The aim of this paper is to study the quasistatic limit of a one-dimensional model of dynamic debonding. We start from a dynamic problem that strongly couples the wave equation in a time-dependent domain with Griffith's criterion for the evolution of the domain. Passing to the limit as inertia tends to zero, we find that the limit evolution satisfies a stability condition; however, the activation rule in Griffith's (quasistatic) criterion does not hold in general, thus the limit evolution is not rate-independent.

%B Journal of Nonlinear Science %V 28 %P 269–304 %8 Feb %G eng %U https://doi.org/10.1007/s00332-017-9407-0 %R 10.1007/s00332-017-9407-0 %0 Journal Article %J Journal of Differential Equations %D 2017 %T Quasi-periodic solutions for quasi-linear generalized KdV equations %A Filippo Giuliani %K KAM for PDE's %K KdV %K Nash–Moser theory %K Quasi-linear PDE's %K Quasi-periodic solutions %XWe prove the existence of Cantor families of small amplitude, linearly stable, quasi-periodic solutions of quasi-linear autonomous Hamiltonian generalized KdV equations. We consider the most general quasi-linear quadratic nonlinearity. The proof is based on an iterative Nash–Moser algorithm. To initialize this scheme, we need to perform a bifurcation analysis taking into account the strongly perturbative effects of the nonlinearity near the origin. In particular, we implement a weak version of the Birkhoff normal form method. The inversion of the linearized operators at each step of the iteration is achieved by pseudo-differential techniques, linear Birkhoff normal form algorithms and a linear KAM reducibility scheme.

%B Journal of Differential Equations %V 262 %P 5052 - 5132 %G eng %U http://www.sciencedirect.com/science/article/pii/S0022039617300487 %R https://doi.org/10.1016/j.jde.2017.01.021 %0 Journal Article %J Nonlinear Differential Equations and Applications NoDEA %D 2017 %T Quasistatic crack growth based on viscous approximation: a model with branching and kinking %A Vito Crismale %A Giuliano Lazzaroni %XEmploying the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions of cracks in brittle materials in the setting of antiplane shear. The crack path is not prescribed a priori and is chosen in an admissible class of piecewise regular sets that allows for branching and kinking.

%B Nonlinear Differential Equations and Applications NoDEA %V 24 %P 7 %8 Jan %G eng %U https://doi.org/10.1007/s00030-016-0426-6 %R 10.1007/s00030-016-0426-6 %0 Journal Article %J Bulletin of the Brazilian Mathematical Society, New Series %D 2016 %T A quadratic interaction estimate for conservation laws: motivations, techniques and open problems %A Stefano Modena %XIn a series of joint works with S. Bianchini [3, 4, 5], we proved a quadratic interaction estimate for general systems of conservation laws. Aim of this paper is to present the results obtained in the three cited articles [3, 4, 5], discussing how they are related with the general theory of hyperbolic conservation laws. To this purpose, first we explain why this quadratic estimate is interesting, then we give a brief overview of the techniques we used to prove it and finally we present some related open problems.

%B Bulletin of the Brazilian Mathematical Society, New Series %V 47 %P 589–604 %8 Jun %G eng %U https://doi.org/10.1007/s00574-016-0171-9 %R 10.1007/s00574-016-0171-9 %0 Journal Article %J Contemporary Mathematics. Fundamental Directions %D 2016 %T Quadratic interaction estimate for hyperbolic conservation laws, an overview %A Stefano Modena %B Contemporary Mathematics. Fundamental Directions %I Peoples' Friendship University of Russia %V 59 %P 148–172 %G eng %0 Thesis %D 2016 %T Qualitative properties and construction of solutions to some semilinear elliptic PDEs %A Matteo Rizzi %K moving planes method, maximum principle, Lyapunov-Schmidt reduction, Willmore surfaces, Otha-Kawasaki functional %X This thesis is devoted to the study of elliptic equations. On the one hand, we study some qualitative properties, such as symmetry of solutions, on the other hand we explicitly construct some solutions vanishing near some fixed manifold. The main techniques are the moving planes method, in order to investigate the qualitative properties and the Lyapunov-Schmidt reduction. %I SISSA %G en %1 35500 %# MAT/05 %$ Submitted by mrizzi@sissa.it (mrizzi@sissa.it) on 2016-07-11T13:48:21Z No. of bitstreams: 1 tesi_finale.pdf: 1142117 bytes, checksum: 481f2b0713509231cd3a6f8ead92bf59 (MD5) %0 Report %D 2016 %T Quasi-static hydraulic crack growth driven by Darcy's law %A Stefano Almi %XIn the framework of rate independent processes, we present a variational model of quasi-static crack growth in hydraulic fracture. We first introduce the energy functional and study the equilibrium conditions of an unbounded linearly elastic body subject to a remote strain ε ∈ R and with a sufficiently regular crack Γ filled by a volume V of incompressible fluid. In particular, we are able to find the pressure p of the fluid inside the crack as a function of Γ, V , and ε. Then, we study the problem of quasi-static evolution for our model, imposing that the fluid volume V and the fluid pressure p are related by Darcy’s law. We show the existence of such an evolution, and we prove that it satisfies a weak notion of the so-called Griffith’s criterion.

%G en %U http://urania.sissa.it/xmlui/handle/1963/35198 %1 35492 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by salmi@sissa.it (salmi@sissa.it) on 2016-06-18T13:27:56Z No. of bitstreams: 1 Almi-16.pdf: 469056 bytes, checksum: eeb8055150115033880a6c206e9d9fa8 (MD5) %0 Journal Article %J Communications in Mathematical Physics %D 2015 %T Quadratic Interaction Functional for General Systems of Conservation Laws %A Stefano Bianchini %A Stefano Modena %XFor the Glimm scheme approximation to the solution of the system of conservation laws in one space dimension with initial data u 0 with small total variation, we prove a quadratic (w.r.t. Tot. Var. ( u 0)) interaction estimate, which has been used in the literature for stability and convergence results. No assumptions on the structure of the flux f are made (apart from smoothness), and this estimate is the natural extension of the Glimm type interaction estimate for genuinely nonlinear systems. More precisely, we obtain the following results: a new analysis of the interaction estimates of simple waves;

%B Communications in Mathematical Physics %V 338 %P 1075–1152 %G eng %R 10.1007/s00220-015-2372-2 %0 Journal Article %J Journal of Hyperbolic Differential Equations %D 2014 %T On a quadratic functional for scalar conservation laws %A Stefano Bianchini %A Stefano Modena %XWe prove a quadratic interaction estimate for approximate solutions to scalar conservation laws obtained by the wavefront tracking approximation or the Glimm scheme. This quadratic estimate has been used in the literature to prove the convergence rate of the Glimm scheme.

%B Journal of Hyperbolic Differential Equations %I World Scientific Publishing %V 11 %P 355-435 %G en %U http://arxiv.org/abs/1311.2929 %N 2 %1 34903 %2 Mathematics %4 1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2015-10-22T08:34:56Z No. of bitstreams: 1 31M_Bianchini_Modena.pdf: 710996 bytes, checksum: 8dd1a009996ca60b6c2f1dc96bb46f43 (MD5) %R 10.1142/S0219891614500118 %0 Journal Article %J Bulletin of the Institute of Mathematics of Academia Sinica (New Series) %D 2014 %T Quadratic interaction functional for systems of conservation laws: a case study %A Stefano Bianchini %A Stefano Modena %B Bulletin of the Institute of Mathematics of Academia Sinica (New Series) %V 9 %P 487-546 %G eng %U https://w3.math.sinica.edu.tw/bulletin_ns/20143/2014308.pdf %0 Journal Article %D 2014 %T Quantum dimension and quantum projective spaces %A Marco Matassa %X We show that the family of spectral triples for quantum projective spaces introduced by D'Andrea and Dbrowski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element K2por its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connection with the well known notion of quantum dimension of quantum group theory is pointed out. %I Institute of Mathematics %G en %U http://urania.sissa.it/xmlui/handle/1963/34764 %1 34991 %2 Physics %4 2 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-10-27T16:56:59Z No. of bitstreams: 1 preprint2014.pdf: 380876 bytes, checksum: 85fd2f00123c6d7110bdd8ba9731b97b (MD5) %R 10.3842/SIGMA.2014.097 %0 Journal Article %D 2014 %T Quantum gauge symmetries in noncommutative geometry %A Jyotishman Bhowmick %A Francesco D'Andrea %A Biswarup Krishna Das %A Ludwik Dabrowski %X We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite-dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms in the framework of compact quantum group theory and spectral triples. The quantum analogue of these groups are defined as universal (initial) objects in some natural categories. After proving the existence of the universal objects, we discuss several examples that are of interest to physics, as they appear in the noncommutative geometry approach to particle physics: in particular, the C*-algebras M n(R), Mn(C) and Mn(H), describing the finite noncommutative space of the Einstein-Yang-Mills systems, and the algebras A F = C H M3 (C) and Aev = H H M4(C), that appear in Chamseddine-Connes derivation of the Standard Model of particle physics coupled to gravity. As a byproduct, we identify a "free" version of the symplectic group Sp.n/ (quaternionic unitary group). %I European Mathematical Society Publishing House %G en %U http://urania.sissa.it/xmlui/handle/1963/34897 %1 35182 %2 Mathematics %4 1 %$ Submitted by gfeltrin@sissa.it (gfeltrin@sissa.it) on 2015-11-04T16:59:52Z No. of bitstreams: 1 preprint2014.pdf: 652206 bytes, checksum: d452b46b1c285be861389dcab56e1f6e (MD5) %R 10.4171/JNCG/161 %0 Journal Article %J Nonlinear Analysis %D 2014 %T Quasi-static crack growth in hydraulic fracture %A Stefano Almi %A Gianni Dal Maso %A Rodica Toader %XWe present a variational model for the quasi-static crack growth in hydraulic fracture in the framework of the energy formulation of rate-independent processes. The cracks are assumed to lie on a prescribed plane and to satisfy a very weak regularity assumption.

%B Nonlinear Analysis %I Elsevier %V 109 %P 301-318 %G en %U http://hdl.handle.net/20.500.11767/17350 %N Nov %9 Journal article %1 34741 %2 Mathematics %4 1 %# MAT/05 %$ Submitted by salmi@sissa.it (salmi@sissa.it) on 2015-09-24T08:10:23Z No. of bitstreams: 1 A-DM-T-070714.pdf: 283645 bytes, checksum: 68056ef27e9dcfa246029148c0016c0f (MD5) %& 301 %R 10.1016/j.na.2014.07.009 %0 Journal Article %J Journal of Dynamics and Differential Equations %D 2014 %T Quasistatic Evolution in Perfect Plasticity as Limit of Dynamic Processes %A Gianni Dal Maso %A Riccardo Scala %XWe introduce a model of dynamic visco-elasto-plastic evolution in the linearly elastic regime and prove an existence and uniqueness result. Then we study the limit of (a rescaled version of) the solutions when the data vary slowly. We prove that they converge, up to a subsequence, to a quasistatic evolution in perfect plasticity.

%B Journal of Dynamics and Differential Equations %V 26 %P 915–954 %8 Dec %G eng %U https://doi.org/10.1007/s10884-014-9409-7 %R 10.1007/s10884-014-9409-7 %0 Journal Article %J Mathematical Models and Methods in Applied Sciences %D 2014 %T Quasistatic evolution models for thin plates arising as low energy Γ-limits of finite plasticity %A Elisa Davoli %XIn this paper we deduce by $\Gamma$-convergence some partially and fully linearized quasistatic evolution models for thin plates, in the framework of finite plasticity. Denoting by $\epsilon$ the thickness of the plate, we study the case where the scaling factor of the elasto-plastic energy is of order $\epsilon^{2 \alpha -2}$, with $\alpha\geq 3$. These scalings of the energy lead, in the absence of plastic dissipation, to the Von Kármán and linearized Von Kármán functionals for thin plates. We show that solutions to the three-dimensional quasistatic evolution problems converge, as the thickness of the plate tends to zero, to a quasistatic evolution associated to a suitable reduced model depending on $\alpha$.

%B Mathematical Models and Methods in Applied Sciences %V 24 %P 2085-2153 %G eng %U https://doi.org/10.1142/S021820251450016X %R 10.1142/S021820251450016X %0 Journal Article %D 2013 %T Quadratic cohomology %A Andrei A. Agrachev %X We study homological invariants of smooth families of real quadratic forms as\r\na step towards a \"Lagrange multipliers rule in the large\" that intends to\r\ndescribe topology of smooth vector functions in terms of scalar Lagrange\r\nfunctions. %I SISSA %G en %1 6456 %2 Mathematics %4 1 %# MAT/03 GEOMETRIA %$ Submitted by Andrei Agrachev (agrachev@sissa.it) on 2013-02-27T15:16:17Z\nNo. of bitstreams: 1\n1301.2059v1.pdf: 236951 bytes, checksum: a481b8e5391a39e5e87402873e4023f7 (MD5) %0 Journal Article %J Journal of the European Mathematical Society %D 2013 %T Quasi-periodic solutions with Sobolev regularity of NLS on Td with a multiplicative potential %A Massimiliano Berti %A Philippe Bolle %X We prove the existence of quasi-periodic solutions for Schrödinger equations with a multiplicative potential on Td , d ≥ 1, finitely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are C∞ then the solutions are C∞. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. The key off-diagonal decay estimates of the Green functions are proved via a new multiscale inductive analysis. The main novelty concerns the measure and "complexity" estimates. © European Mathematical Society 2013. %B Journal of the European Mathematical Society %V 15 %P 229-286 %G eng %R 10.4171/JEMS/361 %0 Journal Article %J Annales de l'Institut Henri Poincare (C) Non Linear Analysis %D 2013 %T A quasistatic evolution model for perfectly plastic plates derived by Γ-convergence %A Elisa Davoli %A Maria Giovanna Mora %K -convergence %K Perfect plasticity %K Prandtl–Reuss plasticity %K Quasistatic evolution %K Rate-independent processes %K Thin plates %XThe subject of this paper is the rigorous derivation of a quasistatic evolution model for a linearly elastic–perfectly plastic thin plate. As the thickness of the plate tends to zero, we prove via Γ-convergence techniques that solutions to the three-dimensional quasistatic evolution problem of Prandtl–Reuss elastoplasticity converge to a quasistatic evolution of a suitable reduced model. In this limiting model the admissible displacements are of Kirchhoff–Love type and the stretching and bending components of the stress are coupled through a plastic flow rule. Some equivalent formulations of the limiting problem in rate form are derived, together with some two-dimensional characterizations for suitable choices of the data.

%B Annales de l'Institut Henri Poincare (C) Non Linear Analysis %V 30 %P 615 - 660 %G eng %U http://www.sciencedirect.com/science/article/pii/S0294144912001035 %R https://doi.org/10.1016/j.anihpc.2012.11.001 %0 Journal Article %J Calculus of variations and partial differential equations 44 (2012) 495-541 %D 2012 %T Quasistatic evolution for Cam-Clay plasticity: properties of the viscosity solution %A Gianni Dal Maso %A Antonio DeSimone %A Francesco Solombrino %XCam-Clay plasticity is a well established model for the description of the mechanics of fine grained soils. As solutions can develop discontinuities in time, a weak notion of solution, in terms of a rescaled time s , has been proposed in [8] to give a meaning to this discontinuous evolution. In this paper we first prove that this rescaled evolution satisfies the flow-rule for the rate of plastic strain, in a suitable measure-theoretical sense. In the second part of the paper we consider the behavior of the evolution in terms of the original time variable t . We prove that the unrescaled solution satisfies an energy-dissipation balance and an evolution law for the internal variable, which can be expressed in terms of integrals depending only on the original time. Both these integral identities contain terms concentrated on the jump times, whose size can only be determined by looking at the rescaled formulation.

%B Calculus of variations and partial differential equations 44 (2012) 495-541 %I Springer %G en_US %U http://hdl.handle.net/1963/3900 %1 809 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-07-26T09:17:50Z\\r\\nNo. of bitstreams: 1\\r\\nSolombrino_46M.pdf: 382911 bytes, checksum: dd118cc7f80d4cb1902713eb18747ac6 (MD5) %R 10.1007/s00526-011-0443-6 %0 Journal Article %J SIAM Journal on Mathematical Analysis 44, nr. 1 (2012) 245-292 %D 2012 %T Quasistatic evolution in non-associative plasticity - the cap models %A Jean-Francois Babadjian %A Gilles A. Francfort %A Maria Giovanna Mora %K Elasto-plasticity %X Non-associative elasto-plasticity is the working model of plasticity for soil and rocks mechanics. Yet, it is usually viewed as non-variational. In this work, we prove a contrario the existence of a variational evolution for such a model under a natural capping assumption on the hydrostatic stresses and a less natural mollification of the stress admissibility constraint. The obtained elasto-plastic evolution is expressed for times that are conveniently rescaled. %B SIAM Journal on Mathematical Analysis 44, nr. 1 (2012) 245-292 %I SIAM %G en %U http://hdl.handle.net/1963/4139 %1 3879 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2011-09-15T12:13:15Z No. of bitstreams: 1 Bab-Fra-Mora_05M.pdf: 420336 bytes, checksum: cf8a2e6bd6c333fb5b6b130ae22de0a7 (MD5) %R 10.1137/110823511 %0 Report %D 2011 %T Q-factorial Laurent rings %A Ugo Bruzzo %A Antonella Grassi %X Dolgachev proved that, for any field k, the ring naturally associated to a\\r\\ngeneric Laurent polynomial in d variables, $d \\\\geq 4$, is factorial. We prove a\\r\\nsufficient condition for the ring associated to a very general complex Laurent\\r\\npolynomial in d=3 variables to be Q-factorial. %I SISSA %G en %U http://hdl.handle.net/1963/4183 %1 3907 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2011-09-22T10:50:24Z\\nNo. of bitstreams: 1\\n1108.4116v1.pdf: 105862 bytes, checksum: e65647872af8c9b70f2fe8466f37669a (MD5) %0 Journal Article %J Commun. Math. Phys. 308 (2011) 567-589 %D 2011 %T Quantum Geometry on Quantum Spacetime: Distance, Area and Volume Operators %A Dorothea Bahns %A Sergio Doplicher %A Klaus Fredenhagen %A Gherardo Piacitelli %X We develop the first steps towards an analysis of geometry on the quantum\\r\\nspacetime proposed in Doplicher et al. (Commun Math Phys 172:187–220, 1995). The homogeneous elements of the universal differential algebra are naturally identified with operators living in tensor powers of Quantum Spacetime; this allows us to compute their spectra. In particular, we consider operators that can be interpreted as distances, areas, 3- and 4-volumes. The Minkowski distance operator between two independent events is shown to have pure Lebesgue spectrum with infinite multiplicity. The Euclidean distance operator is shown to have spectrum bounded below by a constant of the order of the Planck length. The corresponding statement is proved also for both the space-space and space-time area operators, as well as for the Euclidean length of the vector representing the 3-volume operators. However, the space 3-volume operator (the time component of that vector) is shown to have spectrum equal to the whole complex plane. All these operators are normal, while the distance operators are also selfadjoint. The Lorentz invariant spacetime volume operator, representing the 4- volume spanned by five\\r\\nindependent events, is shown to be normal. Its spectrum is pure point with a\\r\\nfinite distance (of the order of the fourth power of the Planck length) away\\r\\nfrom the origin. The mathematical formalism apt to these problems is developed and its relation to a general formulation of Gauge Theories on Quantum Spaces is outlined. As a byproduct, a Hodge Duality between the absolute differential and the Hochschild boundary is pointed out. %B Commun. Math. Phys. 308 (2011) 567-589 %I Springer %G en %U http://hdl.handle.net/1963/5203 %1 5025 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-12-16T08:24:01Z\\nNo. of bitstreams: 1\\n1005.2130v1.pdf: 270027 bytes, checksum: d12021458a91ccdbfdaf8adfbb2ec89d (MD5) %R 10.1007/s00220-011-1358-y %0 Report %D 2011 %T Quantum Hitchin Systems via beta-deformed Matrix Models %A Giulio Bonelli %A Kazunobu Maruyoshi %A Alessandro Tanzini %XWe study the quantization of Hitchin systems in terms of beta-deformations of generalized matrix models related to conformal blocks of Liouville theory on punctured Riemann surfaces. We show that in a suitable limit, corresponding to the Nekrasov-Shatashvili one, the loop equations of the matrix model reproduce the Hamiltonians of the quantum Hitchin system on the sphere and the torus with marked points. The eigenvalues of these Hamiltonians are shown to be the epsilon1-deformation of the chiral observables of the corresponding N=2 four ndimensional gauge theory. Moreover, we find the exact wave-functions in terms of the matrix model representation of the conformal blocks with degenerate field insertions.

%I SISSA %G en %U http://hdl.handle.net/1963/4181 %1 3904 %2 Physics %3 Elementary Particle Theory %4 -1 %$ Submitted by Maria Pia Calandra (calapia@sissa.it) on 2011-09-22T10:22:35Z\\nNo. of bitstreams: 1\\n1104.4016v2.pdf: 317021 bytes, checksum: 92bcb534e0e6fdeb2e5c51517f28d510 (MD5) %0 Journal Article %J Commun. Math. Phys. 307:101-131, 2011 %D 2011 %T Quantum Isometries of the finite noncommutative geometry of the Standard Model %A Jyotishman Bhowmick %A Francesco D'Andrea %A Ludwik Dabrowski %X We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple corresponding to M x F where M is a compact spin manifold. We also prove that the bosonic and fermionic part of the spectral action are preserved by these symmetries. %B Commun. Math. Phys. 307:101-131, 2011 %I Springer %G en %U http://hdl.handle.net/1963/4906 %1 4688 %2 Mathematics %3 Mathematical Physics %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-25T07:42:04Z\\nNo. of bitstreams: 1\\n1009.2850v3.pdf: 383426 bytes, checksum: 9d28d3070f7e7c39ec9486f40fd4f13b (MD5) %R 10.1007/s00220-011-1301-2 %0 Journal Article %J Journal of the Mechanics and Physics of Solids 59 (2011) 787-803 %D 2011 %T Quasiconvex envelopes of energies for nematic elastomers in the small strain regime and applications %A Pierluigi Cesana %A Antonio DeSimone %X We provide some explicit formulas for the quasiconvex envelope of energy densities for nematic elastomers in the small strain regime and plane strain conditions. We then demonstrate their use as a powerful tool for the interpretation of mechanical experiments. %B Journal of the Mechanics and Physics of Solids 59 (2011) 787-803 %G en_US %U http://hdl.handle.net/1963/4065 %1 337 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-09-15T10:43:52Z\\r\\nNo. of bitstreams: 1\\r\\nCesana_DeSimone_62M.pdf: 1146834 bytes, checksum: 514b1b282763c0351f234702f96e73f4 (MD5) %0 Journal Article %J ESAIM: COCV 17 (2011) 1-27 %D 2011 %T Quasistatic crack evolution for a cohesive zone model with different response to loading and unloading: a Young measures approach %A Filippo Cagnetti %A Rodica Toader %X A new approach to irreversible quasistatic fracture growth is given, by means of Young measures. The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases. In the particular situation of constant unloading response, the result contained in [6] is recovered. In this case, the convergence of the discrete time approximations is improved. %B ESAIM: COCV 17 (2011) 1-27 %I Cambridge University Press / EDP Sciences %G en_US %U http://hdl.handle.net/1963/2355 %1 1662 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-11-07T13:38:16Z\\r\\nNo. of bitstreams: 1\\r\\nQuasCrack.pdf: 298737 bytes, checksum: 127034af65efce0ab97317eccb47e76c (MD5) %R 10.1051/cocv/2009037 %0 Journal Article %J {ANNALI DI MATEMATICA PURA ED APPLICATA} %D 2011 %T Quasistatic crack growth in finite elasticity with Lipschitz data %A Giuliano Lazzaroni %K Brittle fracture %K Crack propagation %K Energy minimization %K Finite elasticity %K free-discontinuity problems %K Griffith's criterion %K Non-interpenetration} %K Polyconvexity %K Quasistatic evolution %K Rate-independent processes %K {Variational models %X{We extend the recent existence result of Dal Maso and Lazzaroni (Ann Inst H Poincare Anal Non Lineaire 27:257-290, 2010) for quasistatic evolutions of cracks in finite elasticity, allowing for boundary conditions and external forces with discontinuous first derivatives.}

%B {ANNALI DI MATEMATICA PURA ED APPLICATA} %I {SPRINGER HEIDELBERG} %C {TIERGARTENSTRASSE 17, D-69121 HEIDELBERG, GERMANY} %V {190} %P {165-194} %8 {JAN} %G eng %9 {Article} %R {10.1007/s10231-010-0145-2} %0 Journal Article %J Calculus of Variations and Partial Differential Equations 40 (2011) 125-181 %D 2011 %T Quasistatic evolution for Cam-Clay plasticity: a weak formulation via viscoplastic regularization and time rescaling %A Gianni Dal Maso %A Antonio DeSimone %A Francesco Solombrino %K Cam-Clay plasticity %XCam-Clay nonassociative plasticity exhibits both hardening and softening behaviour, depending on the loading. For many initial data the classical formulation of the quasistatic evolution problem has no smooth solution. We propose here a notion of generalized solution, based on a viscoplastic approximation. To study the limit of the viscoplastic evolutions we rescale time, in such a way that the plastic strain is uniformly Lipschitz with respect to the rescaled time. The limit of these rescaled solutions, as the viscosity parameter tends to zero, is characterized through an energy-dissipation balance, that can be written in a natural way using the rescaled time. As shown in [4] and [6], the proposed solution may be discontinuous with respect to the original time. Our formulation allows to compute the amount of viscous dissipation occurring instantaneously at each discontinuity time.

%B Calculus of Variations and Partial Differential Equations 40 (2011) 125-181 %I Springer %G en_US %U http://hdl.handle.net/1963/3670 %1 635 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-07-20T08:26:03Z\\r\\nNo. of bitstreams: 1\\r\\nDM-DeS-Sol-36_2009_preprint.pdf: 421582 bytes, checksum: 011806364200378d6deec80b88978550 (MD5) %R 10.1007/s00526-010-0336-0 %0 Journal Article %J Arch. Rational Mech. Anal. 202 (2011) 295-348 %D 2011 %T Quasistatic evolution of sessile drops and contact angle hysteresis %A Giovanni Alberti %A Antonio DeSimone %X We consider the classical model of capillarity coupled with a rate-independent dissipation mechanism due to frictional forces acting on the contact line, and prove the existence of quasistatic evolutions with prescribed initial configuration. We also discuss in detail some explicit solutions to show that the model does account for contact angle hysteresis, and to compare its predictions with experimental observations. %B Arch. Rational Mech. Anal. 202 (2011) 295-348 %I Springer %G en %U http://hdl.handle.net/1963/4912 %1 4693 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-25T09:46:50Z\\nNo. of bitstreams: 1\\ngads2.pdf: 620848 bytes, checksum: f5fef67c4e88cd294016f704c59394bd (MD5) %R 10.1007/s00205-011-0427-x %0 Report %D 2010 %T Quantum Spacetime: a Disambiguation %A Gherardo Piacitelli %X We review an approach to non-commutative geometry, where models are constructed by quantisation of the coordinates. In particular we focus on the full DFR model and its irreducible components; the (arbitrary) restriction to a particular irreducible component is often referred to as the \\\"canonical quantum spacetime\\\". The aim is to distinguish and compare the approaches under various points of view, including motivations, prescriptions for quantisation, the choice of mathematical objects and concepts, approaches to dynamics and to covariance. Some incorrect statements as \\\"universality of Planck scale conflicts with Lorentz-Fitzgerald contraction and requires a modification of covariance\\\", or \\\"stability of the geometric background requires an absolute lower bound of (\\\\Delta x^\\\\mu)\\\", or \\\"violations of unitarity are due to time/space non-commutativity\\\" are put in context, and discussed. %G en_US %U http://hdl.handle.net/1963/3864 %1 845 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2010-04-30T11:06:32Z\\r\\nNo. of bitstreams: 1\\r\\n1004.5261v1.pdf: 529975 bytes, checksum: d7fe48852cac62ba9266e61f93413bd0 (MD5) %0 Journal Article %J Arch. Ration. Mech. Anal. 196 (2010) 867-906 %D 2010 %T Quasistatic crack growth in elasto-plastic materials: the two-dimensional case %A Gianni Dal Maso %A Rodica Toader %X We study a variational model for the quasistatic evolution of elasto-plastic materials with cracks in the case of planar small strain associative elasto-plasticity. %B Arch. Ration. Mech. Anal. 196 (2010) 867-906 %G en_US %U http://hdl.handle.net/1963/2964 %1 1736 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-09-23T17:17:34Z\\nNo. of bitstreams: 1\\nDM-Toa-07-preprint.pdf: 320979 bytes, checksum: e4f2a1856f9bd91d63fc45557cbd6a16 (MD5) %R 10.1007/s00205-009-0258-1 %0 Journal Article %J Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 257-290 %D 2010 %T Quasistatic crack growth in finite elasticity with non-interpenetration %A Gianni Dal Maso %A Giuliano Lazzaroni %XWe present a variational model to study the quasistatic growth of brittle cracks in hyperelastic materials, in the framework of finite elasticity, taking\\ninto account the non-interpenetration condition.

%B Ann. Inst. H. Poincare Anal. Non Lineaire 27 (2010) 257-290 %G en_US %U http://hdl.handle.net/1963/3397 %1 935 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-12-15T09:16:46Z\\nNo. of bitstreams: 1\\nDM-Laz-08pre.pdf: 360778 bytes, checksum: 14a35b4647fde0ea0931df8ae6cbfb73 (MD5) %R 10.1016/j.anihpc.2009.09.006 %0 Journal Article %J Netw. Heterog. Media 5 (2010) 97-132 %D 2010 %T Quasistatic evolution for Cam-Clay plasticity: the spatially homogeneous case %A Gianni Dal Maso %A Francesco Solombrino %K Cam-Clay plasticity %XWe study the spatially uniform case of the problem of quasistatic evolution in small strain nonassociative elastoplasticity (Cam-Clay model). Through the introdution of a viscous approximation, the problem reduces to determine the limit behavior of the solutions of a singularly perturbed system of ODE\\\'s in a finite dimensional Banach space. Depending on the sign of two explicit scalar indicators, we see that the limit dynamics presents, under quite generic assumptions, the alternation of three possible regimes: the elastic regime, when the limit equation is just the equation of linearized elasticity, the slow dynamics, when the strain evolves smoothly on the yield surface and plastic flow is produced, and the fast dynamics, which may happen only in the softening regime, where\\nviscous solutions exhibit a jump across a heteroclinic orbit of an auxiliary system.

%B Netw. Heterog. Media 5 (2010) 97-132 %G en_US %U http://hdl.handle.net/1963/3671 %1 634 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2009-07-20T08:35:17Z\\nNo. of bitstreams: 1\\nDM-Sol2-39_2009_preprint.pdf: 345995 bytes, checksum: 073ce622e3320be1f56a1cc9f8904fae (MD5) %R 10.3934/nhm.2010.5.97 %0 Journal Article %J Discrete & Continuous Dynamical Systems - A %D 2010 %T Quasistatic evolution for plasticity with softening: The spatially homogeneous case %A Francesco Solombrino %K plasticity with softening %K rate independent processes %XThe spatially uniform case of the problem of quasistatic evolution in small strain associative elastoplasticity with softening is studied. Through the introdution of a viscous approximation, the problem reduces to determine the limit behaviour of the solutions of a singularly perturbed system of ODE's in a finite dimensional Banach space. We see that the limit dynamics presents, for a generic choice of the initial data, the alternation of three possible regimes (elastic regime, slow dynamics, fast dynamics), which is determined by the sign of two scalar indicators, whose explicit expression is given.

%B Discrete & Continuous Dynamical Systems - A %V 27 %P 1189 %G eng %U http://aimsciences.org//article/id/4c2301d8-f553-493e-b672-b4f76a3ede2f %R 10.3934/dcds.2010.27.1189 %0 Journal Article %J Math. Models Methods Appl. Sci. 19 (2009) 1643-1711 %D 2009 %T Quasistatic evolution for Cam-Clay plasticity: examples of spatially homogeneous solutions %A Gianni Dal Maso %A Antonio DeSimone %X We study a quasistatic evolution problem for Cam-Clay plasticity under a special loading program which leads to spatially homogeneous solutions. Under some initial conditions, the solutions exhibit a softening behaviour and time discontinuities.\\nThe behavior of the solutions at the jump times is studied by a viscous approximation. %B Math. Models Methods Appl. Sci. 19 (2009) 1643-1711 %G en_US %U http://hdl.handle.net/1963/3395 %1 937 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2008-12-10T11:56:05Z\\nNo. of bitstreams: 1\\nDal_Maso-DeSimone.pdf: 1160572 bytes, checksum: fd49c80a369183f1941b0fcf3cd65341 (MD5) %R 10.1142/S0218202509003942 %0 Journal Article %J J. Convex Anal. %D 2009 %T Quasistatic evolution problems for nonhomogeneous elastic plastic materials %A Francesco Solombrino %XThe paper studies the quasistatic evolution for elastoplastic materials when the yield surface depends on the position in the reference configuration. The main results are obtained when the yield surface is continuous with respect to the space variable. The case of piecewise constant dependence is also considered. The evolution is studied in the framework of the variational formulation for rate independent problems developed by Mielke. The results are proved by adapting the arguments introduced for a constant yield surface, using some properties of convex valued semicontinuous multifunctions. A strong formulation of the problem is also obtained, which includes a pointwise version of the plastic flow rule. Some examples are considered, which show that strain concentration may occur as a consequence of a nonconstant yield surface.

%B J. Convex Anal. %V 16 %P 89–119 %G eng %0 Report %D 2007 %T Quasistatic crack growth for a cohesive zone model with prescribed crack path %A Gianni Dal Maso %A Chiara Zanini %X In this paper we study the quasistatic crack growth for a cohesive zone model. We assume that the crack path is prescribed and we study the time evolution of the crack in the framework of the variational theory of rate-independent processes. %B Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 253-279 %G en_US %U http://hdl.handle.net/1963/1686 %1 2447 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2005-06-20T12:05:32Z\\nNo. of bitstreams: 1\\nDM-Zan-04.pdf: 312529 bytes, checksum: 2373a684663d7901c85daad77c7aa8b6 (MD5) %R 10.1017/S030821050500079X %0 Journal Article %J Milan J. Math. 75 (2007) 117-134 %D 2007 %T Quasistatic evolution problems for pressure-sensitive plastic materials %A Gianni Dal Maso %A Alexey Demyanov %A Antonio DeSimone %X We study quasistatic evolution problems for pressure-sensitive plastic materials in the context of small strain associative perfect plasticity. %B Milan J. Math. 75 (2007) 117-134 %G en_US %U http://hdl.handle.net/1963/1962 %1 2231 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-05-15T10:32:12Z\\nNo. of bitstreams: 1\\nSissa27_2007M.pdf: 223083 bytes, checksum: a4cdbb7a1018403355c560749680e47a (MD5) %R 10.1007/s00032-007-0071-y %0 Journal Article %J J. Differential Geom. 73 (2006) 1-44 %D 2006 %T Q-curvature flow on S^4 %A Andrea Malchiodi %A Michael Struwe %B J. Differential Geom. 73 (2006) 1-44 %G en_US %U http://hdl.handle.net/1963/2193 %1 2051 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-10T11:52:21Z\\nNo. of bitstreams: 1\\nQ-curv-Nov-05.pdf: 298052 bytes, checksum: 2916315e6ffcef99b7748745e768d831 (MD5) %0 Journal Article %J Czechoslovak Journal of Physics 56 (2006), n. 10-11, 1143-1148 %D 2006 %T Quantisation of bending flows %A Gregorio Falqui %A Fabio Musso %X We briefly review the Kapovich-Millson notion of Bending flows as an integrable system on the space of polygons in ${\\\\bf R}^3$, its connection with a specific Gaudin XXX system, as well as the generalisation to $su(r), r>2$. Then we consider the quantisation problem of the set of Hamiltonians pertaining to the problem, quite naturally called Bending Hamiltonians, and prove that their commutativity is preserved at the quantum level. %B Czechoslovak Journal of Physics 56 (2006), n. 10-11, 1143-1148 %G en_US %U http://hdl.handle.net/1963/2537 %1 1582 %2 Mathematics %3 Mathematical Physics %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-12-19T10:53:01Z\\nNo. of bitstreams: 1\\n0610003v1.pdf: 113471 bytes, checksum: 34a8a67eda45bff5d2e70aaa0c1edf65 (MD5) %R 10.1007/s10582-006-0415-9 %0 Journal Article %J Comm. Partial Differential Equations 31 (2006) 959 - 985 %D 2006 %T Quasi-periodic solutions of completely resonant forced wave equations %A Massimiliano Berti %A Michela Procesi %X We prove existence of quasi-periodic solutions with two frequencies of completely resonant, periodically forced nonlinear wave equations with periodic spatial boundary conditions. We consider both the cases the forcing frequency is: (Case A) a rational number and (Case B) an irrational number. %B Comm. Partial Differential Equations 31 (2006) 959 - 985 %G en_US %U http://hdl.handle.net/1963/2234 %1 2010 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-16T07:48:45Z\\nNo. of bitstreams: 1\\n0504406v1.pdf: 330239 bytes, checksum: 5dbf59bdd590a6876ea206f70cf0ecc9 (MD5) %R 10.1080/03605300500358129 %0 Journal Article %J Arch. Ration. Mech. Anal. 180 (2006) 237-291 %D 2006 %T Quasistatic evolution problems for linearly elastic-perfectly plastic materials %A Gianni Dal Maso %A Antonio DeSimone %A Maria Giovanna Mora %X The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation. This provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain. %B Arch. Ration. Mech. Anal. 180 (2006) 237-291 %G en_US %U http://hdl.handle.net/1963/2129 %1 2114 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-09-18T10:14:12Z\\nNo. of bitstreams: 1\\n0412212v1.pdf: 419970 bytes, checksum: fa4e3c2a5db22bfa28aa4f682cb55da4 (MD5) %R 10.1007/s00205-005-0407-0 %0 Journal Article %J Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 %D 2005 %T Quasi-periodic oscillations for wave equations under periodic forcing %A Massimiliano Berti %A Michela Procesi %B Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16 (2005), no. 2, 109-116 %I Accademia Nazionale dei Lincei %G en %U http://hdl.handle.net/1963/4583 %1 4350 %2 Mathematics %3 Functional Analysis and Applications %4 -1 %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2011-10-07T10:27:05Z\\nNo. of bitstreams: 1\\nBertiProcesi05-1.pdf: 211758 bytes, checksum: b6c3ae059191cddb5c025aee61a23799 (MD5) %0 Journal Article %J Arch. Ration. Mech. Anal. 176 (2005) 165-225 %D 2005 %T Quasistatic Crack Growth in Nonlinear Elasticity %A Gianni Dal Maso %A Gilles A. Francfort %A Rodica Toader %X In this paper, we prove a new existence result for a variational model of crack growth in brittle materials proposed in [15]. We consider the case of $n$-dimensional finite elasticity, for an arbitrary $n\\\\ge1$, with a quasiconvex bulk energy and with prescribed boundary deformations and applied loads, both depending on time. %B Arch. Ration. Mech. Anal. 176 (2005) 165-225 %G en_US %U http://hdl.handle.net/1963/2293 %1 1723 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-24T09:12:09Z\\nNo. of bitstreams: 1\\n0401196v1.pdf: 664295 bytes, checksum: cb1000c44e6ae356984e24b55ee97117 (MD5) %R 10.1007/s00205-004-0351-4 %0 Journal Article %J Quad. Mat. Dip. Mat. Seconda Univ. Napoli 14 (2004) 245-266 %D 2004 %T Quasi-static evolution in brittle fracture: the case of bounded solutions %A Gianni Dal Maso %A Gilles A. Francfort %A Rodica Toader %X The main steps of the proof of the existence result for the quasi-static evolution of cracks in brittle materials, obtained in [7] in the vector case and for a general quasiconvex elastic energy, are presented here under the simplifying assumption that the minimizing sequences involved in the problem are uniformly bounded in $L^\\\\infty$. %B Quad. Mat. Dip. Mat. Seconda Univ. Napoli 14 (2004) 245-266 %G en_US %U http://hdl.handle.net/1963/2229 %1 2015 %2 Mathematics %3 Functional Analysis and Applications %$ Submitted by Andrea Wehrenfennig (andreaw@sissa.it) on 2007-10-15T15:04:28Z\\r\\nNo. of bitstreams: 1\\r\\n0401198v1.pdf: 166634 bytes, checksum: c21fba2b1fbbaec4fe14c56595b0664e (MD5) %0 Journal Article %J J. Phys. A 36 (2003), no. 13, 3829-3840 %D 2003 %T Quantum spin coverings and statistics %A Ludwik Dabrowski %A Cesare Reina %X SL_q(2) at odd roots of unity q^l =1 is studied as a quantum cover of the complex rotation group SO(3,C), in terms of the associated Hopf algebras of (quantum) polynomial functions. We work out the irreducible corepresentations, the decomposition of their tensor products and a coquasitriangular structure, with the associated braiding (or statistics). As an example, the case l=3 is discussed in detail. %B J. Phys. A 36 (2003), no. 13, 3829-3840 %I IOP Publishing %G en %U http://hdl.handle.net/1963/1667 %1 2451 %2 Mathematics %3 Mathematical Physics %$ Made available in DSpace on 2004-09-01T13:06:10Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2002 %R 10.1088/0305-4470/36/13/314 %0 Journal Article %J Ann.Physics 296 (2002), no.2, 371 %D 2002 %T Quantum mechanics and stochastic mechanics for compatible observables at different times %A Michele Correggi %A Giovanni Morchio %B Ann.Physics 296 (2002), no.2, 371 %I SISSA Library %G en %U http://hdl.handle.net/1963/1577 %1 2541 %2 Mathematics %3 Mathematical Physics %$ Made available in DSpace on 2004-09-01T13:04:48Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2002 %R 10.1006/aphy.2002.6236 %0 Journal Article %J Lagrangian and Hamiltonian Methods for Nonlinear Control : a proc. volume from the IFAC Workshop. Princeton, New Jersey, 16-18 March 2000 / ed. by N.E. Leonard, R. Ortega. - Oxford : Pergamon, 2000 %D 2000 %T Quantized control systems and discrete nonholonomy %A Alessia Marigo %A Benedetto Piccoli %A Antonio Bicchi %B Lagrangian and Hamiltonian Methods for Nonlinear Control : a proc. volume from the IFAC Workshop. Princeton, New Jersey, 16-18 March 2000 / ed. by N.E. Leonard, R. Ortega. - Oxford : Pergamon, 2000 %I Elsevier %@ 0-08-043658-7 %G en %U http://hdl.handle.net/1963/1502 %1 2661 %2 Mathematics %3 Functional Analysis and Applications %$ Made available in DSpace on 2004-09-01T13:03:18Z (GMT). No. of bitstreams: 0\\n Previous issue date: 2000 %0 Book Section %B Quantization, Coherent States and Poisson Structures, Proc. XIVth Workshop on Geometric Methods in Physics, Bialowieza, Poland, 9-15 July 1995, eds. A. Strasburger,\\nS.T. Ali, J.-P. Antoine, J.-P. Gazeau , A. Odzijewicz, Polish Scientific Publisher PWN 1 %D 1995 %T Quantum homogeneous spaces at roots of unity %A Cesare Reina %A Alessandro Zampa %B Quantization, Coherent States and Poisson Structures, Proc. XIVth Workshop on Geometric Methods in Physics, Bialowieza, Poland, 9-15 July 1995, eds. A. Strasburger,\\nS.T. Ali, J.-P. Antoine, J.-P. Gazeau , A. Odzijewicz, Polish Scientific Publisher PWN 1 %I SISSA Library %G en %U http://hdl.handle.net/1963/1022 %1 2834 %2 Mathematics %3 Mathematical Physics %$ Made available in DSpace on 2004-09-01T12:41:48Z (GMT). No. of bitstreams: 1\\n120_95.pdf: 543406 bytes, checksum: bfe024146f8ed75ea27cda0399052977 (MD5)\\n Previous issue date: 1995 %0 Journal Article %J Publ. Res. Inst. Math. Sci. 26 (1990), no. 5, 803--817 %D 1990 %T Quadratic forms for singular perturbations of the Laplacian %A Alessandro Teta %B Publ. Res. Inst. Math. Sci. 26 (1990), no. 5, 803--817 %I SISSA Library %G en %U http://hdl.handle.net/1963/757 %1 3034 %2 Mathematics %3 Mathematical Physics %$ Made available in DSpace on 2004-09-01T12:37:11Z (GMT). No. of bitstreams: 1\\n165_88.pdf: 420094 bytes, checksum: 84ec202153b3f40e2584898048015e2a (MD5)\\n Previous issue date: 1988