We obtain some basic results on helical geodesic immersions in semi-Riemannian geometry. For example, it is shown that, for an indefinite semi-Riemannian submanifold, if any space-like geodesics of the submanifold are helices of order d, curvatures λ₁,...,λ_d-1 and signatures ε₁,...,ε_d in the ambient space, then any time-like geodesics of the submanifold have the same order and curvatures, and signatures (-1)¹ε₁,...,(-1)^dε_d.

For the compact submanifold M immersed in the standard Euclidean sphere S^n+p or the Euclidean space R^n+p, we obtain Simons-type inequalities about the first eigenvalue λ₁ and the squared norm of the second fundamental form S respectively. In particular, for the case of the ambient space is S^n+p, we need not the assumption that M is minimal. Following which, we obtain the estimate about the lower bound for S if it is constant respectively.

In this article, we investigate the uniqueness of meromorphic functions dealing with two shared values and a shared set in an angular domain. Results are obtained extending some results given by W. C. Lin and S. Mori.

In this paper, we introduce a new class of lightlike submanifolds, namely, semi-invariant lightlike submanifolds of a semi-Riemannian product manifold. We investigate totally umbilical, curvature invariant lightlike submanifolds in real space forms M₁(c₁) × M₂(c₂) and discuss integrabilities of distributions on semi-Riemannian product manifold.

It is proved that any non-hyperelliptic Riemann surface of genus five all of whose Weierstrass points have maximal weight has only Weierstrass points with gap sequence (1,2,3,5,9), and a defining equation of such a surface is given.

Let D denote the open unit disc and let p $\in$ (0,1). We consider the family Co(p) of functions f : D → $\overline{{\mathbf C}}$ that satisfy the following conditions:

(i) f is meromorphic in D and has a simple pole at the point p.

(ii) f(0) = f′(0) – 1 = 0.

(iii) f maps D conformally onto a set whose complement with respect to $\overline{{\mathbf C}}$ is convex.

We determine the exact domains of variability of some coefficients a_n (f) of the Laurent expansion

$$f(z)=\sum_{n=-1}^{\infty} a_n(f)(z-p)^n, \quad |z – p|<1 – p,$$

for f $\in$ Co(p) and certain values of p. Knowledge on these Laurent coefficients is used to disprove a conjecture of the third author on the closed convex hull of Co(p) for certain values of p.

Equations for the locus of Riemann Surfaces of genus three with a nonabelian automorphism group generated by involutions are determined from vanishings of Riemann's theta function.

In this paper we introduce the notion of good systems of parameters with respect to certain increasing filtration of submodules of a Noetherian module in order to prove several characterizations of sequentially Cohen-Macaulay modules in terms of good systems of parameters. The sequential Cohen-Macaulayness of Stanley-Reisner rings of small embedding dimension are also examined.

The aim of this paper is to prove that a pair of analytic sets X, Y $\subset$ C_z^m × C_wⁿ is locally regularly separated with a uniform exponent α in the fibres taken over a proper projection π(z,w) = z of X ∩ Y (under the assumption that X ∩ Y has pure dimension): for all z $\in$ π (X ∩ Y) ∩ U, dist(w,Y^z) ≥ const.dist(w,(X ∩ Y)^z)^α when w $\in$ X^z ∩ V, where U × V is a neighbourhood of a point a $\in$ X ∩ Y such that π(a) is regular in π(X ∩ Y). As an application of this we obtain a parameter version of the Łojasiewicz inequality for c-holomorphic mappings. Both results are a complex counterpart of the main result of [ŁW] from the subanalytic case, extended in this paper by a bound on the uniform exponent.

We generalize Nevanlinna's five-value theorem to the cases that two meromorphic functions partially sharing either five or more values, or five or more small functions. In each case, we formulate a way to measure how far these two meromorphic functions are from sharing either values or small functions, and use this measurement to get some uniqueness results.