandro Zagatti AB - We devote the \\\\textit{integro-extremization} method to the study of the Dirichlet problem for homogeneous Hamilton-Jacobi equations \\\\begin{displaymath} \\\\begin{cases} F(Du)=0 & \\\\quad \\\\textrm{in} \\\\quad\\\\O\\\\cr u(x)=\\\\varphi(x) & \\\\quad \\\\textrm{for} \\\\quad x\\\\in \\\\partial \\\\O, \\\\end{cases} \\\\end{displaymath} with a particular interest for non coercive hamiltonians $F$, and to the Cauchy-Dirichlet problem for the corresponding homogeneous time-dependent equations \\\\begin{displaymath} \\\\begin{cases} \\\\frac{\\\\partial u}{\\\\partial t}+ F(\\\\nabla u)=0 & \\\\quad \\\\textrm{in} \\\\quad ]0,T[\\\\times \\\\O\\\\cr u(0,x)=\\\\eta(x) & \\\\quad \\\\textrm{for} \\\\quad x\\\\in\\\\O \\\\cr u(t,x)=\\\\psi(x) & \\\\quad \\\\textrm{for} \\\\quad (t,x)\\\\in[0,T]\\\\times \\\\partial \\\\O. \\\\end{cases} \\\\end{displaymath} We prove existence and some qualitative results for viscosity and almost everywhere solutions, under suitably convexity conditions on the hamiltonian $F$, on the domain $\\\\O$ and on the boundary datum, without any growth assumptions on $F$. PB - Heldermann Verlag UR - http://hdl.handle.net/1963/5538 U1 - 5375 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Invariant manifolds for a singular ordinary differential equation JF - Journal of Differential Equations 250 (2011) 1788-1827 Y1 - 2011 A1 - Stefano Bianchini A1 - Laura Spinolo PB - Elsevier UR - http://hdl.handle.net/1963/2554 U1 - 1565 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - THES T1 - Invariants, volumes and heat kernels in sub-Riemannian geometry Y1 - 2011 A1 - Davide Barilari KW - Sub-Riemannian geometry AB - Sub-Riemannian geometry can be seen as a generalization of Riemannian geometry under non-holonomic constraints. From the theoretical point of view, sub-Riemannian geometry is the geometry underlying the theory of hypoelliptic operators (see [32, 57, 70, 92] and references therein) and many problems of geometric measure theory (see for instance [18, 79]). In applications it appears in the study of many mechanical problems (robotics, cars with trailers, etc.) and recently in modern elds of research such as mathematical models of human behaviour, quantum control or motion of self-propulsed micro-organism (see for instance [15, 29, 34])\\r\\nVery recently, it appeared in the eld of cognitive neuroscience to model the\\r\\nfunctional architecture of the area V1 of the primary visual cortex, as proposed by Petitot in [87, 86], and then by Citti and Sarti in [51]. In this context, the sub-Riemannian heat equation has been used as basis to new applications in image reconstruction (see [35]). PB - SISSA UR - http://hdl.handle.net/1963/6124 U1 - 6005 U2 - Mathematics U3 - Functional Analysis and Applications U4 - -1 ER - TY - JOUR T1 - Invariant Lagrange submanifolds of dissipative systems JF - Russian Mathematical Surveys. Volume 65, Issue 5, 2010, Pages: 977-978 Y1 - 2010 A1 - Andrei A. Agrachev AB - We study solutions of modified Hamilton-Jacobi equations H(du/dq,q) + cu(q) =\\r\\n0, q \\\\in M, on a compact manifold M . PB - SISSA UR - http://hdl.handle.net/1963/6457 U1 - 6403 U2 - Mathematics U4 - 1 U5 - MAT/05 ANALISI MATEMATICA ER - TY - JOUR T1 - Initial value problem of the Whitham equations for the Camassa-Holm equation JF - Physica D 238 (2009) 55-66 Y1 - 2009 A1 - Tamara Grava A1 - Virgil U. Pierce A1 - Fei-Ran Tian AB - We study the Whitham equations for the Camassa-Holm equation. The equations are neither strictly hyperbolic nor genuinely nonlinear. We are interested in the initial value problem of the Whitham equations. When the initial values are given by a step function, the Whitham solution is self-similar. When the initial values are given by a smooth function, the Whitham solution exists within a cusp in the x-t plane. On the boundary of the cusp, the Whitham equation matches the Burgers solution, which exists outside the cusp. PB - Elsevier UR - http://hdl.handle.net/1963/3429 U1 - 906 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups JF - J. Funct. Anal. 256 (2009) 2621-2655 Y1 - 2009 A1 - Andrei A. Agrachev A1 - Ugo Boscain A1 - Jean-Paul Gauthier A1 - Francesco Rossi AB - We present an invariant definition of the hypoelliptic Laplacian on sub-Riemannian structures with constant growth vector, using the Popp\\\'s volume form introduced by Montgomery. This definition generalizes the one of the Laplace-Beltrami operator in Riemannian geometry. In the case of left-invariant problems on unimodular Lie groups we prove that it coincides with the usual sum of squares.\\nWe then extend a method (first used by Hulanicki on the Heisenberg group) to compute explicitly the kernel of the hypoelliptic heat equation on any unimodular Lie group of type I. The main tool is the noncommutative Fourier transform. We then study some relevant cases: SU(2), SO(3), SL(2) (with the metrics inherited by the Killing form), and the group SE(2) of rototranslations of the plane.\\nOur study is motivated by some recent results about the cut and conjugate loci on these sub-Riemannian manifolds. The perspective is to understand how singularities of the sub-Riemannian distance reflect on the kernel of the corresponding hypoelliptic heat equation. UR - http://hdl.handle.net/1963/2669 U1 - 1428 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Investigating the Conformational Stability of Prion Strains through a Kinetic Replication Model JF - PLoS Comput Biol 2009;5(7): e1000420 Y1 - 2009 A1 - Mattia Zampieri A1 - Giuseppe Legname A1 - Claudio Altafini AB - Prion proteins are known to misfold into a range of different aggregated forms, showing different phenotypic and pathological states. Understanding strain specificities is an important problem in the field of prion disease. Little is known about which PrPSc structural properties and molecular mechanisms determine prion replication, disease progression and strain phenotype. The aim of this work is to investigate, through a mathematical model, how the structural stability of different aggregated forms can influence the kinetics of prion replication. The model-based results suggest that prion strains with different conformational stability undergoing in vivo replication are characterizable in primis by means of different rates of breakage. A further role seems to be played by the aggregation rate (i.e. the rate at which a prion fibril grows). The kinetic variability introduced in the model by these two parameters allows us to reproduce the different characteristic features of the various strains (e.g., fibrils\\\' mean length) and is coherent with all experimental observations concerning strain-specific behavior. PB - PLoS UR - http://hdl.handle.net/1963/3989 U1 - 413 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Instanton counting on Hirzebruch surfaces Y1 - 2008 A1 - Ugo Bruzzo A1 - Rubik Poghossian A1 - Alessandro Tanzini AB - We perform a study of the moduli space of framed torsion free sheaves on Hirzebruch surfaces by using localization techniques. After discussing general properties of this moduli space, we classify its fixed points under the appropriate toric action and compute its Poincare\\\' polynomial. From the physical viewpoint, our results provide the partition function of N=4 Vafa-Witten theory on Hirzebruch surfaces, which is relevant in black hole entropy counting problems according to a conjecture due to Ooguri, Strominger and Vafa. UR - http://hdl.handle.net/1963/2852 U1 - 1848 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Invariant Carnot-Caratheodory metrics on S3, SO(3), SL(2) and Lens Spaces JF - SIAM J. Control Optim. 47 (2008) 1851-1878 Y1 - 2008 A1 - Ugo Boscain A1 - Francesco Rossi AB - In this paper we study the invariant Carnot-Caratheodory metrics on SU(2) \\\' S3,\\nSO(3) and SL(2) induced by their Cartan decomposition. Beside computing explicitly geodesics and conjugate loci, we compute the cut loci (globally) and we give the expression of the Carnot-Caratheodory distance as the inverse of an elementary function. We then prove that the metric\\ngiven on SU(2) projects on the so called Lens Spaces L(p; q). Also for Lens Spaces, we compute\\nthe cut loci (globally). UR - http://hdl.handle.net/1963/2144 U1 - 2099 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - RPRT T1 - Invariant Manifolds for Viscous Profiles of a Class of Mixed Hyperbolic-Parabolic Systems Y1 - 2008 A1 - Stefano Bianchini A1 - Laura Spinolo UR - http://hdl.handle.net/1963/3400 U1 - 932 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - The Isospectral Dirac Operator on the 4-dimensional Orthogonal Quantum Sphere JF - Comm. Math. Phys. 279 (2008) 77-116 Y1 - 2008 A1 - Francesco D'Andrea A1 - Ludwik Dabrowski A1 - Giovanni Landi AB - Equivariance under the action of Uq(so(5)) is used to compute the left regular and (chiral) spinorial representations of the algebra of the quantum Euclidean 4-sphere S^4_q. These representations are the constituents of a spectral triple on this sphere with a Dirac operator which is isospectral to the canonical one of the spin structure of the round undeformed four-sphere and which gives metric dimension four for the noncommutative geometry. Non-triviality of the geometry is proved by pairing the associated Fredholm module with an `instanton\\\' projection. A real structure which satisfies all required properties modulo a suitable ideal of `infinitesimals\\\' is also introduced. UR - http://hdl.handle.net/1963/2567 U1 - 1553 U2 - Mathematics U3 - Mathematical Physics ER - TY - RPRT T1 - Infinite Horizon Noncooperative Differential Games Y1 - 2006 A1 - Alberto Bressan A1 - Fabio Simone Priuli AB - For a non-cooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we consider a class of infinite-horizon games with nonlinear costs exponentially discounted in time. By the analysis of the value\\nfunctions, we establish the existence of Nash equilibrium solutions in feedback form and provide results and counterexamples on their uniqueness and stability. JF - J. Differential Equations 227 (2006) 230-257 UR - http://hdl.handle.net/1963/1720 U1 - 2431 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An instability of the Godunov scheme JF - Comm. Pure Appl. Math. 59 (2006) 1604-1638 Y1 - 2006 A1 - Alberto Bressan A1 - Helge Kristian Jenssen A1 - Paolo Baiti AB - We construct a solution to a $2\\\\times 2$ strictly hyperbolic system of conservation laws, showing that the Godunov scheme \\\\cite{Godunov59} can produce an arbitrarily large amount of oscillations. This happens when the speed of a shock is close to rational, inducing a resonance with the grid. Differently from the Glimm scheme or the vanishing viscosity method, for systems of conservation laws our counterexample indicates that no a priori BV bounds or $L^1$ stability estimates can in general be valid for finite difference schemes. UR - http://hdl.handle.net/1963/2183 U1 - 2061 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Ionization for Three Dimensional Time-dependent Point Interactions JF - Comm. Math. Phys. 257 (2005) 169-192 Y1 - 2005 A1 - Michele Correggi A1 - Gianfausto Dell'Antonio A1 - Rodolfo Figari A1 - Andrea Mantile AB - We study the time evolution of a three dimensional quantum particle under the action of a time-dependent point interaction fixed at the origin. We assume that the ``strength\\\'\\\' of the interaction (\\\\alpha(t)) is a periodic function with an arbitrary mean. Under very weak conditions on the Fourier coefficients of (\\\\alpha(t)), we prove that there is complete ionization as (t \\\\to \\\\infty), starting from a bound state at time (t = 0). Moreover we prove also that, under the same conditions, all the states of the system are scattering states. UR - http://hdl.handle.net/1963/2297 U1 - 1719 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Isomonodromic deformation of resonant rational connections JF - IMRP Int. Math. Res. Pap. Y1 - 2005 A1 - Marco Bertola A1 - Mo, M. Y. ER - TY - JOUR T1 - An ill posed Cauchy problem for a hyperbolic system in two space dimensions Y1 - 2003 A1 - Alberto Bressan AB - The theory of weak solutions for nonlinear conservation laws is now well developed in the case of scalar equations [3] and for one-dimensional hyperbolic systems [1, 2]. For systems in several space dimensions, however, even the global existence of solutions to the Cauchy problem remains a challenging open question. In this note we construct a conterexample showing that, even for a simple class of hyperbolic systems, in two space dimensions the Cauchy problem can be ill posed. PB - Università di Padova UR - http://hdl.handle.net/1963/2913 U1 - 1787 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - An interior estimate for a nonlinear parabolic equation JF - J.Math.Anal.Appl. 284 (2003) no.1, 49 Y1 - 2003 A1 - Giuseppe Maria Coclite PB - SISSA Library UR - http://hdl.handle.net/1963/1622 U1 - 2496 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Instanton algebras and quantum 4-spheres JF - Differential Geom. Appl. 16 (2002) 277-284 Y1 - 2002 A1 - Ludwik Dabrowski A1 - Giovanni Landi AB - We study some generalized instanton algebras which are required to describe `instantonic complex rank 2 bundles\\\'. The spaces on which the bundles are defined are not prescribed from the beginning but rather are obtained from some natural requirements on the instantons. They turn out to be quantum 4-spheres $S^4_q$, with $q\\\\in\\\\IC$, and the instantons are described by self-adjoint idempotents e. We shall also clarify some issues related to the vanishing of the first Chern-Connes class $ch_1(e)$ and on the use of the second Chern-Connes class $ch_2(e)$ as a volume form. PB - Elsevier UR - http://hdl.handle.net/1963/3134 U1 - 1199 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Instantons on the Quantum 4-Spheres S^4_q JF - Comm. Math. Phys. 221 (2001) 161-168 Y1 - 2001 A1 - Ludwik Dabrowski A1 - Giovanni Landi A1 - Tetsuya Masuda AB - 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. PB - Springer UR - http://hdl.handle.net/1963/3135 U1 - 1198 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds JF - Mathematical Physics, Analysis and Geometry 4: 245–291, 2001 Y1 - 2001 A1 - Davide Guzzetti KW - Frobenius Manifolds, Painleve Equations, Isomonodromy deformations AB - 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. PB - RIMS, Kyoto University U1 - 6479 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Inverse problem for Semisimple Frobenius Manifolds Monodromy Data and the Painlevé VI Equation Y1 - 2000 A1 - Davide Guzzetti PB - SISSA Library UR - http://hdl.handle.net/1963/1557 U1 - 2561 U2 - Mathematics U3 - Mathematical Physics ER - TY - JOUR T1 - Infinite time regular synthesis JF - ESAIM: COCV 3 (1998) 381-405 Y1 - 1998 A1 - Benedetto Piccoli AB - 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. PB - EDP Sciences UR - http://hdl.handle.net/1963/3517 U1 - 747 U2 - Mathematics U3 - Functional Analysis and Applications ER - TY - JOUR T1 - Integrable functional equations and algebraic geometry JF - Duke Mathematical Journal. Volume: 76, Issue: 2, Pages: 645-668 Y1 - 1994 A1 - Boris Dubrovin A1 - A.S. Fokas A1 - P.M. Santini PB - SISSA UR - http://hdl.handle.net/1963/6482 U1 - 6428 U2 - Mathematics U4 - 1 U5 - MAT/03 GEOMETRIA ER - TY - CHAP T1 - Integrable systems and classification of 2D topological field theories T2 - 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 Y1 - 1993 A1 - Boris Dubrovin AB - 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. JF - 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 PB - SISSA SN - 0817636536 UR - http://hdl.handle.net/1963/6478 U1 - 6432 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Integrable systems in topological field theory JF - Nuclear Physics B. Volume 379, Issue 3, 1992, pages : 627-689 Y1 - 1992 A1 - Boris Dubrovin AB - 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. PB - SISSA UR - http://hdl.handle.net/1963/6477 U1 - 6433 U2 - Mathematics U4 - 1 U5 - MAT/07 FISICA MATEMATICA ER - TY - JOUR T1 - Integral representation of some convex local functionals. JF - Ricerche Mat. 36 (1987), no. 2, 197-214 Y1 - 1987 A1 - Gianni Dal Maso A1 - Gabriella Paderni PB - SISSA Library UR - http://hdl.handle.net/1963/497 U1 - 3407 U2 - Mathematics U3 - Functional Analysis and Applications ER -