We study the nonrelativistic limit of Dirac operators from the viewpoint of the spectral relationship between Dirac operators and Pauli operators. We show that Dirac operators have spectral concentration about eigenvalues of Pauli operators for a large class of magnetic fields and electric potentials diverging at infinity.

We prove a simple optimal relationship between Riemannian submersions and minimal immersions; namely, if a Riemannian manifold admits a non-trivial Riemannian submersion with totally geodesic fibers, then it cannot be isometrically immersed in any Riemannian manifold of non-positive sectional curvature as a minimal manifold. Some related results are also presented. In the last section, we introduce a cohomology class for Riemannian submersions and provide an application.

In 1999, Iwan Duursma defined the zeta function for a linear code as a generating function of its Hamming weight enumerator. It has various properties similar to those of the zeta function of an algebraic curve. This article extends Duursma's theory to the case of formal weight enumerators. It is shown that the zeta function for a formal weight enumerator has a similar structure to that of the weight enumerator of a Type II code. The notion of the extremal formal weight enumerators is introduced and an analogue of the Mallows-Sloane bound is obtained. Moreover the ternary case is considered.

We describe all of smooth complete toric threefolds of Picard number 5 with no nontrivial nef line bundles, and show that no such examples exist with Picard number less than 5.

We show basic properties of zeta functions over the one element field starting from an algebraic set over the integer ring. We calculate several examples and we investigate special values via the associated $K$-group identified as the stable homotopy group of spheres.

We study values of absolute tensor products (multiple zeta functions) at integral arguments. We obtain a simple formula for the absolute value of the double sine function. We express values of the multiple gamma function related to the functional equation.