We refine a formula on values of quadruple sine functions at division points. As applications we prove a formula on a sum of reciprocal trigonometric values, and obtain multiple modularity of a three variable modular function, which concerns a generalization of the Dedekind η function.

Let $(\overline{M},\overline{g})$ be a semi-Riemannain manifold and (M, g, S (TM), S (TM^⊥)) be its lightlike submanifold. We show that if $\overline{M}$ satisfies the axiom of r-planes & spheres then it is a real space form.

We consider Mⁿ, n ≥ 3, a complete, connected submanifold of a space form $\tilde{M}^{n+p}(\tilde{c})$, whose non vanishing mean curvature vector H is parallel in the normal bundle. Assuming the second fundamental form h of M satisfies the inequality <h>² ≤ n² |H|²/(n - 1), we show that for $\tilde{c}$ ≥ 0 the codimension reduces to 1. When M is a submanifold of the unit sphere, then Mⁿ is totally umbilic. For the case $\tilde{c}$ < 0, one imposes an additional condition that is trivially satisfied when $\tilde{c}$ ≥ 0. When M is compact and has non-negative Ricci curvature then it is a geodesic hypersphere in the hyperbolic space. An alternative additional condition, when $\tilde{c}$ < 0, reduces the codimension to 3.

It is well known that the Weierstrass product representation of the Barnes multiple gamma function Γ_r(z) can be calculated concretely. However, there has been no study on its explicit formulation. In this paper, its simple formulation is achieved. It is applicable to the Weierstrass product representation of the Vignéras multiple gamma function also. Moreover, the Weierstrass product representation of the Kurokawa multiple sine function S_r(z) is also formulated explicitly.

We study pseudo-Jacobi operators associated to algebraic curvature maps on lightlike hypersurfaces M and investigate conditions for an induced Riemann curvature tensor to be an algebraic curvature map on M. Two examples are provided with explicit determination of their pseudo-Jacobi operators. Finally, we introduce the notion of lightlike Osserman hypersurfaces and prove some characterization results.

We form, what we call, an afforested surface R over a plantation P by foresting with trees T_n (n $\in$ N: the set of positive integers). If all of P and T_n (n $\in$ N) belong to the class ${\mathscr O}_s$ of hyperbolic Riemann surfaces W carrying no singular harmonic functions on W, then we will show that, under a certain diminishing condition on roots of trees T_n (n $\in$ N), the afforested surface R also belongs to ${\mathscr O}_s$.

The Seiberg-Witten-Floer homology of an oriented closed 3-manifold M with the geometric structure R × H² is computed.

We discuss possible relations between two meromorphic functions f and g when they share some pairs of small functions. By utilizing the generalized Nevanlinna's second main theorem for small functions obtained recently, we have been able to show that two meromorphic functions f and g must be linked by a quasi-Möbius transformation if they share three pairs of small functions CM^* and share another pair of small function IM^*. Moreover, we also improves a known result due to T. Czubiak and G. Gundersen on two meromorphic functions sharing five pairs of values and the results on the unicity of meromorphic functions that share five small functions obtained by Li Bao-Qin and Li Yu-Hua as well.

In this paper, we prove some results for implicit fixed point iterations associated with pseudocontractive mappings.

We study the central value of the triple sine function for a general period. We give an explicit integral expression and an inequality. As an application we obtain an expression for ζ(3).

Let F_q be a finite field of q elements of characteristic p. N. M. Katz and Z. Zheng have shown the uniformity of distribution of the arguments arg G (a, χ) of all (q - 1)(q - 2) nontrivial Gauss sums

$$G(a, \chi) = \sum_{x \in {\mathbf F}_q} \chi(x) \exp(2 \pi i \mathrm{Tr}(ax)/p),$$

where χ is a non-principal multiplicative character of the multiplicative group F_q^* and Tr(z) is the trace of z $\in$ F_q into F_p.

Here we obtain a similar result for the set of arguments arg G(a, χ) when a and χ run through arbitrary (but sufficiently large) subsets ${\mathscr A}$ and ${\mathscr X}$ of F_q^* and the set of all multiplicative characters of F_q^*, respectively.