We compare two different models of noncommutative geometry of the cyclotomic tower, both based on an arithmetic algebra of functions of roots of unity and an action by endomorphisms, the first based on the Bost-Connes (BC) quantum statistical mechanical system and the second on the Habiro ring, where the Habiro functions have, in addition to evaluations at roots of unity, also full Taylor expan...

Given a sphere of radius r > 1 in the n-dimensional Euclidean space, we study the coverings of this sphere with unit spheres. Our goal is to design a covering of the lowest covering density, which defines the average number of unit spheres covering a point in a bigger sphere. For growing n, we obtain the covering density of (n lnn)/2. This new upper bound is half the order n lnn established in ...

We are now embarrassingly rich in knot and 3-manifold invariants. We have to organize these invariants systematically and find out ways to make use of them. The theory of finite type knot invariants, or Vassiliev invariants, has been very successful in accomplishing the first task. Recently, an analogous theory of finite type invariants of integral homology 3-spheres started to emerge. The anal...

Carbon Spheres were fabricated by hydrothermal method and their structural properties were investigated. Carbon Spheres with average sizes around of 230, 320 and 430 nm were synthesized in different concentration of glucose aqueous solution and different hydrothermal reaction time. The temperature of 180 °C by a hydrothermal reaction was fixed in all of glucose concentration and hydrothermal re...

For every rational homology 3–sphere with H1(M, Z) = (Z/2Z) n we construct a unified invariant (which takes values in a certain cyclotomic completion of a polynomial ring), such that the evaluation of this invariant at any odd root of unity provides the SO(3) Witten–Reshetikhin–Turaev invariant at this root and at any even root of unity the SU(2) quantum invariant. Moreover, this unified invari...