Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton

Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

Functional Fiber Junctions for Circuit Routing in E-Textiles: Deterministic Alignment of MEMS Layout With Fabric Structure

Fabrics and fibrous materials offer a soft, porous, and flexible substrate for microelectromechanical systems (MEMS) packaging in breathable, wearable formats that allow airflow. Device-on-fiber systems require developments in the field of E-Textiles including smart fibers, functional fiber intersections, textile circuit routing, and alignment methods that adapt to irregular materials. In this paper, we demonstrate a MEMS-on-fabric layout workflow that obtains fiber intersection locations from high-resolution fabric images. We implement an image processing algorithm to drive the MEMS layout software, creating an individualized MEMS “gripper” layout designed to grasp fibers on a specific fabric substrate during a wafer-to-fabric parallel transfer step. The efficiency of the algorithm in terms of a number of intersections identified on the complete image is analyzed. The specifications of the MEMS layout design such as the length of the MEMS gripper, spatial distribution, and orientation are derivable from the MATLAB routine implemented on the image. Furthermore, the alignment procedure, tolerance, and hardware setup for the alignment method of the framed sample fabric to the wafer processed using the custom gripper layout are discussed along with the challenges of the release of MEMS devices from the Si substrate to the fabric substrate.

The development of stamp points displacement mathematical model by static tests of road structures

In various static stamp tests methods, both foreign and Ukrainian, it is foreseen to use a different number of sensors for stamp settlement measuring. Austrian method with three displacement sensors allows to reveal the stamp warping while loading, but it is unclear how to determine the settlement in the stamp center, when the sensor readings at the moment of warping will be completely different. The German method with a single displacement sensor is much simpler, but does not consider the stamp warping. The use of Ukrainian methodology is not appropriate at all, because two displacement sensors do not allow to observe the stamp warping. The article is focused on mathematical model for the displacement of stamp points in road structures static testing development with theoretical justification of necessary and sufficient number of sensors for stamp settlement measuring. Initially, the stamp displacement under warping in a flat deformed condition is considered as a rotation on a certain angle α and a parallel transfer. It has been established that in this case, the stamp settlement with enough accuracy for engineering practice can be measured with a single displacement sensor installed in the stamp center. In fact, the stamp can rotate not only around the axis Oy, but also around the Ox axis. Therefore, a separate mathematical model is constructed and describes the process of stamp displacement while it’s warping in a volumetric deformed condition. The developed mathematical model provides an opportunity to determine the settlement in the stamp center, necessary for the modulus of elasticity calculation or modulus of deformation in the case when the stamp at the warping moment will rotate both around the axis Oy and around the Ox axis, and the rotation angles α and β will be significant.

Method of applied direction of mathematics in the aviation institution of higher education

The elements of the method of applied direction of mathematics in the aviation institution of higher education developed by us are considered. We use this technique in higher mathematics classes at the Flight Academy of the National Aviation University. Examples of a combination of fundamental mathematical concepts and practical methods of their application are given. We illustrate in detail the coordinate systems used in aviation. Among these systems are mobile and fixed coordinate systems. Problems of higher mathematics related to one or another coordinate system are indicated. For example, to record vector equations of motion in projections, moving coordinate systems are used, the beginning of which is located in the center of mass of the aircraft. Therefore, the study of the topic "Reflection of linear (vector) spaces" acquires a professional orientation. In particular, we present the formulas of coordinate transformations for parallel transfer and rotation of the axes. Note that the transformations of rectangular coordinate systems are used in aviation. Having considered aviation coordinate systems, the teacher is interested in students in the study of equations of motion, determination of accelerations, velocities and displacements. Methodical methods of formation of practical skills and abilities of future aviation specialists contribute to the implementation of the applied direction of mathematics. We have given some examples of methods of applied direction of mathematics in aviation in the sources that are currently published. The prospect of our research is to further improve practical approaches to solving problems of mathematical training of students of aviation institutions of higher education. This should help increase the level of methodological training of scientific and pedagogical staff of higher education institutions. At the same time, it should contribute to the improvement of methods of teaching mathematics in terms of its application. As a result, the graduate of the aviation institution of higher education must be ready for successful professional activity. Key words: applied direction of mathematics, coordinate systems in aviation, parallel transfer, rotation of axes, fundamental mathematical concepts in aviation.

Methods of optimization of Hausdorff distance between convex rotating figures

We studied the problem of optimizing the Hausdorff distance between two convex polygons. Its minimization is chosen as the criterion of optimality. It is believed that one of the polygons can make arbitrary movements on the plane, including parallel transfer and rotation with the center at any point. The other polygon is considered to be motionless. Iterative algorithms for the phased shift and rotation of the polygon are developed and implemented programmatically, providing a decrease in the Hausdorff distance between it and the fixed polygon. Theorems on the correctness of algorithms for a wide class of cases are proved. Moreover, the geometric properties of the Chebyshev center of a compact set and the differential properties of the Euclidean function of distance to a convex set are essentially used. When implementing the software package, it is possible to run multiple times in order to identify the best found polygon position. A number of examples are simulated.

Textbook Theory

Textbook theory seems to be at the farthest level from the most abstract categories, and it is repeatedly associated with in the current study. The connection of the principal fundamental categories of invariance and binarism are not even superficially traced to the traditional textbook theory inventory. The numerous conceptions of predication predominantly deal with its logical frames. The aim of the chapter is to explicate some hidden links between the levels of the conceptual vertical. The idea of invariance is embodied by the continuous iteration of binary predicative units originally incompressible in terms of their constituents and structure. Binary opposition is the category indissolubly active at every stage of speech generation. There have been found four functional patterns produced by a particular predicative operation. Each pattern correlates with all types of speech but to various extents, accenting some of them. They are parallel transfer, inverse transformation, convergent or divergent pivot structuring, and zooming.

Lines on the surface in the quasi-hiperbolic space 11^S1/3

Quasi-hyperbolic spaces are projective spaces with decaying abso­lute. This work is a continuation of the author's work [7], in which surfac­es in one of these spaces are examined by methods of external forms and a moving frame. The semi-Chebyshev and Chebyshev net­works of lines on the surface in quasi-hyperbolic space are considered. In this pa­per we use the definition of parallel transfer adopted in [6]. By analogy with Euclidean geometry, the semi-Chebyshev network of lines on the surface is the network in which the tangents to the lines of one family are carried parallel along the lines of another family. A Che­byshev network is a network in which tangents to the lines of each family are carried parallel along the lines of another family. We proved three theorems. In Theorem 1, we obtain a natural equa­tion for non-geodesic lines that are part of a conjugate semi-Chebyshev network on the surface so that tangents to lines of another family are transferred in parallel along them. In Theorem 2, the natural equation of non-geodesic lines in the Chebyshev network is obtained. In Theorem 3 we prove that conjugate Chebyshev networks, one family of which is nei­ther geodesic lines, nor Euclidean sections, exist on surfaces with the lati­tude of four functions of one argument.

Comparative Study of DDS with Different Types of Phase Accumulators

The paper deals with the problems of delayed transfer signals in the direct digital synthesizer (DDS) phase accumulator adders. Transfer delay is one of the factors that affect the maximum output frequency of the DDS synthesizer. The main types of adders used in DDS synthesizers are described. Separately, attention was paid to the adder with a consistent transfer of the transfer signal, adders with a transmission carry signal with a fixed block length, adders with a signal transmission delay with a variable block length, and a mathematical analysis of the origin and duration of the delay of the transfer signal in them. It was found that the use of a transfer adder with a variable length of a block in the core of a direct digital synthesizer would increase the maximum output frequency by 2.4 times compared to the adder with a parallel transfer, and by 1.43 times as compared with the adder with a fixed length the block.