We show the existence of a symplectic structure on the moduli space of he Seiberg-Witten equations on $\Sigma\times\Sigma$ where $\Sigma$ is a compact Riemann surface. To prequantize the moduli space, we construct a Quillen-type determinant line bundle on it and show its curvature is proportional to the symplectic form.

By means of hypercomplex numbers, in this paper we discuss algebraic equations and obtain some interesting relations. A structure equation$A^2=nA$ of a group is derived. The matrix representation of a group constitutes the basis elements of a hypercomplex number system. By a canonical real matrix representation of a cyclic group, we define the cyclic number system, which is exactly the solution space of the higher order algebraic equations, and thus can be used to solve the roots of algebraic equations. Hypercomplex numbers are linear algebras with definition of multiplication and division, satisfying the associativity and distributive law, which provide a unified, standard, and elegant language for many complex mathematical and physical objects. So, we have one more proof that the hypercomplex numbers are worthy of application in teaching and scientific research.

In this short note we present a new proof of Ekeland's variational principle and Caristi's fixed point theorem using a recently proved constrained variational principle in completely regular topological spaces.

Dynamical orbits of the harmonic oscillator potential in the plane are ellipses which depend on a real parameter. Some time ago in this journal it has been proven by purely geometrical methods that the locus of the focuses of these ellipses are Cassinian ovals. Here we present several explicit analytic parameterizations of these remarkable curves. Nominally, their forms depend on the magnitude of the initial distance from the center of attraction and the magnitude of the initial velocity. We have found a few parameterizations in which the roles of the size and shapes can be clearly distinguished.

A construction methods of noncommutative locally symmetric K{\"a}hler manifolds via a deformation quantization with separation of variables was proposed by Sako-Suzuki-Umetsu and Hara-Sako. This construction gives the recurrence relations to determine the star product. These recurrence relations were solved for the case of the arbitrary one-dimensional ones, $N$-dimensional complex space, complex projective space and complex hyperbolic space. For any two-dimensional case, authors found the solution of the recurrence relations. In this paper, we review the solution and make the star product for two-dimensional complex projective space as a concrete example of this solution.

In this paper we survey Delaunay surfaces in $\mathbb{R}^{3}$ spanning two coaxial circles which appear as capillary surfaces supported on different solid supports in the absence of gravity. We classify these surfaces based on contact angles and the geometry of the support. Numerical solutions of the Euler Lagrange equation are provided using numerical methods.