Texture parameter variation region for orthotropic polycrystals with cubic symmetry of the crystal lattice

The variation region of texture parameters (which are integral characteristics of the preferable orientation of crystallographic axes) allows solutions to be found for managing anisotropic properties, illustrating all possible textured states of an orthotropic polycrystalline material with a crystal lattice of cubic symmetry. Each point in this region is matched by certain anisotropy of both elastic and plastic properties. The region of texture parameter variation is defined both analytically and using a numerical experiment of statistic simulation. Analytically, the solution is found via determining the effective eigenvalues of the elasticity operator for a textured anisotropic cubic polycrystal. The algorithm to be followed for visualizing the region-forming elements implies determining the lines of intersection of planes with a conical surface. The numerical solution is based on the determination of texture parameters, i.e, on the starting assumption that the variation region is bounded and lies in the first octant. The task of constructing the variation region is solved via finding the texture parameters using the Monte-Carlo method according to the density of distribution of crystallographic axes in space. When modeling the variation region, octets are used, which are symmetrical reflections of randomly taken orientations in all the octants of space. The constructed regions have the required symmetry. The numerically obtained cloud of textured states and the analytically constructed variation region have geometric centers coinciding at the point corresponding to the non-textured state. At various stages of thermal and mechanical treatment of metallic materials, texture evolution can be represented geometrically as a texture state trajectory that is seen to be within the determined texture parameter variation region. © 2018 Author(s)

Geometric representation of polycrystalline material texture by axis-angle parametrization

Texture is the preferential orientation of crystallographic axes in polycrystal. For its mathematical modeling, the orientation distribution function of the crystallographic axes is used. Traditionally, the orientation distribution function is written with the help of directional cosine matrices, Miller indices or Euler-Krylov angles. Recently, texture has increasingly often been described using quaternions, Rodrigues parameters and the vector space of axis-angle parameters. Axis-angle parameters allow us to describe all possible rotations of the SO(3) group, which corresponds to all possible orientations of crystallographic axes in polycrystalline materials. The SO(3) group is a set of rotations to all possible angles around all possible axes given by all vectors of the unit sphere. The set of such rotations corresponds to points set on a ball of radius π in three-dimensional Euclidean space. The description of the crystallographic texture using axis-angle parameters made it possible to visualize the distribution of crystallographic axes and obtain a new geometric representation of the texture. © 2018 Author(s)

Models of shifts and correlation links of hematological and immunological parameters of patients with chronic generalized periodontitis

The goal of this study is to examine and to evaluate hematological and immunological parameters of patients with chronic generalized periodontitis based on the models of shifts and correlations. At that, the assessment of carried-out modeling and correlations allows to identify the leading parameters of diagnostics and intrasystemic blood parameters and cell-mediated immunit

Mathematical model of building envelope element insolation

In this paper, the problem of estimation of building envelope oriented element insolation by direct sunlight is solved using mathematical model of Earth's rotation. Quaternions are used as mathematical tool for description of rotation. The model allows to obtain automatically estimation of building element insolation by direct sunlight for given latitude and given time interval (month, week, day, etc.). The model takes into account schedule of change of day and night, change of direct sunlight angle on a given element of the surface caused by sun motion over horizon (changes of its height and azimuth during time of day). Distinctive feature of the model is simplicity of program-algorithmic implementation due to using description of rotation by means of quaternions. © Published under licence by IOP Publishing Ltd

Proof Theory and Ordered Groups

Ordering theorems, characterizing when partial orders of a group extend to total orders, are used to generate hypersequent calculi for varieties of lattice-ordered groups (l-groups). These calculi are then used to provide new proofs of theorems arising in the theory of ordered groups. More precisely: an analytic calculus for abelian l-groups is generated using an ordering theorem for abelian groups; a calculus is generated for l-groups and new decidability proofs are obtained for the equational theory of this variety and extending finite subsets of free groups to right orders; and a calculus for representable l-groups is generated and a new proof is obtained that free groups are orderable

Models of shifts and correlation links of hematological and immunological parameters of patients with chronic generalized periodontitis

The goal of this study is to examine and to evaluate hematological and immunological parameters of patients with chronic generalized periodontitis based on the models of shifts and correlations. At that, the assessment of carried-out modeling and correlations allows to identify the leading parameters of diagnostics and intrasystemic blood parameters and cell-mediated immunit