On Fundamental Group of a Certain Class of Welded Knots

In this paper, a certain class of welded knots K_(2n) is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K_(2n),n ∈ Z~+, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.     

Free Subgroups of Groups with Nontrivial Floyd Boundary

Abstract

We prove that when a countable group admits a nontrivial Floyd-type boundary, then every nonelementary and metrically proper subgroup contains a noncommutative free subgroup. This generalizes the corresponding well-known results for hyperbolic groups and groups with infinitely many ends. It also shows that no finitely generated amenable group admits a nontrivial boundary of this type. This improves on a theorem by Floyd (Floyd, W. J. (1980). Group completions and limit sets of Kleinian groups. Invent. Math. 57: 205–218) as well as giving an elementary proof of a conjecture stated in that same paper. We also show that if the Floyd boundary of a finitely generated group is nontrivial, then it is a boundary in the sense of Furstenberg and the group acts on it as a convergence group.     

A new method for the stereoselective construction of angular methyl group of fuzed cyclic ethers

A convenient method for the stereoselective construction of angular methyl group of fuzed cyclic ethers is described. Reactions of mixed thioacetals with Me₂Zn/Zn(OTf)₂ afforded the corresponding methylated products in good yields. Various protective groups such as MOM ether, benzylidene acetal, TBS ether, and pivaloyl group were stable under the reaction conditions.     

The nonexistence of sensitive commutative group actions on dendrites

In this paper, using the integral method observed by Mai Jiehua recently, we show that no dendrite admits a sensitive commutative group action.     

Asymptotic behavior of intermediate points in the differential mean value theorem of divided differences with repetitions

By the explicit representation for the divided differences with repetitions involving cycle index of symmetric groups, this paper deals with the asymptotic behavior of the intermediate point of the mean value theorem of divided differences with repetitions. Our results generalize some recent interesting results such as the asymptotic expansions of the immediate points of the Taylor formula and the differential mean value theorem of divided differences with distinct points.     

Distributional solutions of Burgers' equation and intrinsic regular graphs in Heisenberg groups

In the present paper we will characterize the continuous distributional solutions of Burgers' equation as those which induce intrinsic regular graphs in the first Heisenberg group H¹≡R³, endowed with a left-invariant metric d_∞ equivalent to its Carnot-Carathéodory metric. We will also extend the characterization to higher Heisenberg groups Hn≡R2n+1.     

What is control of turbulence in crossed fields?

Convective instability in the cross-field system of thermonuclear reactors can be overridden by poloidal drifts. While in crossed fields, a long-time, large-scale turbulent regime, in which the eddies of some particular size are destined to persist longer than usual, would come into being. Perhaps, we may keep such vortexes using them as tools for maintaining the stability of still an illusory construct of plasma fusion.     

Blowing-ups and valuations on surfaces

In this paper, we show that all the nontrivial valuations on surfaces can be given by the infinite sequences of blowing-ups, and give the process of blowing-ups.