Revista de la
Unión Matemática Argentina

Volumen 58, número 1 (2017)

June 2017
An interpolation theorem between Calderón-Hardy spaces. Sheldy Ombrosi, Alejandra Perini, Ricardo Testoni
We obtain a complex interpolation theorem between weighted Calderón-Hardy spaces for weights in a Sawyer class. The technique used is based on the method obtained by J.-O. Strömberg and A. Torchinsky; however, we must overcome several technical difficulties associated with considering one-sided Calderón-Hardy spaces. Interpolation results of this type are useful in the study of weighted weak type inequalities of strongly singular integral operators.
Quotient $p$-Schatten metrics on spheres. Esteban Andruchow, Andrea C. Antunez
Let $S(H)$ be the unit sphere of a Hilbert space $H$ and $U_p(H)$ the group of unitary operators in $H$ such that $u-1$ belongs to the $p$-Schatten ideal $B_p(H)$. This group acts smoothly and transitively in $S(H)$ and endows it with a natural Finsler metric induced by the $p$-norm $\Vert z \Vert_p= \operatorname{tr}\left((z z^*)^{p/2}\right)^{1/p}$. This metric is given by \[ \Vert v \Vert_{x,p} = \min \lbrace \Vert z-y \Vert_p: y \in \mathfrak{g} _x \rbrace, \] where $z \in \mathcal{B} _p(H)_ {ah}$ satisfies that $(d \pi_x)_1(z)=z \cdot x = v$ and $ \mathfrak{g} _x$ denotes the Lie algebra of the subgroup of unitaries which fix $x$. We call $z$ a lifting of $v$. A lifting $z_0$ is called a minimal lifting if additionally $ \Vert v \Vert_{x,p} = \Vert z_0 \Vert_p$. In this paper we show properties of minimal liftings and we treat the problem of finding short curves $ \alpha$ such that $ \alpha(0)=x $ and $ \dot{\alpha} (0)= v$ with $x \in S(H)$ and $v \in T_xS(H)$ given. Also we consider the problem of finding short curves which join two given endpoints $x,y \in S(H)$.
Affine Szabó connections on smooth manifolds. Abdoul Salam Diallo, Fortuné Massamba
We introduce a new structure, called affine Szabó connection. We prove that, on $2$-dimensional affine manifolds, the affine Szabó structure is equivalent to one of the cyclic parallelisms of the Ricci tensor. A characterization for locally homogeneous affine Szabó surfaces is obtained. Examples of two- and three-dimensional affine Szabó manifolds are also given.
Connectedness of the algebraic set of vectors generating planar normal sections of homogeneous isoparametric hypersurfaces. Cristián U. Sánchez
Let $M\subset \mathbb{S}^{n+1}\subset $ $\mathbb{R}^{n+2}$ be a homogeneous isoparametric hypersurface and consider the algebraic set of unit tangent vectors generating planar normal sections at a point $E\in M$ (denoted by $\widehat{X}_{E}[M] \subset T_{E}(M)$). The present paper is devoted to prove that $\widehat{X}_{E}[M]$ is connected by arcs. This in turn proves that its projective image $X[M] \subset \mathbb{RP}(T_{E}(M))$ also has this property.
Vanishing, Bass numbers, and cominimaxness of local cohomology modules. Jafar A'zami
Let $(R,m)$ be a commutative Noetherian regular local ring and $I$ be a proper ideal of $R$. It is shown that $H^ {d-1} _ \mathfrak{p} (R)=0$ for any prime ideal $ \mathfrak{p} $ of $R$ with $ \dim(R/ \mathfrak{p})=2$, whenever the set $ \{ n \in \mathbb{N} : R/ \mathfrak{p} ^ {(n)}$ is Cohen-Macaulay$\}$ is infinite. Now, let $(R,m)$ be a commutative Noetherian unique factorization local domain of dimension $d$, $I$ an ideal of $R$, and $M$ a finitely generated $R$-module. It is shown that the Bass numbers of the $R$-module $H^i_I(M)$ are finite, for all integers $i \geq 0$, whenever $ \operatorname{height} (I)=1$ or $d \leq 3$.
Soft ideals in ordered semigroups. E. H. Hamouda
The notions of soft left and soft right ideals, soft quasi-ideal and soft bi-ideal in ordered semigroups are introduced. We show here that in ordered groupoids the soft right and soft left ideals are soft quasi-ideals, and in ordered semigroups the soft quasi-ideals are soft bi-ideals. Moreover, we prove that in regular ordered semigroups the soft quasi-ideals and the soft bi-ideals coincide. We finally show that in an ordered semigroup the soft quasi-ideals are just intersections of soft right and soft left ideals.
On the Betti numbers of filiform Lie algebras over fields of characteristic two. Ioannis Tsartsaflis

An $n$-dimensional Lie algebra $ \mathfrak{g} $ over a field $ \mathbb{F} $ of characteristic two is said to be of Vergne type if there is a basis $e_1, \dots,e_n$ such that $ [e_1,e_i] =e_ {i+1} $ for all $2 \leq i \leq[e_i,e_j] n-1$ and $ = c_ {i,j} e_ {i+j} $ for some $c_ {i,j} \in \mathbb{F} $ for all $i,j \ge 2$ with $i+j \le n$. We define the algebra $ \mathfrak{m} _0$ by its nontrivial bracket relations: $ [e_1,e_i] =e_ {i+1} $, $2 \leq i \leq n-1$, and the algebra $ \mathfrak{m} _2$: $ [e_1,e_i ] =e_ {i+1} $, $2 \le i \le[e_2, e_j ] n-1$, $ =e_ {j+2} $, $3 \le j \le n-2$.

We show that, in contrast to the corresponding real and complex cases, $ \mathfrak{m} _0(n)$ and $ \mathfrak{m} _2(n)$ have the same Betti numbers. We also prove that for any Lie algebra of Vergne type of dimension at least $5$, there exists a non-isomorphic algebra of Vergne type with the same Betti numbers.

Certain curves on some classes of three-dimensional almost contact metric manifolds. Avijit Sarkar, Ashis Mondal
The object of the present paper is to characterize three-dimensional trans-Sasakian generalized Sasakian space forms admitting biharmonic almost contact curves with respect to generalized Tanaka Webster Okumura (gTWO) connections and to give illustrative examples. The mean curvature vector of almost contact curves has been analyzed on trans-Sasakian manifolds with gTWO connections. Some properties of slant curves on the same manifolds have been established. Finally curvature and torsion, with respect to gTWO connections, of C-parallel and C-proper slant curves in three-dimensional almost contact metric manifolds have been deduced.
Generalizations of Cline's formula for three generalized inverses. Yong Jiang, Yongxian Wen, Qingping Zeng
It is shown that an element $a$ in a ring is Drazin invertible if and only if so is $a^ {n} $; the Drazin inverse of $a$ is given by that of $a^ {n} $, and vice versa. Using this result, we prove that, in the presence of $aba=aca$, for any natural numbers $n$ and $m$, $(ac)^ {n} $ is Drazin invertible in a ring if and only if so is $(ba)^ {m} $; the Drazin inverse of $(ac)^ {n} $ is expressed by that of $(ba)^ {m} $, and vice versa. Also, analogous results for the pseudo Drazin inverse and the generalized Drazin inverse are established on Banach algebras.
Harmonic analysis associated with the modified Cherednik type operator on the real line and Paley-Wiener theorems for its Hartley transform. Hatem Mejjaoli
We consider a new differential-difference operator $\Lambda$ on the real line. We study the harmonic analysis associated with this operator. Next, we establish the Paley-Wiener theorems for its Hartley transform on $\mathbb{R}$.
Elementary proof of the continuity of the topological entropy at $\theta=\underline{1001}$ in the Milnor-Thurston world. Andrés Jablonski, Rafael Labarca
In 1965, Adler, Konheim and McAndrew introduced the topological entropy of a given dynamical system, which consists of a real number that explains part of the complexity of the dynamics of the system. In this context, a good question could be if the topological entropy $H_{\mathrm{top}} (f)$ changes continuously with $f$. For continuous maps this problem was studied by Misisurewicz, Slenk and Urbański. Recently, and related with the lexicographic and the Milnor-Thurston worlds, this problem was studied by Labarca and others. In this paper we will prove, by elementary methods, the continuity of the topological entropy in a maximal periodic orbit ($ \theta= \underline{1001} $) in the Milnor-Thurston world. Moreover, by using dynamical methods, we obtain interesting relations and results concerning the largest eigenvalue of a sequence of square matrices whose lengths grow up to infinity.