Here we study the deformations of associative submanifolds inside a G2 manifold M 7 with a calibration 3-form φ. A choice of 2-plane field Λ on M (which always exists) splits the tangent bundle of M as a direct sum of a 3-dimensional associate bundle and a complex 4-plane bundle TM = E ⊕ V, and this helps us to relate the deformations to SeibergWitten type equations. Here all the surveyed resul...

We determine, in every finite characteristic p, those hypergeometric sheaves of type (7,m) with 7 ≥ m whose geometric monodromy group Ggeom lies in G2, cf. Theorem 3.1 and Theorem 6.1. For each of these we determine Ggeom exactly, cf. Theorem 9.1. Each of the five primitive irreducible finite subgroups of G2, namely L2(8), U3(3), U3(3).2 = G2(2), L2(7).2 = PGL2(7), L2(13) turns out to occur as ...

compositions of biotite from three different rock types of mashhad granitoids, i.e., granodiorite, monzogranite and leucogranite in ne of iran have been documented by electron microprobe and wet chemistry for fe3+ and fe2+. mashhad granitoids have been geochronologically and petrologically grouped into g1 and g2 phases. microprobe data show that the total fe contents in biotite from g2 leucogra...

Let V be the 7-dimensional irreducible representations of G2. We decompose the tensor power V ⊗n into irreducible representations of G2 and obtain all irreducible representations of G2 in the decomposition. This generalizes Weyl’s work on the construction of irreducible representations and decomposition of tensor products for classical groups to the exceptional group G2.

DNA replication in G2 does not normally occur due to the checkpoint control. To elucidate its mechanism, the functions of the escargot and Dmcdc2 genes of Drosophila were studied. When escargot function was eliminated, diploid imaginal cells that were arrested in G2 lost Cyclin A, a regulatory subunit of G2/M cdk, and entered an endocycle. escargot genetically interacted with Dmcdc2 which encod...

A set of vertices W resolves a graph G if every vertex of G is uniquely determined by its vector of distances to the vertices in W . The metric dimension for G, denoted by dim(G), is the minimum cardinality of a resolving set of G. In order to study the metric dimension for the hierarchical product G2 2 uG1 1 of two rooted graphs G2 2 and G u1 1 , we first introduce a new parameter, the rooted ...

In 1978, Bollobás and Eldridge [5] made the following two conjectures. (C1) There exists an absolute constant c > 0 such that, if k is a positive integer and G1 and G2 are graphs of order n such that ∆(G1),∆(G2) n− k and e(G1), e(G2) ckn, then the graphs G1 and G2 pack. (C2) For all 0 < α < 1/2 and 0 < c < √ 1/8, there exists an n0 = n0(α, c) such that, if G1 and G2 are graphs of order n > n0 s...