In this paper, we shall describe the equations of Codazzi-Mainardi for a surface without any umbilical point, using the gradient vector fields of the mean curvature function and the Gaussian curvature function. In addition, based on this description, we shall obtain a homogeneous polynomial, which is an analogue of a Codazzi-Mainardi polynomial obtained in [3], and we shall find a relation between two polynomials.

We study the Grassmann geometries of surfaces when the ambient spaces are the Lie groups of rigid motions on the Euclidean and the Minkowski planes, furnished with left invariant metrics.

Let $\varphi: A \to B$ be a homological epimorphism of $k$-algebras. We investigate the relationship of the Hochschild cohomologies $H^{i}(A)$ and $H^{i}(B)$ of $A$ and $B$, and show that they can be connected by a long exact sequence. In particular, if $A$ is a quasihereditary algebra and $B$ is the quotient of $A$ by a minimal heredity ideal, then the long exact sequence provides information on $H^{i}(A)$, $H^{i}(B)$ and the extension groups between costandard modules and standard modules, thus one can actually compute $H^{i}(A)$ inductively. As a consequence, we obtain the Hochschild cohomology of all nonsemisimple Temperley-Lieb algebras and representation-finite Schur algebras.

We show that space-like isothennic surfaces in the pseudo-riemannian space $\bf{R}^{n-j,j}$ are associated to $O(n - j + 1,j + 1)/O(n - j,j) x 0(1, I)$-system, and that the action of a rational map with two simple poles on the space of local solutions of $O(n - j + I,j + l)/O(n - j,j) \times 0(1, I)$-system correspond to Ribaucour and Darboux transformations to space-like surfaces in $\bf{R}^{n-j,i}$.

The Dirichlet-Neumann problem for the dissipative Helmholtz equation in a connected plane region bounded by closed curves and open arcs (cuts) is studied. The Dirichlet condition is specified on the closed curves, while the Neumann condition is specified on the cuts. The existence of a classical solution is proved by potential theory. The problem is reduced to a Fredholm equation of the second kind, which is uniquely solvable. An integral representation for the solution of the problem is obtained. Our approach holds for both interior and exterior domains.

We will explicitly give the defining equation of the moduli space of symmetric minimal tori with one end.

In this paper, we show the existence of strongly universal spaces of non-separable Borel class $\alpha \geq 2$. By combining this with the result of Sakai and Yaguchi, we can extend the results concerning absorbing sets due to Bestvina and Mogilski to every non-separable absolute Borel classes.

We study the first order linear system $u_{\tilde{z}} + \bar{v}_{w} = 0, u_{\bar{w}} - \bar{v}_{z} = 0$ in a domain $\Omega \subset \mathbb{C}^{2}$ (first considered by G. Cimmino, [3]). We prove a Morera type theorem, emphasizing the analogy to the Cauchy-Riemann system, and a representation formula yielding a result on removable singularities of solutions to (2). We derive (by a Hilbert space technique outlined in [5]) compatibility relations among the free terms and boundary data in the boundary value problem $u_{\tilde{z}}+ \bar{v}_{w} = f, u_{\bar{w}} - \bar{v}_{z} = g$ in $\Omega$, and $u = \varphi,v = \psi$ on $\partial \Omega$. If $F = (u, v) : \Omega \to \mathbb{C}^{2}$ is a solution to (2) such that $\sup_{\varepsilon \gt O}$\int_{\partial \Omega_{\varepsilon}}|F(z, w)|^{p} d\sigma_{\varepsilon}(z, w} \lt \infty$ for some $p \geq 2$, then we show that $F$ admits non tangential limits at almost every $(\xi, \omega) \in \partial\Omega$.

In this paper we give a characterization of the kernel of the bordism intersection map and we present some related results as the following. The set of bordism classes of $C^{\infty}$ maps $f : M \to N$ such that rank $df(x) \leq p$ for all $x$ is contained in $J_{p,m-p}(N)$, where $M$ is a smooth closed manifold of dimension $m$, $N$ is a smooth closed manifold, $df$ is the differential of $f$, $J_{p,m-p}(N)$ is the image of the homomorphism $\ell_{\ast}: \mathfrak{N}_{m}(N^{(p)}) \to \mathfrak{N}_{m}(N)$ induced by the inclusion, $0 \leq p \leq m$, and $N^{(p)}$ is the $p$-skeleton of $N$.

Let $\ell_{2}(\tau)$ be the Hilbert space with weight $\tau$ and $\ell_{2}^{f}$ be the linear span of the canonical orthonormal basis of the separable Hilbert space $\ell_{2}$. In this paper, we prove that if a metric space $X$ ishomeomorphic to $\ell_{2}(\tau)$ or $\ell_{2](\tau) \times \ell_{2}^{f}$ then the hyperspace $Fin_{H}(X)$ of non-empty finite subsets of $X$ with the Hausdorff metric is homeomorphic to $\ell_{2}(\tau) \times \ell_{2}^{f}$.

The atomic surface $X_q$ from an unimodular Pisot substitution $\sigma$ usually has the fractal boundary and it generates a selfaffine tiling. In this paper, we study the boundary $\partial X_{q}$ as the graph directed self-affine fractal and estimate the Hausdorff dimension of the boundary.

An interpolation function for a set of finite input-output data is a function which fits the data. Let us say that a topological space $X$ has a continuous interpolation if interpolation functions can be selected continuously, more precisely, if there is a continuous map from a certain subspace of the hyperspace $F(X x \mathbb{R})$ of finite subsets of $X x \mathbb{R}$ to the Banach space $C(X)$ of bounded real-valued continuous functions on $X$. The concept of weakly continuous interpolation is also introduced. The real line has a continuous interpolation. Every metrizable space has a weakly continuous interpolation. On the other hand, $\omega_{1}$ and $\beta \omega$ do not have weakly continuous interpolations.