Generalized 3G theorem and application to relativistic stable process on non-smooth open sets

Let G(x,y) and GD(x,y) be the Green functions of rotationally invariant symmetric α-stable process in Rd and in an open set D, respectively, where 0<α<2. The inequality GD(x,y)GD(y,z)/GD(x,z)?c(G(x,y)+G(y,z)) is a very useful tool in studying (local) Schrödinger operators. When the above inequality is true with c=c(D)∈(0,∞), then we say that the 3G theorem holds in D. In this paper, we establish a generalized version of 3G theorem when D is a bounded κ-fat open set, which includes a bounded John domain. The 3G we consider is of the form GD(x,y)GD(z,w)/GD(x,w), where y may be different from z. When y=z, we recover the usual 3G. The 3G form GD(x,y)GD(z,w)/GD(x,w) appears in non-local Schrödinger operator theory. Using our generalized 3G theorem, we give a concrete class of functions belonging to the non-local Kato class, introduced by Chen and Song, on κ-fat open sets. As an application, we discuss relativistic α-stable processes (relativistic Hamiltonian when α=1) in κ-fat open sets. We identify the Martin boundary and the minimal Martin boundary with the Euclidean boundary for relativistic α-stable processes in κ-fat open sets. Furthermore, we show that relative Fatou type theorem is true for relativistic stable processes in κ-fat open sets. The main results of this paper hold for a large class of symmetric Markov processes, as are illustrated in the last section of this paper. We also discuss the generalized 3G theorem for a large class of symmetric stable Lévy processes.     

The two-dimensional quantum Euclidean algebra

The algebra dual to Woronowicz's deformation of the two-dimensional Euclidean group is constructed. The same algebra is obtained from SU _q (2) via contraction on both the group and algebra levels.     

Constructions of rank five geometries for the Mathieu group M 22

We construct nine rank five incidence geometries that are firm and residually connected and on which the Mathieu group M²² acts flag-transitively. The constructions use mainly objects arising from the Steiner systemS(3, 6, 22). One of these geometries was constructed by Meixner and Pasini in [10]. Three of them are obtained from the geometry of Meixner and Pasini using doubling (see [8] or [12]) or similar constructions. The remaining five are new and four of them have a star diagram. These latter four geometries are constructed using special partitions of the 22 points of the Steiner system S(3, 6, 22).     

Direct, organocatalytic α-sulfenylation of aldehydes and ketones

A method for direct sulfenylation of aldehydes and ketones, catalyzed by a novel pyrrolidine trifluoromethanesulfonamide organocatalyst, has been developed. This process serves as an efficient and mild approach to the preparation of α-phenylthio-ketones and -aldehydes.     

On two mean value properties and functional equations associated with them

Summary Motivated by different mean value properties, the functional equationsf(x) – f(y)/x–y=[(x, y)], (i)xf(y) – yf(x)/x–y=[(x, y)] (ii) (x y) are completely solved when, are arithmetic, geometric or harmonic means andx, y elements of proper real intervals. In view of a duality between (i) and (ii), three of the results are consequences of other three.The equation (ii) is also solved when is a (strictly monotonic) quasiarithmetic mean while the real interval contains 0 and when is the arithmetic mean while the domain is a field of characteristic different from 2 and 3. (A result similar to the latter has been proved previously for (i).)     

Continued fractions associated with the Newton-Padé table

Summary We discuss first the block structure of the Newton-Padé table (or, rational interpolation table) corresponding to the double sequence of rational interpolants for the data{(z _k, h(z_k)} _k =0. (The (m, n)-entry of this table is the rational function of type (m,n) solving the linearized rational interpolation problem on the firstm+n+1 data.) We then construct continued fractions that are associated with either a diagonal or two adjacent diagonals of this Newton-Padé table in such a way that the convergents of the continued fractions are equal to the distinct entries on this diagonal or this pair of diagonals, respectively. The resulting continued fractions are generalizations of Thiele fractions and of Magnus'sP-fractions. A discussion of an some new results on related algorithms of Werner and Graves-Morris and Hopkins are also given.Dedicated to the memory of Helmut Werner (1931–1985)     

24-亚甲基胆甾-5-烯-3β,19-二醇的合成

从豆甾醇出发,通过10步反应,合成得到具有显著细胞毒活性的24-亚甲基胆甾-5-烯-3β,19-二醇(1),总产率为16%.目标产物1的熔点、[α]值和波谱数据与天然产物一致.化合物1的合成是首次报道,它对人体鼻咽癌(CN2)和胃癌(Mgc803)细胞有显著抑制活性.     

冠醚型电极对局部麻醉药响应功能的研究

用二苯并24-冠-8作为活性物质、邻苯二甲酸二(2-乙基)己酯为增塑剂制得PVC膜电极。电极对局麻药丁卡因、普鲁卡因和利多卡因具有能斯特响应,用于局麻药针剂的测定取得满意的结果。     

New catalytic effect of thiourea on the oxidative cyanation of N-aryltetrahydroisoquinolines

Thiourea itself has been introduced as a mild and efficient organocatalyst for the oxidative α -cyanation of N-aryltetrahydroisoquinolines (THIQs) with trimethylsilyl cyanide (TMSCN), giving the corresponding products in good to excellent yields. Experimental investigations demonstrated that thiourea acts as a radical initiator by abstracting hydroxyl radical (OH) from tert-butyl hydroperoxide (TBHP) directly instead of non-covalent hydrogen bondings (H-bondings) activation. The use of thiourea as a radical initiator offers a new avenue for innovative chemical transformations in organocatalyzed radical chemistry.