We consider the symplectic quotient for a direct product of several integral coadjoint orbits of $SU(3)$ and investigate its symplectic volume. According to a fundamental theorem for symplectic quotients, it is equivalent to studying the dimension of the trivial part in a tensor product of several irreducible representations for $SU(3)$, and its asymptotic behavior. We assume that either all of coadjoint orbits are flag manifolds of $SU(3)$, or all are complex projective planes. As main results, we obtain an explicit formula for the symplectic volume in each case.

This article shows explicit relations between fractional expressions of Schottky-Klein type for hyperelliptic $\sigma$-functions and a product of differences of the algebraic coordinates on each stratum of a natural stratification in a hyperelliptic Jacobian.

This paper analytically describes the local geometry of a generalized $(\kappa,\mu )$-manifold $M(\eta,\xi,\phi,g)$ with $\kappa<1$ which satisfies the condition ``the function $\mu$ is constant along the integral curves of the characteristic vector field $\xi$''. This class of manifolds is especially rich, since it is possible to construct in $R^3$ two families of such manifolds, for any smooth function $\kappa$ ($\kappa<1$) of one variable. Every family is determined by two arbitrary functions of one variable.

We give explicit formulas of the number of rational points and those of the congruence zeta functions for the hyperelliptic curves over a finite field defined by affine equations $y^2=x^6+a$, $y^2=x^{12}+a$ and $y^2=x(x^6+a)$.

In this article, we define $M$-matrices of the ternary Golay code and build fundamental properties of the ternary Golay code on $M$-matrices. Moreover, using four $M$-matrices of the ternary Golay code, we give order three elements, in the Mathieu group $M_{12}$, which generate $M_{11}$ and $M_{12}$.

We obtain some uncertainty inequalities for the Jacobi transform $\hat f_{\alpha,\beta}(\lambda)$, where we suppose $\alpha, \beta\in\mathbb{R}$ and $\rho=\alpha+\beta+1\geq 0$. As in the Euclidean case, analogues of the local and global uncertainty principles hold for $\hat f_{\alpha,\beta}$. In this paper, we shall obtain a new type of an uncertainty inequality and its equality condition: When $\beta\leq 0$ or $\beta\leq\alpha$, the $L^2$-norm of $\hat f_{\alpha,\beta}(\lambda)\lambda$ is estimated below by the $L^2$-norm of $\rho f(x)(\cosh x)^{-1}$. Otherwise, a similar inequality holds. Especially, when $\beta>\alpha+1$, the discrete part of $f$ appears in the Parseval formula and it influences the inequality. We also apply these uncertainty principles to the spherical Fourier transform on $SU(1,1)$. Then the corresponding uncertainty principle depends, not uniformly on the $K$-types of $f$.

We show that the moduli space $\mathcal{M}_{g,1}^N$ of pointed algebraic curves of genus $g$ with a given numerical semigroup $N$ is an irreducible rational variety if $N$ is generated by less than five elements for low genus ($ g \leq 6$) except one case. As a corollary to this result, we get a computational proof of the rationality of the moduli space $\mathcal{M}_{g,1}$ of pointed algebraic curves of genus $g$ for $1 \leq g \leq 3$. If $g \leq 5$, we also have that $\mathcal{M}_{g,1}^N$ is an irreducible rational variety for any semigroup $N$ except two cases. It is known that such a moduli space $\mathcal{M}_{g,1}^N$ is non-empty for $g \leq 7$.

We prove a theorem about elliptic operators with symmetric potential functions, defined on a function space over a closed loop. The result is similar to a known result for a function space on an interval with Dirichlet boundary conditions. These theorems provide accurate numerical methods for finding the spectra of those operators over either type of function space. As an application, we numerically compute the Morse index of constant mean curvature tori of revolution in the unit $3$-sphere $\mathbb{S}^3$, confirming that every such torus has Morse index at least five, and showing that other known lower bounds for this Morse index are close to optimal.

One of Silverman's results gives a relationship between the number of integral points and the rank of elliptic curves over $\mathbb{Q}$. This paper generalizes this result for all imaginary quadratic fields.

We introduce notions of Morita equivalence for both twisted symplectic groupoids and integrable twisted Poisson manifolds without terms of groupoids. We show that two integrable twisted Poisson manifolds are Morita equivalent if and only if their associated groupoids are Morita equivalent as twisted symplectic groupoids.

The paper deals with the following nonlinear partial differential equation $(t \partial/\partial t)^m u = F (t,x, \{(t \partial/\partial t)^j (\partial/\partial x)^{\alpha} u\}_{j+\alpha \leq m, j<m} )$ with $(t,x) \in \mathbb{C}^2$ in the complex domain. Under the assumption that the equation is of totally characteristic type, the uniqueness of the solution was first proved in [4]. The present paper gives a sharp form of this uniqueness theorem.

We discuss the interaction between the geometry of a quaternion-K\"{a}hler manifold $M$ and that of the Grassmannian $\mathbb{G}_3(\mathfrak{g})$ of oriented $3$-dimensional subspaces of a compact Lie algebra $\mathfrak{g}$. This interplay is described mainly through the moment mapping induced by the action of a group $G$ of quaternionic isometries on $M$. We give an alternative expression for the imaginary quaternionic endomorphisms $I,J,K$ in terms of the structure of the Grassmannian's tangent space. This relies on a correspondence between the solutions of respective twistor-type equations on $M$ and $\mathbb{G}_3(\mathfrak{g})$.

A duality of weight systems which corresponds to Batyrev's toric mirror symmetry is given. It is shown that Arnold's strange duality for exceptional unimodal singularities reduces to this duality.

It is known that any genus one hyperbolic knot in the 3-dimensional sphere admits at most seven mutually disjoint and mutually non-parallel genus one Seifert surfaces. In this note, it is shown that for any integers $g>1$ and $n>0$, there is a hyperbolic knot of genus $g$ in the 3-dimensional sphere which bounds $n$ mutually disjoint and mutually non-parallel genus $g$ Seifert surfaces.