We know, after [P1], that, given a tame algebra $\Lambda$, the Tits form $q_\Lambda$ is weakly non negative. Moreover, the converse has been shown for some families of algebras, but it is not true in general. In the same article [P1], De la Peña proved that if $\Lambda$ is a tame concealed algebra, not of type $\tilde{A}_{n}$ and $M$ is an indecomposable $\Lambda$-module then $\Lambda[M]$ is tame if and only if $q_{\Lambda[M]}$ is weakly non negative. The purpose of this work is to show the same result for $\Lambda$ a strongly simply connected tilted algebra of euclidean type.

We show how to take a canonical homology basis and a basis of the space of holomorphic $1$-forms on the curve $x^{m}+y^{n}=1$, and we show how to calculate its explicit period matrix.

In this paper we establish an isometric imbedding of a complex Grassmann manifold $G(m, n)$ into a Euclidean space. Then we use this isometric imbedding to study $1$-type minimal surfaces in $G(m,n)$ and its Gauss map, and obtain some results.

Let $F$ be an algebraic number field of a finite degree, and let $K$ be an extension of $F$ of a finite degree. Denote by $\delta(K/F)$ the greatest common divisor of the discriminants of integers of $K$ with respect to $K/F$. Then, $\delta(K/F)$ is divisible by the discriminant $d(K/F)$ of $K/F$. Let $\mathfrak{p}$ be an arbitrary prime ideal of $F$, let $\mathfrak{p}=\mathfrak{q}_{1}^{e_1} \mathfrak{q}_{2}^{e_2}\cdots \mathfrak{q}_{g}^{e_g}$ be the decomposition of $\mathfrak{p}$ in $K$ into primes, and let $f_i$ be the degree of $\mathfrak{q}_i$. The set of indices $\{1, 2, \ldots, g\}$ is then divided into the union of maximal subsets $I$ such that $f_i = f_j$ whenever $i$ and $j$ belong to a common $I$. We write $f_I$ instead of $f_i$ for $i \in I$, and denote by $g_I$ the number of elements in $I$. Put on the other hand $c(I)= \sum_{d|f_{I}}\mu(f_{I}/d)N\mathfrak{p}^{d}$, where $\mu$ is the Möbius function. Then, $\mathfrak{p}$ divides $\delta(K/F)d(K/F)^{-1}$ if and only if there exists an $I$ such that $c(I) \lt f_{I}g_{I}$.

In previous papers [2, 3], B. Y. Chen introduced a Riemannian invariant $\delta_M$ for a Riemannian $n$-manifold $M^n$. He proved in [3] that every submanifold $M^n$ in the complex hyperbolic $m$-space $CH^{m}(-4)$ satisfies the sharp inequality: $\delta_{M}\leq(n^{2}(n-2)/2(n-1))H^{2}-2(n+1)(n-2)$, where $H^2$ is the squared mean curvature. In this paper, we study Lagrangian submanifolds in $CH^{n}(-4)$ which satisfy the equality case of the inequality.

In this paper we study the Weierstrass points of singular Gorenstein curves. We need to analyze separately the cases in which the trigonal pencil is induced by a line bundle or not, in which the Weierstrass point, $P$, is a smooth point or not, in which $P$ is a smooth ordinary or total ramification point or not.

In this paper, we first characterize those compatible metrics $d$ on a metrizable space $X$ which give rise to a connected $d$-proximal hyperspace. We show that the space of irrational numbers, in particular, admits a complete metric with this property and, as a consequence, we get a negative answer to a question of [11] about selections for hyperspace topologies. Next, we characterize the compatible metrics on $X$ which are uniformly equivalent to ultrametrics showing that this is equivalent to the zero-dimensionality of the corresponding proximal hyperspaces. Applications and related results about other disconnectedness-like properties of proximal hyperspaces are obtained.

The purpose of this paper is to determine explicitly the Euler Lagrange equations of our conformal gauge invariant functional on the space of all Weyl structures.