We consider a see-saw pair consisting of a Hermitian symmetric pair $(G_R,K_R)$ and a compact symmetric pair $(M_R,H_R)$ , where $(G_R,H_R)$ and $(K_R,M_R)$ form a real reductive dual pair in a large symplectic group. In this setting, we get Capelli identities which explicitly represent certain $K_C$ -invariant elements in $U({\mathfrak{g}}_C)$ in terms of $H_C$ -invariant elements in $U({\mathfrak{m}}_C)$ . The corresponding $H_C$ -invariant elements are called Capelli elements.

We also give a decomposition of the intersection of $O_{2n}$ -harmonics and $S{p}_{2n}$ -harmonics as a module of $G{L}_n=O_{2n}\cap S{p}_{2n}$ , and construct a basis for the $G{L}_n$ highest weight vectors. This intersection is in the kernel of our Capelli elements.

Takamura constructed a theory on splitting families of degenerations of Riemann surfaces. We call them Takamura splitting families. In a Takamura splitting family, there appear two kinds of singular fibers, called a main fiber and subordinate fibers. In this paper, when the original singular fiber is stellar and the core is a projective line, we determine the number of subordinate fibers and describe the types of singular points, which are nodes.

By using the generalized sigma function of a $C_{rs}$ curve $y^r=f(x)$ , we give a solution to the Jacobi inversion problem over the stratification in the Jacobian given by the Abel image of the symmetric products of the curve. We show that determinants consisting of algebraic functions on the curve, whose zeros give the Abelian pre-image of the strata, are written by ratios of certain derivatives of the sigma function on the strata. We also discuss the order of vanishing of abelian functions on the strata in terms of intersection theory.

Let $R$ be a Noetherian local ring with the maximal ideal $\mathfrak{m}$ and $\dim R=1$ . In this paper, we shall prove that the module ${\text{Ext}}_R^1(R/Q,R)$ does not vanish for every parameter ideal $Q$ in $R$ , if the embedding dimension $\text{v}(R)$ of $R$ is at most $4$ and the ideal ${\text{m}}^2$ kills the $0^{\underset{¯}{th}}$ local cohomology module $H_{\mathfrak{m}}^0(R)$ . The assertion is no longer true unless $\text{v}(R)\le 4$ . Counterexamples are given. We shall also discuss the relation between our counterexamples and a problem on modules of finite G-dimension.

We are interested in a global version of Lê-Ramanujam $\mu$ -constant theorem from the Newton polyhedron point of view. More precisely, we prove a stability theorem which says that the global monodromy fibration of a polynomial function with Newton non-degenerate is uniquely determined by its Newton boundary at infinity. Furthermore, the continuity of atypical values for a family of complex polynomial functions also is considered.

We determine the structure of the Mordell-Weil lattice, Néron-Severi lattice and the lattice of transcendental cycles for certain elliptic K3 surfaces. We find that such questions from algebraic geometry are closely related to the sphere packing problem, and a key ingredient is the use of the sphere packing bounds in establishing geometric results.

When a homogeneous convex cone is given, a natural partial order is introduced in the cone. We shall show that a homogeneous convex cone is a symmetric cone if and only if Vinberg´s $\ast$ -map and its inverse reverse the order. Actually our theorem is formulated in terms of the family of pseudoinverse maps including the $\ast$ -map, and states that the above order-reversing property is typical of the $\ast$ -map of a symmetric cone which coincides with the inverse map of the Jordan algebra associated with the symmetric cone.

The aim of this article is to present a simplified proof of a global existence result for systems of nonlinear wave equations in an exterior domain. The novelty of our proof is to avoid completely the scaling operator which would make the argument complicated in the mixed problem, by using new weighted pointwise estimates of a tangential derivative to the light cone.

We prove that a ‘small’ extension of a minimal AF equivalence relation on a Cantor set is orbit equivalent to the AF relation. By a ‘small’ extension we mean an equivalence relation generated by the minimal AF equivalence relation and another AF equivalence relation which is defined on a closed thin subset. The result we obtain is a generalization of the main theorem in [GMPS2]. It is needed for the study of orbit equivalence of minimal $Z^d$ -systems for $d>2$ [GMPS3], in a similar way as the result in [GMPS2] was needed (and sufficient) for the study of minimal $Z^2$ -systems [GMPS1].

This paper is the second part of our study on limiting behavior of characters of wreath products ${\mathfrak{S}}_n(T)$ of compact group $T$ as $n\to \infty$ and its connection with characters of ${\mathfrak{S}}_{\infty }(T)$ . Contrasted with the first part, which has a representation-theoretical flavor, the approach of this paper is based on probabilistic (or ergodic-theoretical) methods. We apply boundary theory for a fairly general branching graph of infinite valencies to wreath products of an arbitrary compact group $T$ . We show that any character of ${\mathfrak{S}}_{\infty }(T)$ is captured as a limit of normalized irreducible characters of ${\mathfrak{S}}_n(T)$ as $n\to \infty$ along a path on the branching graph of ${\mathfrak{S}}_{\infty }(T)$ . This yields reconstruction of an explicit character formula for ${\mathfrak{S}}_{\infty }(T)$ .

The nonlinear evolution problem for a crack with a kink in elastic body is considered. This nonlinear formulation accounts the condition of mutual non-penetration between the crack faces. The kinking crack is presented with the help of two unknown shape parameters of the kink angle and of the crack length, which minimize an energy due to the Griffith hypothesis. Based on the obtained results of the shape sensitivity analysis, solvability of the evolutionary minimization problem is proved, and the necessary conditions for the optimal crack are derived.

In this paper, we give a geometric characterization for developing mappings such that the asymptotic class of its Schwarzian derivative is in the image of the asymptotic Bers map from the asymptotic Teichmüller space of the unit disk $D$ . We also give a characterization of points in the closure of the image, and discuss the density problem for the asymptotic Teichmüller space.