The Second Pluri-Genus of Surface Singularities

This paper studies the second pluri-genus of surface singularities. We give a formula for this invariant of a Gorenstein singularity, and several inequalities relating the invariant with the Milnor number, Tjurina number and the modality of a hypersurface singularity.     

The slice genus and the Thurston-Bennequin invariant of a knot

For any knot , .

     

Nepetanudone, a dimer of the α-pyrone 5,9-dehydronepetalactone

The title compound C₂₀H₂₄O₄ crystallizes in the orthorhombic space group P2₁2₁2, with a = 10.245(3), b = 17.103(3), c = 9.879(3) Å, D _c = 1.260 g cm ^–3, and Z = 4. Nepetanudone is postulated to be formed by a [4+4] cycloaddition of two 5,9-dehydronepetalactone moieties to give an asymmetric dimer in which each pair of atoms is magnetically nonequivalent. The compound was recovered from extracts of Nepeta nuda ssp. albiflora.     

Diels-Alder Reaction of Hedychenone and Maleic Anhydride

SeveraIditerPenoidswithantitum0ractivity,includinghedychenonel,afiJranoidcompound,werepreviouslyis0latedfr0mHetychiumgenus'.Inattemptt00btains0mediterPen0idderivativeswithhigherantitum0ractivity,Diels-Alderreactionoflwithmaleicacidanhydride2wasintr0ducedt0getsomeanalogues0fcantharidin3,anantitUm0rmedicineinclinicaluse.Reaction0flandexcessive2at0Cinchloroformgavetw0thermodynamicaIlydominatedexoadducts4and5,togetherwithamixtureoftWokinetical1ydominatedendoadducts6and7.Whenthereacti0nwasc0ndu…     

Quadratic perturbations of a quadratic reversible center of genus one

In this paper, we study a reversible and non-Hamitonian system with a period annulus bounded by a hemicycle in the Poincaré disk. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. This verifies some results of the conjecture given by Gautier et al.     

Classification of torsion-free genus zero congruence groups

We study and classify all torsion-free genus zero congruence subgroups of the modular group.

     

A note on the maximum genus of 3-edge-connected nonsimple graphs

Let G be a 3-edge-connected graph (possibly with multiple edges or loops), and let γM(G) and β(G) be the maximum genus and the Betti number of G, respectively. Then γM(G)≥β(G)/3 can be proved and this answers a question posed by Chen, et al. in 1996.     

3,4跳图的平面性

For a graph G of size ε≥1 and its edge-induced subgraphs H1 and H2 of size r(1≤r≤ε), H1 is said to be obtained from H2 by an edge jump if there exist four distinct vertices u,v,w and x in G such that (u,v)∈E(H2), (w,x)∈ E(G)-E(H2) and H1=H2-(u,v)+(w,x). In this article, the r-jump graphs (r≥3) are discussed. A graph H is said to be an r-jump graph of G if its vertices correspond to the edge induced graph of size r in G and two vertices are adjacent if and only if one of the two corresponding subgraphs can be obtained from the other by an edge jump. For k≥2, the k-th iterated r-jump graph Jrk(G) is defined as Jr(Jrk-1(G)), where Jr1(G)=Jr(G).An infinite sequence{Gi} of graphs is planar if every graph Gi is planar. It is shown that there does not exist a graph G for which the sequence {J3k(G)} is planar, where k is any positive integer. Meanwhile,lim gen(J3k(G))=∞,where gen(G) denotes the genus of a graph G, if the sequencek→∞J3k(G) is defined for every positive integer k. As for the 4-jump gra