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

APPROACHES TO THE ESTIMATION OF THE LEVEL OF FORMATION OF PROFESSIONAL COMPETENCES OF GRADUATES OF THE SPO IN MODERN CONDITIONS THE IMPLEMENTATION OF FGOS SPO

В статье рассматриваются возможности использования технологий активного обучения для формирования профессиональной компетентности будущего педагогаThe article describes the potentiality of active learning technologies for forming of professional-pedagogical competenc

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

ATF2 Proposal

ATF2 Group Accelerator Physic

A systematic approach towards the design of a multi-layered woven fabric: modelling the structure of a multi-layered woven fabric

This paper introduced an innovative method of description of the structure of multi-layered woven fabrics (MLWF). The method enables a complicated design procedure for the MLWF to be reduced to the much simpler design process of a single-layer woven fabric using conventional CAD systems. The method can be applied to all types of common MLWF including orthogonal and angle-interlock woven structures. This method will be widely used in the textile industry for the design of multi-layered woven fabrics and by the software companies which provide CAD/CAM systems for the industry