SOME SUBGROUPS OF THE EXTENDED HECKE GROUPS H^－（λq）

The authors consider the extended Hecke groups H(γq) generated by T(z) = -1 / z, S(z) = -1(z +γq) and R(z) = 1 / z with A, = 2 cos(π/q) for q≥3 an integer. In this paper, the even subgroup He(γq), the second commutator subgroup H''(γq) and the principal congruence subgroups Hp(λq) of the extended Hecke groups .H(γq) are studied. Also, relations between them are given.

Extended Group of the Subgroup Gr in Linear Group Over the Principal Ideal Ring

Suppose R is a principal ideal ring, R^＊ is a multiplicative group which is composed of all reversible elements in R, and Mn（R）, GL（n,R）, SL（n,R） are denoted by,Mn（R）={A=（aij）n×n｜aij∈R,i,j=1,2…,n},GL（n,R）={g｜g∈Mn（R）,det g∈R^＊},SL（n,R）={g∈GL（n,R）｜det g=1},SL（n,R）≤G≤GL（n,R）（n≥3）,respectively,then basing on these facts, this paper mainly focus on discussing all extended groups of Gr={（O D^A B）∈G｜A∈GL（r,R）,（1≤r〈n）}in G when R is a principal ideal ring.

A Length Function for Weyl Groups of Extended Aitine Root Systems of Type A1

. In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.

Geometric Parameters Corresponding to Representations of the Classical Groups with Integral Infinitesimal Characters

In this paper we calculate the orbits of flag manifolds of the complex classical groups under the action of the sets of fixed points of Cartan involutions, and determine all the geometric parameters corresponding to representations of the classical groups with integral infinitesimal char- acters, which are used to discuss Arthur conjecture and the Langlands classification of the irreducible admissible representations of real classical groups(see[1])

Enhance the performance of polymer solar cells via extension of the flanking end groups of fused ring acceptors

Two new fused ring electron acceptors(FREAs)IDT-IC-T and IDT-IC-B with thienyl or phenyl substituents at the terminal INCN unit are synthesized.Theoretical calculations indicate that the two acceptors dominantly favor an intermolecularπ-πstacking between the flanking terminal groups.The twist angle between the aryl substituent and INCN unit has a significant influence on theπ-πstacking distance of terminal unit.IDT-IC-T with a smaller twist angle has a shorterπ-πstacking distance than that of IDT-IC-B with a larger twist angle.In addition,extending the conjugation also affects the blend film morphology.IDT-IC-T and IDT-IC-B based photoactive films show appropriate nanoscale phase separations;whereas,blend films based on the parent compound IDT-IC show large-size acceptor domains.As expected,PBDB-T:IDT-IC-T blend films show higher and more balanced electron and hole mobilities.Moreover,these two acceptors present a good charge-transport connectivity arising from the extended conjugation and the increased intermolecular overlapping.Ultimately,IDT-IC-T demonstrates the highest electron mobility(1.47×10^(-4)cm^2V^(-1)s^(-1))and the best power conversion efficiency(PCE)of 9.43%.As for IDT-IC,which only shows an electron mobility of 7.33×10^(-5)cm^2V^(-1)s^(-1)and a PCE of 5.82%.These findings provide a facile and effective way to improve the photovoltaic performance.

Crossing-changeable braids from chromatic configuration spaces

Motivated by the work in Li et al.(2019),this paper deals with the theory of the braids from chromatic configuration spaces.These kinds of braids possess the property that some strings of each braid may intersect together and can also be untangled,so they are quite different from the ordinary braids in the sense of Artin(1925).This enriches and extends the theory of ordinary braids.