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We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. The basis of our model is a lesser known characterization of developable surfaces as manifolds that can be parameterized through orthogonal geodesics. Our model is simple, local, and, unlike previous works, it does not directly encode the surface rulings. This allows us to model continuous deformations of discrete developable surfaces independently of their decomposition into torsal and planar patches or the surface topology. We prove and experimentally demonstrate strong ties to smooth developable surfaces, including a theorem stating that every sampling of the smooth counterpart satisfies our constraints up to second order. We further present an extension of our model that enables a local definition of discrete isometry. We demonstrate the effectiveness of our discrete model in a developable surface editing system, as well as computation of an isometric interpolation between isometric discrete developable shapes.

The incommensurate-commensurate phases reported in cupric oxide below 230 K are shown theoretically to realize an inverted sequence of symmetry-breaking mechanisms with respect to the usual sequence occurring in low-temperature multiferroic compounds. The sequence inversion results from a strong triggering-coupling mechanism between two antiferromagnetic order parameters inducing a first-order transition to the multiferroic phase. Such mechanism is favored by the large antiferromagnetic superexchange interactions, responsible of the high-T(N) temperature, and implies a preeminence of these interactions on the magnetocrystalline anisotropy. The magnetic structures of the equilibrium phases and the microscopic interactions giving rise to the polarization are determined.

It is shown that the factorization relation on simple Lie groups with standard Poisson Lie structure restricted to Coxeter symplectic leaves gives an integrable dynamical system. This system can be regarded as a discretization of the Toda flow. In case of SL n the integrals of the factorization dynamics are integrals of the relativistic Toda system. A substantial part of the paper is devoted to the study of symplectic leaves in simple complex Lie groups, its Borel subgroups and their doubles.

Bäcklund transformations for smooth and "space discrete" Hashimoto surfaces are discussed and a geometric interpretation is given. It is shown that the complex curvature of a discrete space curve evolves with the discrete nonlinear Schrödinger equation (NLSE) of Ablowitz and Ladik, when the curve evolves with the Hashimoto or smoke ring flow. A doubly discrete Hashimoto flow is derived and it is shown, that in this case the complex curvature of the discrete curve obeys Ablovitz and Ladik's doubly discrete NLSE. Elastic curves (curves that evolve by rigid motion only under the Hashimoto flow) in the discrete and doubly discrete case are shown to be the same.There is an online version of this paper, that can be viewed using any recent web browser that has JAVA support enabled. It includes two additional java applets. It can be found at

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We show that the electric-field-induced reversal of the magnetic order parameter in multiferroic MnWO 4 occurs on the time scale of milliseconds. Throughout the switching process the magnetic order and the magnetically induced electric polarization remain rigidly coupled. The temporal progression of the domain structure was imaged with nanosecond resolution by an electrical-pump-optical-probe technique using optical second harmonic generation. These nonequilibrium domain states significantly differ from the quasi-static domain reversal. A qualitative model gives an estimate of why the magnetoelectric order-parameter reversal in the magnetically induced ferroelectrics is not inherently ultrafast.

We define discrete flat surfaces in hyperbolic 3-space H 3 from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean curvature 1 surfaces in H 3 , and we also describe discrete focal surfaces (discrete caustics) that can be used to define singularities on discrete flat surfaces. Along the way, we also examine discrete linear Weingarten surfaces of Bryant type in H 3 , and consider an example of a discrete flat surface related to the Airy equation that exhibits swallowtail singularities and a Stokes phenomenon.