In [21] we have counted indefinite metrics (two-dimensional, integrally defined, over Gauss numbers) with a fixed norm (discriminant). We would like to call them also indefinite class numbers. In this article we change from Gauss to Eisenstein numbers. We have to work on the complex two-dimensional unit ball, an Eisenstein lattice on it and the quotient surface. It turns out that the compactified quotient is the complex plane $$. In the first part we present a new proof of this fact. In the second part we construct explicitly a Heegner series with the help of Legendre-symbol coefficients. They can be interpreted as “indefinite class numbers” we look for. Geometrically they appear also as number of plane curves with (normed) Eisenstein disc uniformization.

We discover the appearance of interesting Cassinian ovals in the motion of a two-dimensional harmonic oscillator. The trajectories of the oscillating points are ellipses depending on a parameter. The locus of the foci of these ellipses is a Cassini oval. The form of this oval depends on the magnitude of the initial velocity.

In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate. The time coordinate can be presented according to practical evolving process and keep synchronous with the evolution of the realistic world. In this coordinate system, it is convenient to express the physical laws and to calculate physical variables with clear geometrical meaning. We call it “natural coordinate system”. The constructing method for the natural coordinate system is concretely provided, and its physical and geometrical meanings are discussed in detail. In natural coordinate system, we make classical approximation of spinor equation to get Newtonian mechanics, and then make weak field approximation of Einstein's equation and low speed approximation of particles moving in the space-time. From the analysis and examples we find it is helpful to understand the nature of space-time.

We study deformations of symplectic structures on a smooth manifold $$ via the quasi-Poisson theory. We can deform a given symplectic structure $$ with a Hamiltonian $$-action to a new symplectic structure $$ parametrized by some element $$ in $$. We can obtain concrete examples for the deformations of symplectic structures on the complex projective space and the complex Grassmannian. Moreover applying the deformation method to any symplectic toric manifold, we show that manifolds before and after deformations are isomorphic as a symplectic toric manifold.

Given a generic anticanonical hypersurface $$ of a toric variety determined by a reflexive polytope, we define a line bundle $$ on $$ that generates a spanning class in the bounded derivative category $$. From this fact, we deduce properties of some spaces of strings related with the brane $$. We prove a vanishing theorem for the vertex operators associated to strings stretching from branes of the form $$ to nonzero objects in $$. We also define a gauge field on $$ which minimizes the corresponding Yang-Mills functional.