Anisotropic multiplicative elastoplasticity for the prediction of earings in sheet forming

In this work, the simulation of earings in cup drawing by means of a recently developed anisotropic combined hardening material model is discussed. The model represents a multiplicative formulation of anisotropic elastoplasticity in the finite strain regime with nonlinear kinematic and isotropic hardening. Plastic anisotropy is described by the use of second-order structure tensors as additional arguments in the representation of the yield function and the plastic flow rule. The evolution equations are integrated by a form of the exponential map that preserves the plastic volume and the symmetry of the internal variables. Finite element simulations of cylindrical cup drawing processes are performed by means of ABAQUS/Standard where the discussed material model has been implemented into a user-defined reduced integration solid-shell element. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Remanent quantum correlations in dissipative qubits

Starting from the exact evolution of a Markovian dissipative quantum walk, a non-Markovian decoherence of two qubits interacting with a phonon thermal bath has been investigated analytically using quantum information tools. Concurrence and quantum discord are affected in a complex way, showing that entanglement decreases with dissipation. At the limit where dissipation dominates, quantum correlations survive in time as ∝t^−1/2$\propto t^{-1/2}$. Thus, even under the influence of dissipation, two qubits retain their quantumness for a long time. Quantum correlations could be therefore observed for a long time in related photonic experiments.     

Universality for the Focusing Nonlinear Schrödinger Equation at the Gradient Catastrophe Point: Rational Breathers and Poles of the Tritronquée Solution to Painlevé I

The semiclassical (zero‐dispersion) limit of solutions $q=q(x,t,\epsilon)$ to the one‐dimensional focusing nonlinear Schrödinger equation (NLS) is studied in a scaling neighborhood D of a point of gradient catastrophe ($x_0,t_0$) . We consider a class of solutions, specified in the text, that decay as $|x| \rightarrow \infty$ . The neighborhood D contains the region of modulated plane wave (with rapid phase oscillations), as well as the region of fast‐amplitude oscillations (spikes). In this paper we establish the following universal behaviors of the NLS solutions q near the point of gradient catastrophe: (i) each spike has height $3|q{_0}(x_0,t_0)|$ and uniform shape of the rational breather solution to the NLS, scaled to the size ${\cal O}(\epsilon)$ ; (ii) the location of the spikes is determined by the poles of the tritronquée solution of the Painlevé I (P1) equation through an explicit map between D and a region of the Painlevé independent variable; (iii) if $(x,t)\in D$ but lies away from the spikes, the asymptotics of the NLS solution $q(x,t, \epsilon)$ is given by the plane wave approximation $q_0(x,t, \epsilon)$ , with the correction term being expressed in terms of the tritronquée solution of P1. The relation with the conjecture of Dubrovin, Grava, and Klein about the behavior of solutions to the focusing NLS near a point of gradient catastrophe is discussed. We conjecture that the P1 hierarchy occurs at higher degenerate catastrophe points and that the amplitudes of the spikes are odd multiples of the amplitude at the corresponding catastrophe point. Our technique is based on the nonlinear steepest‐descent method for matrix Riemann‐Hilbert problems and discrete Schlesinger isomonodromic transformations. © 2013 Wiley Periodicals, Inc.     

Recovering plane curves of low degree from their inflection lines and inflection points

In this paper we consider the following problem: is it possible to recover a smooth plane curve of degree d ≥ 3 from its inflection lines? We answer the posed question positively for a general smooth plane quartic curve, making the additional assumption that also one inflection point is given, and for any smooth plane cubic curve.     

A 3-Slope Theorem for the infinite relaxation in the plane

In this paper we consider the infinite relaxation of the corner polyhedron with 2 rows. For the 1-row case, Gomory and Johnson proved in their seminal paper a sufficient condition for a minimal function to be extreme, the celebrated 2-Slope Theorem. Despite increased interest in understanding the multiple row setting, no generalization of this theorem was known for this case. We present an extension of the 2-Slope Theorem for the case of 2 rows by showing that minimal 3-slope functions satisfying an additional regularity condition are facets (and hence extreme). Moreover, we show that this regularity condition is necessary, unveiling a structure which is only present in the multi-row setting.     

Symplectically degenerate maxima via generating functions

We provide a simple proof of a theorem due to Nancy Hingston, asserting that symplectically degenerate maxima of any Hamiltonian diffeomorphism $\phi $ of the standard symplectic $2d$ -torus are non-isolated contractible periodic points or their action is a non-isolated point of the average-action spectrum of $\phi $ . Our argument is based on generating functions.     

Smoothness of holonomy covers for singular foliations and essential isotropy

Given a singular foliation, we attach an “essential isotropy” group to each of its leaves, and show that its discreteness is the integrability obstruction of a natural Lie algebroid over the leaf. We show that a condition ensuring discreteness is the local surjectivity of a transversal exponential map associated with the maximal ideal of vector fields prescribed to be tangent to the foliation. The essential isotropy group is also shown to control the smoothness of the holonomy cover of the leaf (the associated fiber of the holonomy groupoid), as well as the smoothness of the associated isotropy group. Namely, the (topological) closeness of the essential isotropy group is a necessary and sufficient condition for the holonomy cover to be a smooth (finite-dimensional) manifold and the isotropy group to be a Lie group. These results are useful towards understanding the normal form of a singular foliation around a compact leaf. At the end of this article we briefly outline work of ours on this normal form, to be presented in a subsequent paper.