Drawing bobbin lace graphs, or, Fundamental cycles for a subclass of periodic graphs

In this paper, we study a class of graph drawings that arise from bobbin lace patterns. The drawings are periodic and require a combinatorial embedding with specific properties which we outline and demonstrate can be verified in linear time. In addition, a lace graph drawing has a topological requirement: it contains a set of non-contractible directed cycles which must be homotopic to $(1,0)$, that is, when drawn on a torus, each cycle wraps once around the minor meridian axis and zero times around the major longitude axis. We provide an algorithm for finding the two fundamental cycles of a canonical rectangular schema in a supergraph that enforces this topological constraint. The polygonal schema is then used to produce a straight-line drawing of the lace graph inside a rectangular frame. We argue that such a polygonal schema always exists for combinatorial embeddings satisfying the conditions of bobbin lace patterns, and that we can therefore create a pattern, given a graph with a fixed combinatorial embedding of genus one.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Doubly periodic textile patterns

Knitted and woven textile structures are examples of doubly periodic structures in a thickened plane made out of intertwining strands of yarn. Factoring out the group of translation symmetries of such a structure gives rise to a link diagram in a thickened torus. Such a diagram on a standard torus is converted into a classical link by including two auxiliary components which form the cores of the complementary solid tori. The resulting link, called a kernel for the structure, is determined by a choice of generators u and v for the group of symmetries. A normalised form of the multi-variable Alexander polynomial of a kernel is used to provide polynomial invariants of the original structure which are essentially independent of the choice of generators. It gives immediate information about the existence of closed curves and other topological features in the original textile structure. Because of its natural algebraic properties under coverings we can recover the polynomial for kernels based on a proper subgroup from the polynomial derived from the full symmetry group of the structure. This enables two structures to be compared at similar scales, even when one has a much smaller minimal repeating cell than the other. Examples of simple traditional structures are given, and their Alexander data polynomials are presented to illustrate the techniques and results.Comment: 27 pages, 22 figure

Development of financial management in Russian companies in the context of digitalization of the economy

The research is devoted to the study of trends and prospects for the development of digital financial management technologies in the business environment. The process of transition from the classical business system to the use of digital resources and the role of financial management in their effective formation is analyzed. The necessity of introducing digital technologies in the field of financial management is substantiated, their advantages are systematized, including a significant reduction in material, labor and financial costs. The problems of financial management development in the context of digital transformation of companies are formulated and recommendations for their solution are developed

Actual problems and prospects of the Russian stock market in the context of geopolitical tensions

Introduction. In a market economy, the dynamics of the securities market development is the most important tool for stimulating economic growth through the mechanism of attracting investment resources and their redistribution between sectors of the economy. In the context of geopolitical tensions, the Russian stock market has experienced enormous pressure, which has led to a deterioration in the prospects for its further development.Goal. Analysis of the state of the Russian securities market under the conditions of sanctions pressure and identification of the main problems hindering its stable development.Materials and methods. The study is based on the analysis of the economic situation with a focus on the segment of the Russian stock market, determining its competitiveness and growth drivers in conditions of economic uncertainty. The complex consideration of many factors in a dynamically changing environment has necessitated the integrated use of general scientific methods (analysis, synthesis, generalization), comparative analysis, abstract thinking, the method of expert assessments and the graphical method.Results and discussion. In the course of the work, the dynamics of the Moscow Exchange index and the market capitalization of the Russian market were analyzed, statistical data on the number of private investors on the exchange, the volume of their investments in the period 2020–2021 were given. Special attention is paid to the largest companies-issuers of the oil and gas sector. The study of the questions raised made it possible to determine that the Russian stock market shows one of the highest dividend yields compared to developing and developed countries.Conclusion. Based on the results of the study, it can be concluded that the Russian economy has adapted to the new economic realities and is ready for new challenges. From a fundamental point of view, Russian issuers show high operating results. This means that as the economy recovers, we should expect an increase in liquidity on the stock market and a significant increase in the quotations of many companies in the medium term