L. Blum, M. Shub and S. Smale defined a kind of computation model having parameters that take real values [2]. In this paper we will extend their computation theory on the basis of the conventional recursion theory which is constructed on the domain of integers. In particular, we will define the Turing degree for the set of reals similarly to the Turing degree for the set of integers, so as to derive some related results.

For a bounded domain in a complete Riemannian manifold $M^n$ isometrically immersed in a Euclidean space, we derive extrinsic estimates for eigenvalues of the Dirichlet eigenvalue problem of the Laplace operator, which depend on the mean curvature of the immersion. Further, we also obtain an upper bound for the ${(k+1)}^{\text{th}}$ eigenvalue, which is best possible in the meaning of order on $k$.

Let $E$ be an ample vector bundle on a projective manifold $X$, with a section vanishing on a smooth subvariety $Z$ of the expected dimension, and let $H$ be an ample line bundle on $X$ inducing a very ample ample line bundle $H_Z$ on $Z$. Triplets $(X,E,H)$ as above are classified assuming that $Z$, embedded by $\left|H_Z\right|$, is a variety of small degree with respect to codimension.

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold $M$ endowed with the Webster metric hence consider a version of the CR Yamabe problem for CR manifolds with boundary. This occurs as the Yamabe problem for the Fefferman metric (a Lorentzian metric associated to a choice of contact structure $\theta$ on $M$, [20]) on the total space of the canonical circle bundle $S^1\to C(M)\stackrel{\pi }{\to }M$ (a manifold with boundary $\partial C(M)={\pi }^{-1}(\partial M)$ and is shown to be a nonlinear subelliptic problem of variational origin. For any real surface $N=\{\phi =0\}\subset H_1$ we show that the mean curvature vector of $N\rightharpoonup H_1$ is expressed by $H=-\frac{1}{2}{\displaystyle {\sum }_{j=1}^2{X}_j}(|X\phi {|}^{-1}{X}_j\phi )\xi$ provided that $N$ is tangent to the characteristic direction $T$ of $(H_1,{\theta }_0)$, thus demonstrating the relationship between the classical theory of submanifolds in Riemannian manifolds (cf. e.g.[7]) and the newer investigations in [1], [6], [8] and [16]. Given an isometric immersion $\Psi :N\to {\text{H}}_n$ of a Riemannian manifold into the Heisenberg group we show that $\Delta \Psi =2J{T}^{\perp }$ hence start a Weierstrass representation theory for minimal surfaces in $H_n$.

We aim to study local rigidity and deformations for the following class of groups: the semidirect product $\Gamma =Z^n{⋊}_{A}Z$ where $n\ge 2$ is an integer and $A$ is a hyperbolic matrix in $SL\left(n,Z\right)$, considered first as a lattice in the solvable Lie group $G=R^n{⋊}_{A}R$, then as a subgroup of the semisimple Lie group $SL\left(n+1,R\right)$. We will notably show that, although $\Gamma$ is locally rigid neither in $G$ nor in $H$, it is locally $SL\left(n+1,R\right)$-rigid in $G$ in the sense that every small enough deformation of $\Gamma$ in $G$ is conjugated to $\Gamma$ by an element of $SL\left(n+1,R\right)$.

Let $V$ be a total valuation ring of a division ring $K$ with an automorphism $\sigma$ and let $A={\oplus }_{i\in Z}{A}_i{X}^i$ be a graded extension of $V$ in $K[X,X^{-1};\sigma ]$, the skew Laurent polynomial ring. We classify $A$ by distinguishing four different types based on the properties of $A_1$ and $A_{-1}$. A complete description of $A_i$ for all $i\in \text{Z}$ is given in the case where $A_1$ is a finitely generated left $O_l(A_1)$-ideal.

We address the question of the classification under blow-Nash equivalence of simple Nash function germs. We state that this classification coincides with the real analytic classification. We prove moreover that a simple germ can not be blow-Nash equivalent to a nonsimple one. The method is based on the computation of relevant coefficients of the real zeta functions associated to a Nash germ via motivic integration.

We show that a second main theorem of Nevanlinna theory holds for meromorphic functions on general complete Kähler manifolds. It is well-known in classical Nevanlinna theory that a meromorphic function whose image grows rapidly enough can omit at most two points. Our second main theorem implies this fact holds for meromorphic functions on general complete Kähler manifolds.

We give a generalization of the Shestakov-Umirbaev inequality which plays an important role in their solution of the Tame Generators Problem on the automorphism group of a polynomial ring. As an application, we give a new necessary condition for endomorphisms of a polynomial ring to be invertible, which implies Jung's theorem in case of two variables.

The completion of a (normed) $C^{*}$-algebra $A_0[\parallel \parallel ·{\parallel \parallel }_0]$ with respect to a locally convex topology $\tau$ on $A_0$ that makes the multiplication of $A_0$ separately continuous is, in general, a quasi $*$-algebra, and not a locally convex $*$-algebra [10], [15]. In this way, one is led to consideration of locally convex quasi $C^{*}$-algebras, which generalize $C^{*}$-algebras in the context of quasi $*$-algebras. Examples are given and the structure of these relatives of $C^{*}$-algebras is investigated.

The Kirchhoff elastic rod is one of the mathematical models of thin elastic rods, and is characterized as a critical point of the energy functional obtained by adding the effect of twisting to the bending energy. In this paper, we investigate Kirchhoff elastic rods in three-dimensional space forms. In particular, we give explicit formulas of Kirchhoff elastic rods in the three-sphere and in the three-dimensional hyperbolic space in terms of Jacobi sn function and the elliptic integrals.

Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent.

We construct a family of singular solutions to some nonlinear partial differential equations which have resonances in the sense of a paper due to T.Kobayashi. The leading term of a solution in our family contains a logarithm, possibly multiplied by a monomial. As an application, we study nonlinear wave equations with quadratic nonlinearities. The proof is done by the reduction to a Fuchsian equation with singular coefficients.