Fixed-point tile sets and their applications

An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many fields, ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene?s fixed-point construction instead of geometric arguments. This construction is similar to J. von Neumann?s self-reproducing automata; similar ideas were also used by P. Gács in the context of error-correcting computations. This construction is rather flexible, so it can be used in many ways. We show how it can be used to implement substitution rules, to construct strongly aperiodic tile sets (in which any tiling is far from any periodic tiling), to give a new proof for the undecidability of the domino problem and related results, to characterize effectively closed one-dimensional subshifts in terms of two-dimensional subshifts of finite type (an improvement of a result by M. Hochman), to construct a tile set that has only complex tilings, and to construct a “robust” aperiodic tile set that does not have periodic (or close to periodic) tilings even if we allow some (sparse enough) tiling errors. For the latter, we develop a hierarchical classification of points in random sets into islands of different ranks. Finally, we combine and modify our tools to prove our main result: There exists a tile set such that all tilings have high Kolmogorov complexity even if (sparse enough) tiling errors are allowed. Some of these results were included in the DLT extended abstract (Durand et al., 2008 [9]) and in the ICALP extended abstract (Durand et al., 2009 [10]).     

Computational Investigation of the Effect of Reinforcement Schemes and Angles on the Stress-Strain State and Strength of Composite Cylinders Under Axisymmetric Internal Explosion. Part 1. Effect of the Discretization Spacings of the Computational Domain on the Accuracy of Determination of Stress-Strain State and Strength

Strength of Materials - The effect of two-dimensional regular finite-difference mesh spacings on the accuracy of calculation of the dynamic axisymmetric stress-strain state and strength of hollow...     

Numerical-analytical method for 3D dynamics study of spirally orthotropic multilayer cylinders

Based on the combination of the Fourier series expansions of variables with respect to the angular coordinate and a two-dimensional Wilkins algorithm, a numerical analytical method is developed to study the three-dimensional non-axisymmetric dynamics of multilayer thick-walled elastic spirally orthotropic cylinders of finite length. Verification of the proposed method is performed based on the comparison of the results with those obtained using the computational kernel of the LS-DYNA combined with the ANSYS/ED (an educational version of ANSYS finite element software). Some features of the dynamic behavior of a two-layer spirally reinforced cylinder under internal pulse loading produced by concentrated explosive charges are studied numerically.     

Method of solving three-dimensional nonlinear dynamic problems for compound bodies of rotation

A mathematical formulation is given for the three-dimensional nonstationary geometrically linear state of stress and strain in a compound body of rotation in a physically nonlinear setting. This incorporates failure and plasticity in the material. The method is based on expanding the unknowns as Fourier series and applying a numerical technique, which is shown by numerical experiments to be suitable for treatments in this class.Translated from Problemy Prochnosti, No. 6, pp. 60–70, June 1993.     

Exact solution of problems of flow theory with isotropic-kinematic hardening. Part 1. Setting the loading trajectory in the space of stresses

For an arbitrary isotropic and linear kinematic hardening and loading paths given in the form of arbitrary multisection polygonal lines in the five-dimensional deviator space of stresses, we studied analytically an initially isotropic elastoplastic von Mises material and the associated flow rule. The solutions obtained are valid for arbitrary relationships governing the variation of the spherical component of the stress tensor. Explicit solutions are obtained for several important cases of material behavior. Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 6, pp. 81–92, November–December, 1999.     

On the Applicability of One-Dimensional Models to the Dynamic Strength Estimation for Axially Symmetric Bodies

A comparative analysis of the characteristics-difference method and the Wilkins algorithm with reference to problems of nonstationary deformation of one-dimensional cylindrical thick-walled shells has been carried out. On the basis of comparing one- and two-dimensional solutions of boundary problems of the dynamic theory of elasticity obtained by the Wilkins method, conclusions have been drawn about the validity limit of one-dimensional models for the estimation of the strength of female dies for impulse-pressure forming of closed cylindrical and conical shells.     

The elastic theory dynamic problem for a transtropic multilayer sphere

The analytical solution is obtained in quadratures of the elastic theory dynamic centrally-symmetric boundary problems of the same class for multilayer transtropic spheres, the sonic speed in the radial direction of which tends to infinity. Convergence and accuracy of the derived solutions are analyzed. It is shown that the analytical solutions obtained can be used as the first approximation for the cases where the sonic speed in materials is finite.