In this paper, we introduce the notion of the simplest cubic fields, and the associated elliptic curves, hyperelliptic curves. Next, we compute ranks of the elliptic curves by using hyperelliptic curves.

In this paper we determine the galois group $\operatorname{Gal}(F_1/\mathbf{Q})$ where $F_1$ is the compositum of first layers of all $\mathbf{Z}_2$-extensions over an imaginary quadratic field. Moreover, we construct $F_1$ explicitly when $k$ has class number one.

A duality is introduced between a pair of polynomial twisted de Rham cohomology groups associated with a generalized Airy function in several variables. Natural bases of the twisted de Rham groups are constructed in terms of Schur polynomials. Then the intersection matrix relative to these bases is calculated explicitly in terms of skew-Schur polynomials.

In this paper, it is proved that, over certain real quadratic fields, there are no elliptic curves having everywhere good reduction and cubic discriminant.

Following Wada's construction [4] on the twisted Alexander polynomial, we introduce a new twisted invariant for finitely presentable groups.

Eguchi-Oki-Matsumura (EOM) equations are introduced to model theoretically the kinetics of ordering which accompanies phase separation in some binary alloys. Numerical analysis shows that EOM equations admit several steady states. Employing the variational structure associated with the free energy, we prove that there really exist non-trivial steady solutions for EOM equations.

We show that there is no blow-up solutions, for positive viscosity constant $\nu$, to the equation $f_{xxt} - \nu f_{xxxx} + f f_{xxx} - f_xf_{xx} =0$, $x \in (0,1)$, $t > 0$ with (i) periodic boundary condition, or (ii) Dirichlet boundary condition $f = f_x = 0$ or (iii) Neumann boundary condition $f = f_{xx} = 0$ on the boundary $x = 0, 1$. Furthermore we show that every solution decays to the trivial steady state as $t$ goes to infinity.

We present four variants of Fan's type inequality for vector-valued multifunctions in topological vector spaces with respect to a cone preorder in the target space, when the functions and the cone possess various kinds of semicontinuity and convexity properties. Using the classical scalar Fan inequality, we prove directly a two-function result of Simons, which is used to establish our main tool for proving the presented results.