A theorem,like f⊘lner’s theorem on amenability,concerning residual properties of free groups

Let PL(F _q) denote the projective line over a Galois field F _q. Consider PSL (2, Z ) as a free product of two cyclic groups <x> and <y> of orders 2 and 3. We have shown that any homomorphism from PSL(2,Z) into PGL(2,q) can be extended to a homomorphism from PGL(2Z) into PGL(2q) except in the case where the order of the image of xyis 6 but the images of xand ydo not commute in PGL(2q). It has been shown also that every element in PGL(2,q), not of order 1,2 , or 6, is the image of xyunder some non-degenerate homomorphism. We have parametrized the conjugacy classes of non-degenerate homomorphisms α with the non-trivial elements of F _q. Due to this parametrization we have developed a useful mechanism by which one can construct.

a unique coset diagram (attributed to G. Higman) for each conjugacy class, depicting the action of PGL(2Z) on PL( F _q).     

Cancellation Problems and Dimension Theory

In this note, we study the global dimension of coalgebras and discuss the class of coalgebras of global dimension less or equal to 1. The coalgebras in this class, which contains all the cosemisimple coalgebras, are called hereditary coalgebras. If C is a finite dimensional coalgebra, then C is hereditary if and only if C (the convolution algebra of C) is a hereditary algebra. Any direct sum of hereditary coalgebras is hereditary too. This gives us many examples of infinite dimensional hereditary coalgebras. A coalgebra is left hereditary if and only if it is right hereditary. Moreover, there do not exist hereditary Hopf algebras of finite dimension which are not cosemisimple.     

Highly Robust but Surface‐Active: An N‐Heterocyclic Carbene‐Stabilized Au25 Nanocluster

Surface organic ligands play a critical role in stabilizing atomically precise metal nanoclusters in solutions. However, it is still challenging to prepare highly robust ligated metal nanoclusters that are surface‐active for liquid‐phase catalysis without any pre‐treatment. Now, an N‐heterocyclic carbene‐stabilized Au₂₅ nanocluster with high thermal and air stabilities is presented as a homogenous catalyst for cycloisomerization of alkynyl amines to indoles. The nanocluster, characterized as [Au₂₅(ⁱPr₂‐bimy)₁₀Br₇]²⁺ (ⁱPr₂‐bimy=1,3‐diisopropylbenzimidazolin‐2‐ylidene) ( 1 ), was synthesized by direct reduction of AuSMe₂Cl and ⁱPr₂‐bimyAuBr with NaBH₄ in one pot. X‐ray crystallization analysis revealed that the cluster comprises two centered Au₁₃ icosahedra sharing a vertex. Cluster 1 is highly stable and can survive in solution at 80 °C for 12 h, which is superior to Au₂₅ nanoclusters passivated with phosphines or thiols. DFT computations reveal the origins of both electronic and thermal stability of 1 and point to the probable catalytic sites. This work provides new insights into the bonding capability of N‐heterocyclic carbene to Au in a cluster, and offers an opportunity to probe the catalytic mechanism at the atomic level.     

1,3,4-(口恶)二唑硫醚酰胺为核心骨架的新型多杂环分子的合成及对Cdc25B与PTP1B的抑制活性

首先利用含有三嗪的芳香酰肼(3)构筑了1,3,4-噁二唑衍生物(5), 然后将化合物5与含有1,3,4-噻二唑的衍生物(6)拼合合成了18个目标分子. 利用红外光谱(IR)、 核磁共振波谱(NMR)和高分辨质谱(HRMS)等技术对其结构进行了表征. 考察了目标分子对细胞分裂周期25磷酸酯酶B(Cdc25B)和蛋白酪氨酸磷酸酯酶1B(PTP1B)的抑制活性. 结果表明, 有8个目标分子的抑制活性优于其阳性对照物, 有望成为潜在的Cdc25B抑制剂; 有12个目标分子的抑制活性优于其对照物, 有望成为潜在的PTP1B抑制剂.     

Coherent hole-burnings induced by a bichromatic laser field

In a Doppler-broadened three-level Λ-type system driven simultaneously by a coupling laser and two equal-amplitude saturating laser fields with a frequency difference 2δ, the absorption spectrum of a weak probe laser exhibits multiple deep spectral holes through coherent hole-burning CHB with controllable numbers, widths, depths, and positions. More significant, changing δ or lasers directions, CHBs can degenerate into narrower and deeper spectral holes where the slope of the refractive index is very steep. The multiple narrow spectral holes in a single absorption profile are expected to have potential applications in high density storage, optical information processes, and slow-light.     

Ground-state properties of the one-dimensional extended Hubbard model at half filling

The density-matrix renormalization group (DMRG) technique is used to study the ground-state properties of the one-dimensional half-filled Hubbard model with on-site (nearest-neighbor) repulsive interaction U (V) and nearest-neighbor hopping t. We calculate the static spin structure factor to consider the spin degrees of freedom. We notice a striking difference of the static spin structure factor among the spin-density-wave, charge-density-wave (CDW), and bond-order-wave (BOW) phases. Based on the results, we identify the BOW-CDW transition at small (large) U value as continuous (of first order). We also calculate the double occupancy to consider the charge degrees of freedom. For large U, the double occupancy show a discontinuous jump at the BOW-CDW critical point and it implies first-order transition. With decreasing U, the jump becomes smaller and vanishes at the tricritical point U_t≈5.961t. This value is close to our previous estimation U_t=5.89t obtained with other quantities. Consequently, the results of static spin structure factor and double occupancy support the accuracy of our ground-state phase diagram.     

Laser-Duration Dependence of Emission Properties of High-Order Harmonic Generation

Quantitative investigations are made for the laser-duration dependence of the emission properties of high-order harmonic generation （HHG）. HHG emission properties produced by few-cycle lasers show some useful characteristics. The cutoff energy is less than that by laser for infinite duration. The single energy distribution pulse decreases much faster than its duration as the laser duration grows. A two-cycle laser with carrier-envelope phase of 0° can produce a single distribution pulse peaked at the laser carrier phase 1.22 rad and spanned 1.18 rad with the cutoff energy 2.9Up ＋ Ip and a bandwidth 0.63Up, where Up is the ponderomotive potential of the laser field and Ip is the atomic ionization potential.     

Property of Image-Object Using TE/TM-polarized Wave in the Near-Field Optics with CCGM-FFT

Discretization of the Lippmann-Schwinger integral equation with complex conjugate gradient method and fast Fourier transform （CCGM-FFT） is solved, which can reduce the memory storage and the CPU time compared with the traditional method, MOM. Thus objects with large size and multiple scattering objects could be simulated with CCGM-FFT. The total intensity and the distribution of each field component of the dielectric and metallic objects under the excitation of the TE//TM-polarized wave are calculated with photon scanning tunnelling microscopy （PSTM） at the constant height. The simulating results are analysed and explained reasonably. The results show that the polarization plays an important role for imaging of PSTM.     

Existence of almost periodic solutions for SICNNs with time-varying delays

In this Letter, the shunting inhibitory cellular neural networks with time-varying delays are considered. We establish some new results about the existence and stability of almost periodic solutions for SICNNs without the global Lipschitz conditions on the activity functions. An example is given to illustrate the effectiveness of our results.     

Polyakov Soldering and Second-Order Frames: The Role of the Cartan Connection

The so-called ‘soldering’ procedure performed by Polyakov (Int J Math Phys A5, 833–842, 1990) for a -gauge theory is geometrically explained in terms of a Cartan connection on second-order frames of the projective space P¹. The relationship between a Cartan connection and the usual (Ehresmann) connection on a principal bundle allows to gain an appropriate insight into the derivation of the genuine ‘diffeomorphisms out of gauge transformations’ given by Polyakov himself. Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix-Marseille I, Aix-Marseille II et de l’Université du Sud Toulon-Var. Unité affiliée à la FRUMAM Fédération de Recherche 2291.