Calculating Complete Lists of Belyi Pairs

Belyi pairs constitute an important element of the program developed by Alexander Grothendieck in 1972–1984. This program related seemingly distant domains of mathematics; in the case of Belyi pairs, such domains are two-dimensional combinatorial topology and one-dimensional arithmetic geometry. The paper contains an account of some computer-assisted calculations of Belyi pairs with fixed discrete invariants. We present three complete lists of polynomial-like Belyi pairs: (1) of genus 2 and (minimal possible) degree 5; (2) clean ones of genus 1 and degree 8; and (3) clean ones of genus 2 and degree 8. The explanation of some phenomena we encounter in these calculations will hopefully stimulate further development of the dessins d’enfants theory.

Asynchronous Computability Theorem in Arbitrary Solo Models

In this paper, we establish the asynchronous computability theorem in d-solo system by borrowing concepts from combinatorial topology, in which we state a necessary and sufficient conditions for a task to be wait-free computable in that system. Intuitively, a d-solo system allows as many d processes to access it as if each were running solo, namely, without detecting communication from any peer. As an application, we completely characterize the solvability of the input-less tasks in such systems. This characterization also leads to a hardness classification of these tasks according to whether their output complexes hold a d-nest structure. As a byproduct, we find an alternative way to distinguish the computational power of d-solo objects for different d.

Development of ordinary harness for aircraft onboard cable networks

The article is devoted to one of the most important components of the onboard complex of aircraft equipment - the onboard cable network. The contents of the design documentation for the aircraft onboard cable network are disclosed. The statement of the problem of designing harnesses is defined in general terms. The main stages of designing aircraft onboard cable networks are described on the verbal level, as well as in the form of logical algorithms and graph-algorithms. Some theoretical aspects of designing aircraft onboard cable networks are presented. The concepts of topological space, topological structure, and continuous mapping of the harness structure into the aircraft structure are introduced. Geometric research of an ordinary cable harness of the onboard cable network led to the need to consider the harnesses as a geometric complex in the framework of combinatorial topology. An example of compiling a table of connections of ordinary harnesses for the aircraft onboard system of ultra-short wave communication is given. The rules and requirements for the information content of the table of connections of an ordinary harness to the aircraft on-board system are emphasized. Mention is made of the need to integrate ordinary harnesses into a complex one consisting of tens or even hundreds of ordinary harnesses to simplify the process of installation of the onboard cable network in the aircraft.

5. Flavours of topology

From the mid-19th century, topological understanding progressed on various fronts. ‘Flavours of topology’ considers other areas such as differential topology, algebraic topology, and combinatorial topology. Geometric topology concerned surfaces and grew out of the work of Euler, Möbius, Riemann, and others. General topology was more analytical and foundational in nature; Hausdorff was its most significant progenitor and its growth mirrored other fundamental work being done in set theory. The chapter introduces the hairy ball theorem, and the work of great French mathematician and physicist Henri Poincaré, which has been rigorously advanced over the last century, making algebraic topology a major theme of modern mathematics.

As-exact-as-possible repair of unprintable STL files

Purpose The class of models that can be represented by STL files is larger than the class of models that can be printed using additive manufacturing technologies. Stated differently, there exist well-formed STL files that cannot be printed. This paper aims to formalize such a gap and describe a fully automatic procedure to turn any such file into a printable model. Design/methodology/approach Based on well-established concepts from combinatorial topology, this paper provide an unambiguous description of all the mathematical entities involved in the modeling-printing pipeline. Specifically, this paper formally defines the conditions that an STL file must satisfy to be printable, and, based on these, an as-exact-as-possible repairing algorithm is designed. Findings It has been found that, to cope with all the possible triangle configurations, the algorithm must distinguish between triangles that bind solid parts and triangles that constitute zero-thickness sheets. Only the former set can be fixed without distortion. Research limitations/implications Owing to the specific approach used that tracks the so-called “outer hull,” models with inner cavities cannot be treated. Practical implications Thanks to this new method, the shift from a 3D model to a printed prototype is faster, easier and more reliable. Social implications The availability of this easily accessible model preparation tool has the potential to foster a wider diffusion of home-made 3D printing in non-professional communities. Originality/value Previous methods that are guaranteed to fix all the possible configurations provide only approximate solutions with an unnecessary distortion. Conversely, this procedure is as exact as possible, meaning that no visible distortion is introduced unless it is strictly imposed by limitations of the printing device. Thanks to such unprecedented flexibility and accuracy, this algorithm is expected to significantly simplify the modeling-printing process, in particular within the continuously emerging non-professional “maker” communities.

On tangent cones at infinity of algebraic varieties

In this paper, we define the geometric and algebraic tangent cones at infinity of algebraic varieties and establish the following version at infinity of Whitney’s theorem [Local properties of analytic varieties, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (Princeton University Press, Princeton, N. J., 1965), pp. 205–244; Tangents to an analytic variety, Ann. of Math. 81 (1965) 496–549]: The geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of the geometric tangent cone at infinity using the global Łojasiewicz inequality with explicit exponents for complex algebraic varieties. Moreover, we show that the tangent cone at infinity of a complex algebraic variety is actually the part at infinity of this variety [G.-M. Greuel and G. Pfister, A Singular Introduction to Commutative Algebra, 2nd extended edn. (Springer, Berlin, 2008)]. We also show that the tangent cone at infinity of a complex algebraic variety can be computed using Gröbner bases.