Topological polymers have various topological types, and they are expressed by graphs. However, the Jones polynomial, we a difficulty to compute it; computational time is growing exponentially with respect crossing number. The simplest Vassiliev invariant linking number thus will seek next simple one as Milnor's triple In this paper, introduce Gauss diagram formulas of invariants Milnor type. T...

It is proved that classifiable simple separable nuclear purely infinite C∗algebras having finitely generated K-theory and torsion-free K1 are semiprojective. This is accomplished by exhibiting these algebras as C∗-algebras of infinite directed graphs.

We generalize a theorem of Knuth relating the oriented spanning trees of a directed graph G and its directed line graph LG. The sandpile group is an abelian group associated to a directed graph, whose order is the number of oriented spanning trees rooted at a fixed vertex. In the case when G is regular of degree k, we show that the sandpile group of G is isomorphic to the quotient of the sandpi...

The directed power graph G?(G) of a group G is the simple digraph with vertex set such that x?y if y x. G, denoted by G(G), underlying graph. In this paper, for groups and H, following proved. If has no quasicyclic subgroup Cp? which trivial intersection every cyclic K K??Cp?, then G(G)?G(H) implies G?(G)?G?(H). particular, any two torsion-free having isomorphic graphs have graphs.

This paper extends the previous work of generating monotonic-rhythm log-aesthetic space curve segments so that compound-rhythm log-aesthetic space curve segments are interactively generated. A compound-rhythm log-aesthetic space curve segment is composed of two monotonic-rhythm log-aesthetic space curve segments whose logarithmic curvature and torsion graphs are represented by two connected lin...

Methods Overview.Our computational approaches involve two stages: estimation of knowledge-based statistical potentials and Monte Carlo/ Simulated Annealing (MC/SA) sampling of 3D graphs. In the first stage, we develop knowledge-based statistical potentials based on 3D graph representations of non-redundant solved RNAs and statistical analysis of their 3D geometrical features, from parts to whol...

Contact path geometries are curved geometric structures on a contact man-ifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of the family of solutions of a system of an odd number of second order ODE's s...

In this paper, knots are considered as smooth embeddings of a circle into ℝ3 defined by their flat diagrams. We propose new method coding T-graphs describing the torsion structure on diagram. For coding, we introduce conceptions cycle and block describe transformations under first third Reidemeister moves applied to diagram knot.

We investigate the possible character values of torsion units of thenormalized unit group of the integral group ring of Mathieu sporadic groupM22. We confirm the Kimmerle conjecture on prime graphs for this group andspecify the partial augmentations for possible counterexamples to the strongerZassenhaus conjecture.

We investigate the possible character values of torsion units of the normalized unit group of the integral group ring of Mathieu sporadic groupM22. We confirm the Kimmerle conjecture on prime graphs for this group andspecify the partial augmentations for possible counterexamples to the stronger Zassenhaus conjecture.