Representations of Free Groups on Rational Line Q

A.M.W. Glass and S.H.McCleary have given the 2 transitive representation of the countable free l group F η(1<η≤ω 0 ).In this paper we shall give the highly ordered transitive representation of countable free groups on the rational line Q, which generalizes their results. As applications,we obtain the highly ordered transitive representation for the direct product of countable free groups,and the inverse limit of countable free groups must be an action on the set Q.

Diophantine Equations and the Freeness of Mobius Groups

Let p and q be two fixed non zero integers verifying the condition gcd(p,q) = 1. We check solutions in non zero integers a1,b1,a2,b2 and a3 for the following Diophantine equations: (B1) (B2) . The equations (B1) and (B2) were considered by R.C. Lyndon and J.L. Ullman in [1] and A.F. Beardon in [2] in connection with the freeness of the M?bius group generated by two matrices of namely and where .?They proved that if one of the equations (B1) or (B2) has solutions in non zero integers then the group is not free. We give algorithms to decide if these equations admit solutions. We obtain an arithmetical criteria on p and q for which (B1) admits solutions. We show that for all p and q the equations (B1) and (B2) have only a finite number of solutions.

A Simple Method of Solving First-order Indefinite Equa tion

The current method of solving first order indefinite equatio n is changing the equation to first order indefinite equation gr oup to solve. But according this method, if variables are very many, it will be difficult to solve the equation using the current method. In this paper, it prov ides a simple method by discussing the structure of solution based on the theory of free abelian group. In addition, this method makes it easy to get the genera lized solution of the equation using the computer.

Correlation of Gluten Molecular Conformation with Dough Viscoelastic Properties during Storage

Significant correlations exist for the total sample between the dough viscoelastic properties and quality, but little information about these links is available. This study investigates the relationship between the gluten molecular conformation and the dough viscoelastic properties of the Zhengmai 379 wheat variety with different relative humidity(RH) values during storage.The results showed that protein extractability, free sulfhydryl(SH) groups, and wet gluten contents decreased to a certain degree during storage, while the glutenin quantity significantly varied. Significant negative correlation coefficients were found between the dough viscosity at 50% RH and the β-turn contents at 40% RH(-0.918) or 60% RH(-0.949)(P < 0.01), the dough viscosity at50% RH and the β-turn contents at 50% RH(-0.912)(P < 0.05), and the dough viscosity at 50% RH and the α-helix/β-sheet ratio at 40% RH(-0.875), 50% RH(-0.869), or 60% RH(-0.843)(P < 0.05). Significant correlation coefficients were further observed between the dough viscosity at 50% RH and the β-sheet contents at 60% RH(0.927)(P < 0.01) and between the dough viscosity at50% RH and the β-sheet contents at 40% RH(0.910) or 50% RH(0.908)(P < 0.05). A decrease in the free SH groups of gluten and an increase in the low-molecular weight contents suggested that gliadin was incorporated via SS crosslinking to the glutenin-like protein, which did not dissolve in ethanol.

A one-step procedure for facile, selective synthesis of protected d-glucopyranosides with free 2-C hydroxyl group starting from peracetylated β-D-glucopyranose

An important field in carbohydrate chemistry is to look for regioselective re-actions of saccharides to simplify the chemical synthesis of oligosaccharides. Oneof such efforts nowadays is directed to the convenient preparation of protected D-

Fixed Subgroups are not Compressed in Direct Products of Surface Groups

By constructing counterexamples,the authors show that the fixed subgroups are not compressed in direct products of free and surface groups,and hence negate a conjecture in[Zhang,Q.,Ventura,E.and J.Wu,Fixed subgroups are compressed in surface groups,Internat.J.Algebra Comput.,25,2015,865–887].

A necessary and sufficient condition for a surface sum of two handlebodies to be a handlebody

The main results of the paper are that we give a necessary and sufficient condition for a surface sum of two handlebodies along a connected surface to be a handlebody as follows:(1)The annulus sum H=H1∪AH2 of two handlebodies H1 and H2 is a handlebody if and only if the core curve of A is a longitude for either H1 or H2;(2)Let H=H1∪Sg,b H2 be a surface sum of two handlebodies H1 and H2 along a connected surface S=Sg,b,b 1,ni=g(Hi)2,i=1,2.Suppose that S is incompressible in both H1 and H2.Then H is a handlebody if and only if there exists a basis J={J1,...,Jm}with a partition(J1,J2)of J such that J1 is primitive in H1 and J2 is primitive in H2.

Methodology-driven efficient synthesis of cytotoxic(±)-piperarborenine B

The evolution of synthetic design toward the efficient synthesis of cyclobutane natural product(±)-piperarborenine B is demonstrated.Taking the advantages of good functional group compatibility of contractive synthesis of cyclobutanes from pyrrolidines,stereoselective synthesis of unsymmetric highly functionalized cyclobutanes core of(±)-piperarborenine B was realized in one step.Also,an unprecedented carboxylic acid assisted-diastereoselective Kracho decarboxylation/transmethylation features a new strategy for a non-symmetrical cyclobutane core.The synthesis of(±)-piperarborenine B illustrates the advancement of methodology resulting in the improvement in synthetic efficiency.

Native amine-directed site-selective C(sp^3)-H arylation of primary aliphatic amines with aryl iodides

Direct C(sp3)-H functionalization of N-unprotected aliphatic amines represents one of the most efficient and straightforward strategies for amine synthesis.Despite some recent progress in this field,the NH2-directed y-C(sp3)-H arylation of primary aliphatic amines exceptα-amino esters remained an unmet challenge.In this report,we established a simple and efficient method for site-selective C(sp3)-H arylation of primary aliphatic amines by aryl iodides.In the presence of only 5 mol%Pd(OAc)2,a wide range of aliphatic amines including O-benzyl and O-silyl amino alcohols were arylated at y-orδ-positions by aryl iodides containing a broad scope of functional groups.The synthetic application of this method had also been demonstrated by large-scale synthesis,the synthesis of a fingolimod analogue,and the conjugation with natural D-menthol and fluorescent 1,8-naphthalimide.