On some properties of normally-inflective complexes with simple inflective center in $E_3$

In the paper, we consider the special degenerate class of mormally-inflective complexes with simple inflective center in three-dimensional Euclidean space $E_3$. We prove that to construct this class of complexes one should take an arbitrary curve and draw sheaf of straight lines through each point of this curve. For arbitrary normally-inflective complex with simple inflective center we establish that such complex is fibered into two one-parametric families of congruences.

Stress–temperature equations of motion of Ignaczak and Beltrami–Michell types in arbitrary curve coordinate system: معادلات الحركة بلغة الإجهادات والحرارة من نوعي إغناتشاك وبيلترامي– ميشيل في أي نظام احداثي منحني

This paper relates to the mathematical linear model of the elastic, homogeneous and isotropic body, with neglected structure and infinitesimal elastic strains, subjected to temperature field; discussed by Hooke, and shortly called (H). We firstly introduce the variable tensorial forms of the traditional and Lame descriptions of the coupled dynamic state of considerable Hooke body, in an arbitrary curve coordinate system. We study the variable tensorial forms in an arbitrary curve coordinate system, of the generalized Beltrami–Michell stress-temperature equations, and of the stress-temperature Ignaczak equations and its completeness problem for the (H) thermoelastic body.

Algorithm to obtain inverse kinematics matrix from the 3D curve and to apply to glue shoe sole

Nowadays, manipulator is widely used in industrial applications. The trajectories of manipulator are more and more complicated. In order to do good tracking performance, the end effector position and orientation have to be determined. This paper describes a method to determine position and orientation of manipulator’s end effector base on a reference path. This method will be applied for manipulator 6 DOF to glue shoe sole. Firstly, assume the reference path is arbitrary curve, the path was then discrete to become multi-point. Secondly, the roll – pitch – yaw vectors of the end effector will be determined at each point. Finally, Euler angles and interpolation method in 3D space will be applied to determine inverse kinematics matrix of manipulator for each point on the reference path. In addition, this paper also gives an example of reference path of shoe sole to apply the presented method. To verify the tracking performance of manipulator and reference path, a PID controller was designed for simulation. The result of simulation proved the correction of the algorithm.

HOMOLOGICAL INDEX FOR 1-FORMS AND A MILNOR NUMBER FOR ISOLATED SINGULARITIES

We introduce a notion of a homological index of a holomorphic 1-form on a germ of a complex analytic variety with an isolated singularity, inspired by Gómez-Mont and Greuel. For isolated complete intersection singularities it coincides with the index defined earlier by two of the authors. Subtracting from this index another one, called radial, we get an invariant of the singularity which does not depend on the 1-form. For isolated complete intersection singularities this invariant coincides with the Milnor number. We compute this invariant for arbitrary curve singularities and compare it with the Milnor number introduced by Buchweitz and Greuel for such singularities.