Consider the group SL(2;Z) acting on the circle consisting of rays from the origin in R². The action of parabolic elements or transvections X $\in$ SL(2;Z) (Tr X = 2) have 2 fixed points on the circle. A half transvection is the restriction of the action of a parabolic element to one of the invariant arcs extended by the identity on the other arc. We show that the group G generated by half transvections is isomorphic to the Higman-Thompson group T, which is a finitely presented infinite simple group. A finite presentation of the group T has been known, however, we explain the geometric way to obtain a finite presentation of the group T by the Bass-Serre-Haefliger theory. We also give a finite presentation of the group T by the generators which are half transvections.

In this paper, we prove appropriate L^p bounds for a class of maximal operators $\mathscr{S}_\Omega$ related to singular integrals with kernels which belong to block spaces and are supported by subvarieties. Also, we show that our condition on the kernel is optimal for the L² boundedness of $\mathscr{S}_\Omega$. Our results improve substantially the main result obtained by L. K. Chen and H. Lin in [CL].

We classify nondegenerate holomorphic maps of a two-dimensional Stein manifold M to C² by study about the value distribution of them.

It is known that a complete linear system on a projective variety in a projective space is generated from the linear system of the projective space by restriction if its degree is sufficiently large. We obtain a bound of degree of linear systems on weighted projective spaces when they are generated from those of the projective spaces. In particular, we show that a weighted projective 3-space embedded by a complete linear system is projectively normal. We treat more generally Q-factorial toric varieties with the Picard number one, and obtain the same bounds for them as those of weighted projective spaces.

Let M^m = Ω be a bounded domain of R^m. Jost and Xu in [7] introduced the subelliptic harmonic map from it and proved the regularity up to the boundary of subelliptic harmonic maps into small balls supposing that the boundary of the domain is noncharacteristic and smooth. In this note, we investigate the boundary regularity of this kind of maps from Ω with non-smooth and partial noncharacteristic boundary.

By using normalized multiple sine functions we show expressions for special values of zeta functions and L-functions containing ζ(3), ζ(5), etc. Our result reveals the importance of division values of normalized multiple sine functions. Properties of multiple Hurwitz zeta functions are crucial for the proof.

In this paper, we find all the possible forms of two nonconstant entire functions f and g that share two values counting multiplicities. As applications, we generalize some known results and confirm a conjecture proposed by Osgood-Yang.

Let φ: X → W be an elementary divisorial Fano-Mori contraction from a smooth projective variety, defined by a linear system |m(K_X + τL)|, with L a φ-ample line bundle in Pic(X), τ a positive integer and m » 0.

General fibers of such contractions are known to be irreducible if τ ≥ dim X − 3 (and so if dim X ≤ 4). We prove that, if τ ≥ dim X − 4, except possibly for one case, a general non trivial fiber is irreducible.

The special case, which can occur when dim X = 5, is effective, as we show by an example in the last section of the paper.

Let M be a compact hypersurface in the Euclidean space with cohomogeneity 2. We show that if the scalar curvature is constant, then M is a standard sphere. This gives a partial positive answer to a problem presented by Yau.