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    Homogenisation problems for free discontinuity functionals with bounded cohesive surface terms
    (2023-07-11) Dal Maso, Gianni; Toader, Rodica; mathematics
    We study stochastic homogenisation problems for free discontinuity func- tionals under a new assumption on the surface terms, motivated by cohesive fracture models. The results are obtained using a characterization of the limit functional by means of the asymptotic behaviour of suitable minimum problems on cubes with very simple boundary conditions. An important role is played by the subadditive ergodic theorem.
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    Discrete approximation of nonlocal-gradient energies
    (2023-01-22) Braides, Andrea; Causin, Andrea; Solci, Margherita; mathematics
    We study a discrete approximation of functionals depending on nonlocal gradients. The discretized functionals are proved to be coercive in classical Sobolev spaces. The key ingredient in the proof is a formulation in terms of circulant Toeplitz matrices.
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    Another look at elliptic homogenization
    (2023-06-21) Braides, Andrea; Cosma Brusca, Giuseppe; Donati, Davide; mathematics
    We consider the limit of sequences of normalized (s, 2)-Gagliardo seminorms with an oscillating coefficient as s → 1. In a seminal paper by Bourgain, Brezis and Mironescu (subsequently extended by Ponce) it is proven that if the coefficient is constant then this sequence Γ-converges to a multiple of the Dirichlet integral. Here we prove that, if we denote by ε the scale of the oscillations and we assume that 1−s << ε2, this sequence converges to the homogenized functional formally obtained by separating the effects of s and ε; that is, by the homogenization as ε → 0 of the Dirichlet integral with oscillating coefficient obtained by formally letting s → 1 first.
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    Validity and failure of the integral representation of Γ-limits of convex non-local functionals
    (2023-05-09) Braides, Andrea; Dal Maso, Gianni; mathematics
    We prove an integral-representation result for limits of non-local quadratic forms on H1 0 pΩq, with Ω a bounded open subset of Rd, extending the representation on C8c pΩq given by the Beurling-Deny formula in the theory of Dirichlet forms. We give a counterexample showing that a corresponding representation may not hold if we consider analogous functionals in W1,p0 pΩq, with p ‰ 2 and 1 ă p ď d.
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    Γ-CONVERGENCE AND INTEGRAL REPRESENTATION FOR A CLASS OF FREE DISCONTINUITY FUNCTIONALS
    (2023-05-05) Dal Maso, Gianni; Toader, Rodica; mathematics
    We study the Γ -limits of sequences of free discontinuity functionals with linear growth, assuming that the surface energy density is bounded. We determine the relevant properties of the Γ -limit, which lead to an integral representation result by means of integrands obtained by solving some auxiliary minimum problems on small cubes.
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    Parker Bound and Monopole Pair Production from Primordial Magnetic Fields
    (2023-04-11) Kobayashi, Takeshi; Perri, Daniele; physics
    We present new bounds on the cosmic abundance of magnetic monopoles based on the survival of primordial magnetic fields during the reheating and radiation-dominated epochs. The new bounds can be stronger than the conventional Parker bound from galactic magnetic fields, as well as bounds from direct searches. We also apply our bounds to monopoles produced by the primordial magnetic fields themselves through the Schwinger effect, and derive additional conditions for the survival of the primordial fields.
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    UNIVERSALITY IN THE 2D QUASI-PERIODIC ISING MODEL AND HARRIS-LUCK IRRELEVANCE
    (2023-04-04) Gallone, Matteo; Mastropietro, Vieri; mathematics
    We prove that in the 2d Ising Model with a weak bidimensional quasi-periodic disorder in the interaction, the critical behavior is the same as in the non-disordered case, that is the critical exponents are identical and no logarithmic corrections are present. The result establishes the validity of the prediction based on the Harris-Luck criterion and it provides the first rigorous proof of universality in the Ising model in presence of quasi-periodic disorder. The proof combines Renormalization Group approaches with direct methods used to deal with small divisors in KAM theory.
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    Gamma-convergence of quadratic functionals perturbed by bounded linear functionals
    (2022-12-14) Dal Maso, Gianni; Donati, Davide; mathematics
    We study the asymptotic behavior of solutions to elliptic equations of the form (􀀀div(Akruk) = fk in ;uk = wk on @; where Rn is a bounded open set, wk is weakly converging in H1(), fk is weakly converging in H􀀀1(), and Ak is a sequence square matrices satisfying some uniform ellipticity and boundedness conditions, and H-converging in . In particular, we characterize the weak limits of the solutions uk and of their momenta Akruk . When Ak is symmetric and wk = w = 0, we characterize the limits of the energies for the solutions.
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    Compactness for a class of integral functionals with interacting local and non-local terms
    (2022-12-20) Braides, Andrea; Dal Maso, Gianni; mathematics
    We prove a compactness result with respect to 􀀀-convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the 􀀀-limit depends on the interaction between the local and non-local terms of the converging subsequence. The result is applied to concentration and homogenization problems.
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    Asymptotic behaviour of the capacity in two-dimensional heterogeneous media
    (2022-06-13) Braides, Andrea; Brusca, G.C.; mathematics
    We describe the asymptotic behaviour of the minimal inhomogeneous two-capacity of small sets in the plane with respect to a fixed open set Ω. This problem is gov erned by two small parameters: ε, the size of the inclusion (which is not restrictive to assume to be a ball), and δ, the period of the inhomogeneity modelled by oscillating coefficients. We show that this capacity behaves as C| log ε| −1. The coefficient C is ex plicitly computed from the minimum of the oscillating coefficient and the determinant of the corresponding homogenized matrix, through a harmonic mean with a proportion depending on the asymptotic behaviour of | log δ|/| log ε|.
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    A note on the homogenization of incommensurate thin films
    (2022-12-21) Anello, Irene; Braides, Andrea; Caragiulo, Fabrizio; mathematics
    Dimension-reduction homogenization results for thin films have been obtained under hy potheses of periodicity or almost-periodicity of the energies in the directions of the mid-plane of the film. In this note we consider thin films, obtained as sections of a periodic medium with a mid-plane that may be incommensurate; that is, not containing periods other than oggi si 0. A geometric almost-periodicity argument similar to the cut-and-project argument used for quasicrystals allows to prove a general homogenization result.
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    Dissipative solutions to Hamiltonian systems
    (2022-09-12) Bianchini, Stefano; Leccese, Giacomo Maria; mathematics
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    An example of a weakly mixing BV vector field which is not strongly mixing
    (2022-08-14) Zizza, Martina; mathematics
    We give an example of a weakly mixing vector field b ∈ L∞([0, 1], BV(T2)) which is not strongly mixing, in the setting first introduced in [3]. The example is based on a work of Chacon [4] who constructed a weakly mixing automorphism which is not strongly mixing on ([0, 1], B([0, 1]), | · |), where B([0, 1]) are the Borel subsets of [0, 1] and | · | is the one-dimensional Lebesgue measure.
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    Algebra of Operators in AdS-Rindler
    (2022-08-08) Bahiru, Eyoab D.; physics
    We discuss the algebra of operators in AdS-Rinlder wedge, particularly in AdS5/CFT4. We explicitly construct the algebra at N = 8 limit and discuss its Type III1 nature. We will consider 1/N corrections to the theory and describe how several divergences can be renormalized and the algebra becomes Type II8. This will make it possible to associate a density matrix to any state in the Hilbert space and thus a von Neumann entropy.
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    Continuity of some non-local functionals with respect to a convergence of the underlying measures
    (2022-04-04) Braides, Andrea; Dal Maso, Gianni; mathematics
    We study some non-local functionals on the Sobolev space W1,p0(Ω) involving a double integral on Ω × Ω with respect to a measure µ. We introduce a suitable notion of convergence of measures on product spaces which implies a stability property in the sense of Γ-convergence of the corresponding functionals.
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    Asymptotic behavior of the dirichlet energy on poisson point clouds
    (2022-03-23) Braides, Andrea; Caroccia, Marco; mathematics
    We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the Dirichlet energy by a deterministic constant. This is achieved by scaling the Poisson cloud and the corresponding energies and computing a compact discrete-to-continuum limit. In order to avoid the effect of exceptional regions of the Poisson cloud, with an accumulation of sites or with ‘disconnected’ sites, a suitable ‘coarse-grained’ notion of convergence of functions defined on scaled Poisson clouds must be given.
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    On the Dirichlet problem associated to bounded perturbations of positively-(p, q)-homogeneous Hamiltonian systems
    (2022-02-21) Fonda, Alessandro; Klun, Giuliano; Obersnel, Franco; Sfecci, Andrea; mathematics
    The existence of solutions for the Dirichlet problem associated to bounded perturbations of positively-(p, q)-homogeneous Hamiltonian sys tems is considered both in nonresonant and resonant situations. In order to deal with the resonant case, the existence of a couple of lower and up per solutions is assumed. Both the well-ordered and the non-well-ordered cases are analysed. The proof is based on phase-plane analysis and topo logical degree theory
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    Controllability of the Schrodinger Equation via Intersection of Eigenvalues
    (2005) Adami, R.; Boscain, U.; mathematics
    We introduce two models of controlled infinite dimensional quantum system whose Hamiltonian operator has a purely discrete spectrum. For any couple of eigenstates we construct a path in the space of controls that approximately steers the system from one eigenstate to the other. To this purpose we use the adiabatic theory for quantum systems, and therefore the strategy requires large times.
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    Canonical and pertubative quantum gravity
    (1993) Esposito, Giampiero; physics
    After a review of Dirac's theory of iconstrained Hamiltonian systems and their quantization, canonical quantum gravity is studied relying on the Arnowitt-Deser-Misner formalism. First-class constraints of the theory are studied in some detail following De Witt's work, and geometrical and topological properties of Wheeler's superspace are dis cussed following the mathematical work of Fisher. Perturbative quantum gravity is then formulated in terms of amplitudes of going from a three-metric and a matter-field configuration on a spacelike surface ~ to a three-metric and a field configuration on a spacelike surface ~'. The Wick-rotated quantum amplitudes are here studied under the assumption that the analytic continuation to the real Rieman- I nian section of the complexified space-time is possible, but this is not a generic property. Within the background-field method, one then expands both the four-metric g and mat ter fields
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    Algebraic structure of Lorentz and diffeomorphism anomalies
    (1993-04-03) Werneck de Oliveira, M.; Sorella, S.P.; physics
    The Wess-Zumino consistency conditions for Lorentz and diffeomor phism anomalies are discussed by introducing an operator δ whichallows to decompose the exterior space-time derivative as a BRS com mutator.