The spectral curve theory for (k,l)-symmetric CMC surfaces

Constant mean curvature surfaces in $S^3$ can be studied via their associated family of flat connections. In the case of tori this approach has led to a deep understanding of the moduli space of all CMC tori. For compact CMC surfaces of higher genus the theory is far more involved due to the non abelian nature of their fundamental group. In this paper we extend the spectral curve theory for tori developed in Hitchin (1990), Pinkall and Sterling (1989) and for genus 2 surfaces (Heller, 2014) to CMC surfaces in $S^3$ of genus $g=k\cdot l$ with commuting ${\mathbb{Z}}_{k+1}$ and ${\mathbb{Z}}_{l+1}$ symmetries. We determine their associated family of flat connections via certain flat line bundle connections parametrized by the spectral curve. We generalize the flow on spectral data introduced in Heller (2015) and prove the short time existence of this flow for certain families of initial surfaces. In this way we obtain countably many $1-$parameter families of new CMC surfaces of higher genus with prescribed branch points and prescribed umbilics.     

A comparative study of kinematic hardening rules at finite deformations

Kinematic hardening models describe a specific kind of plastic anisotropy which evolves with the deformation process. It is well known that the extension of constitutive relations from small to finite deformations is not unique. This applies also to well-established kinematic hardening rules like that of Armstrong-Frederick or Chaboche. However, the second law of thermodynamics offers some possibilities for generalizing constitutive equations so that this ambiguity may, in some extent, be moderated. The present paper is concerned with three possible extensions, from small to finite deformations, of the Armstrong-Frederick rule, which are derived as sufficient conditions for the validity of the second law. All three models rely upon the multiplicative decomposition of the deformation gradient tensor into elastic and plastic parts and make use of a yield function expressed in terms of the so-called Mandel stress tensor. In conformity with this approach, the back-stress tensor is defined to be of Mandel stress type as well. In order to compare the properties of the three models, predicted responses for processes with homogeneous and inhomogeneous deformations are discussed. To this end, the models are implemented in a finite element code (ABAQUS).     

Large deformations of homogeneous and fiber-reinforced cylinders under the action of centrifugal forces

The problem on large deformations of round cylinders made of homogeneous and fiber-reinforced elastomeric materials under the action of inertia forces caused by rotatory motion is solved. The solution is given in a plane axisymmetric statement, where the deformation parameters of the cylinders depend only on the radial coordinate. Based on the mathematical model presented, the deformation of revolving cylinders made of homogeneous elastomeric materials and reinforced with fibers in the axial, circumferential, and radial directions is investigated. The ultimate rotational speeds for cylinders with different reinforcement schemes and tight and loose fits are found at which their limiting configurations are reached. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 3, pp. 347-366, May-June, 2009.     

Quantum duality in quantum deformations

In accordance with the quantum duality principle, the twisted algebra is equivalent to the quantum group and has two preferred bases: one inherited from the universal enveloping algebra and the other generated by coordinate functions of the dual Lie group . We show howthe transformation can be explicitly obtained for any simple Lie algebra and a factorable chain of extended Jordanian twists. In the algebra , we introduce a natural vector grading , compatible with the adjoint representation of the algebra. Passing to the dual-group coordinates allows essentially simplifying the costructure of the deformed Hopf algebra , considered as a quantum group . The transformation can be used to construct new solutions of the twist equations. We construct a parameterized family of extended Jordanian deformations and study it in terms of ; we find new realizations of the parabolic twist. Dedicated to the birthday of my teacher, Yurii Novozhilov __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 1, pp. 112–125, July, 2006.     

Amplitude modulation of a low-coherence source, applications to distance and dynamic displacement sensing

The possibilities of the radio-frequency amplitude modulation of a low-coherence source for distance and displacement measurements are discussed. This intensity-based method, used within a Michelson interferometer, is a novel alternative to perform dynamic deformation measurements. The theoretical background is presented, together with the experimental verification of the principle. Besides, the results of the first quasi-dynamic tests are shown and the perspectives of the technique discussed. This method is specially useful for the dynamic monitoring of civil structures, where large measurements bases are needed. Furthermore, the application of the amplitude modulation for distance measurements is demonstrated experimentally.     

Steps in the Classification of Cohen–Macaulay Modules Over Singularities of Type xt + y3

Let k be an algebraically closed field of characteristic 3 and i, j, t some positive integers such that 1 i < j < t, i + j t. Then there exist a finite number of nonisomorphic indecomposable maximal Cohen–Macaulay modules N over k[[x, y]] /(x^t + y³) such that N / y N is a direct sum of copies of k[[x]] /(xⁱ), k[[x]] /(x^j) and we describe them completely.     

On Continuous and Differential Hochschild Cohomology

We show that there are canonical isomorphisms between Hochschild cohomology spaces , where is the algebra of smooth functions on a manifold M and the space of skew multivector fields over M. This implies that continuous and differential deformation theories of coincide.     

Discrete Structure of Some Schlesinger Systems on the Riemann Sphere

In the present work we investigate the group structure of the Schlesinger transformations for isomonodromic deformations of the Fuchsian differential equations. We perform these transformations as isomorphisms between the moduli spaces of the logarithmic sl(N)-connections with fixed eigenvalues of the residues at singular points. We give a geometrical interpretation of the Schlesinger transformations and perform our calculations using the techniques of the modifications of bundles with connections, or, the Hecke correspondences for the loop group SL(N)C(z).     

Theoretical Modeling of Mechanical Resistance and Stability and Related Characteristics of Polymeric Systems with Branching/Cross-Linking

A theoretical (thermodynamic) method for the estimation of mechanical characteristics of polymeric systems is proposed. This method uses the statistical polymer method for modeling of branched/cross-linked structures. The weak interaction between macromolecules is modeled in the approach of their mutual interpenetration. The proposed method is used for the estimation of mechanical resistance and stability of microporous polymeric materials. An engineer method for the evaluation of mechanical stability and resistance of polymeric materials is derived.