Finding a topological ordering for a directed graph is one of the fundamental problems in computer science. Several textbook-standard algorithms using linear memory have been discovered and utilized to solve several other problems for many decades, especially in resolving dependencies and solving other graph connectivity problems. However, these algorithms are becoming less practical nowadays a...

‎The Mod $2$ Steenrod algebra is a Hopf algebra that consists of the primary cohomology operations‎, ‎denoted by $Sq^n$‎, ‎between the cohomology groups with $mathbb{Z}_2$ coefficients of any topological space‎. ‎Regarding to its vector space structure over $mathbb{Z}_2$‎, ‎it has many base systems and some of the base systems can also be restricted to its sub algebras‎. ‎On the contrary‎, ‎in ...

The structural description of disordered systems has been a longstanding challenge in physical science. We propose an atomic cluster alignment method to reveal the development of three-dimensional topological ordering in a metallic liquid as it undercools to form a glass. By analyzing molecular dynamic (MD) simulation trajectories of a Cu(64.5)Zr(35.5) alloy, we show that medium-range order (MR...

Topological defects easily form in liquid crystals (LCs) as a result of frustrations in spatially dependent anisotropic molecular ordering, and have been regarded as promising tools for facilitating manipulation of relatively large non-LC materials such as colloids. However, it remains unclear whether low-molecular-weight (LMW) impurities that do not aggregate or self-assemble in bulk LCs becau...

We study central configurations lying on a common circle in the Newtonian four-body problem. Using topological argument we prove that there is at most one co-circular configuration for each cyclic ordering of masses circle.

Bogdan and Kolumbán [3] gave sufficient conditions for closedness of the solution map defined on the set of parameters. They considered the parametric equilibrium problems governed by topological pseudomonotone maps depending on a parameter. In this paper we generalize this result for parametric vector equilibrium problems with trifunctions. Let X and Y be Hausdorff topological spaces and P , t...

In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications. Richar...

The effect of a helicoidal magnetic field on the dispersion conduction electrons in one- and two-dimensional systems is investigated. In periodic field, there special symmetry with respect to reversal time, which leads peculiarities band structure. Using an example one-dimensional model system, topological properties structure effective corresponding 120 o -ordering are PdCrO 2 , ordering diele...