Revista de la
Unión Matemática Argentina

Volumen 56, número 1 (2015)

Published online: June 24, 2015
Luis Amadeo Piovan, 1959-2015. Pablo A. Panzone
Quasilinear eigenvalues. Julián Fernández Bonder, Juan P. Pinasco, and Ariel M. Salort
We review and extend some well known results for the eigenvalues of the Dirichlet $p$-Laplace operator to a more general class of monotone quasilinear elliptic operators. As an application we obtain some homogenization results for nonlinear eigenvalues.
Three dimensional real Lie bialgebras. Marco A. Farinati and A. Patricia Jancsa
By different methods, we classify the real three dimensional Lie bialgebras and give their automorphism groups; in case of coboundary Lie bialgebras, the corresponding coboundaries $r\in\Lambda ^2\mathfrak{g}$ are listed.
Computation of the canonical lifting via division polynomials. Altan Erdoğan
We study the canonical lifting of ordinary elliptic curves over the ring of Witt vectors. We prove that the canonical lifting is compatible with the base field of the given ordinary elliptic curve which was first proved in Finotti, J. Number Theory 130 (2010), 620-638. We also give some results about division polynomials of elliptic curves defined over the ring of Witt vectors.
Local bounds, Harnack's inequality and Hölder continuity for divergence type elliptic equations with non-standard growth. Noemi Wolanski
We obtain a Harnack type inequality for solutions to elliptic equations in divergence form with non-standard $p(x)$-type growth. A model equation is the inhomogeneous $p(x)$-Laplacian. Namely, \[ \Delta_{p(x)}u:=\operatorname{div}\big(|\nabla u|^{p(x)-2}\nabla u\big)=f(x)\quad\text{in }\Omega, \] for which we prove Harnack's inequality when $f\in L^{q_0}(\Omega)$ if $\max\{1,\frac N{p_1}\} < q_0\le \infty$. The constant in Harnack's inequality depends on $u$ only through $\||u|^{p(x)}\|_{L^1(\Omega)}^{p_2-p_1}$. Dependence of the constant on $u$ is known to be necessary in the case of variable $p(x)$. As in previous papers, log-Hölder continuity on the exponent $p(x)$ is assumed. We also prove that weak solutions are locally bounded and Hölder continuous when $f\in L^{q_0(x)}(\Omega)$ with $q_0\in C(\Omega)$ and $\max\{1,\frac N{p(x)}\} < q_0(x)$ in $\Omega$. These results are then generalized to elliptic equations \[ \operatorname{div} A(x,u,\nabla u)=B(x,u,\nabla u) \] with $p(x)$-type growth.
A note on the reversibility of the elementary cellular automaton with rule number 90. A. Martín del Rey
The reversibility properties of the elementary cellular automaton with rule number 90 are studied. It is shown that the cellular automaton considered is not reversible when periodic boundary conditions are considered, whereas when null boundary conditions are stated, the reversibility appears when the number of cells of the cellular space is even. The DETGTRI algorithm is used to prove these statements. Moreover, the explicit expressions of inverse cellular automata of reversible ones are computed.

Volumen 56, número 2 (2015)

Published online: December 1, 2015
Inequalities for submanifolds of a Riemannian manifold of nearly quasi-constant curvature with a semi-symmetric non-metric connection. Pan Zhang, Xulin Pan, and Liang Zhang
By using two new algebraic lemmas we obtain Chen’s inequalities for submanifolds of a Riemannian manifold of nearly quasi-constant curvature endowed with a semi-symmetric non-metric connection. Moreover, we correct a result of C. Özgür and A. Mihai’s paper (Chen inequalities for submanifolds of real space forms with a semi-symmetric non-metric connection, Canad. Math. Bull. 55 (2012), 611-622).
Semi-slant lightlike submanifolds of indefinite Kaehler manifolds. S. S. Shukla and Akhilesh Yadav
We introduce the notion of semi-slant lightlike submanifolds of indefinite Kaehler manifolds giving a characterization theorem with some nontrivial examples of such submanifolds. Integrability conditions of distributions D1, D2 and Rad TM on semi-slant lightlike submanifolds of indefinite Kaehler manifolds have been obtained. We also obtain necessary and sufficient conditions for foliations determined by the above distributions to be totally geodesic.
Prime ideals of skew PBW extensions. Oswaldo Lezama, Juan Pablo Acosta, and Milton Armando Reyes Villamil
We describe the prime ideals of some important classes of skew PBW extensions, using the classical technique of extending and contracting ideals. Skew PBW extensions include as particular examples Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and their quantizations), Artamonov quantum polynomials, diffusion algebras, and Manin algebra of quantum matrices, among many others.
The quasi-state space of a $C^*$-algebra is a topological quotient of the representation space. Sergio A. Yuhjtman
We show that for any $C^*$-algebra $A$, a sufficiently large Hilbert space $H$ and a unit vector $\xi \in H$, the natural application $\operatorname{rep}(A\mathrm{:}H) \xrightarrow{\theta_\xi} Q(A)$, $\pi \mapsto \langle \pi(-)\xi,\xi \rangle$ is a topological quotient, where $\operatorname{rep}(A\mathrm{:}H)$ is the space of representations on $H$ and $Q(A)$ the set of quasi-states, i.e. positive linear functionals with norm at most $1$. This quotient might be a useful tool in the representation theory of $C^*$-algebras. We apply it to give an interesting proof of Takesaki–Bichteler duality for $C^*$-algebras which allows to drop a hypothesis.
A generalized version of Fubini's theorem on $C_{a,b}[0,T]$ and applications . Il Yong Lee, Hyun Soo Chung, and Seung Jun Chang
We define a transform with respect to the Gaussian process, the $\diamond$-product and the first variation on function space. We then establish a generalized Fubini theorem rather than the Fubini theorem introduced in H. S. Chung, J. G. Choi and S. J. Chang, Banach J. Math. Anal. 7 (2013), 173-185. Also, we examine the various relationships of the transform with respect to the Gaussian process, the $\diamond$-product and the first variation for functionals on function space.
Detection of the torsion classes in the Brieskorn modules of homogeneous polynomials. Khurram Shabbir
Let $f\in \mathbb{C}[X_1,\dots, X_n]$ be a homogeneous polynomial and $B(f)$ be the corresponding Brieskorn module, which is the quotient of the polynomial ring by some specific $\mathbb{C}$-vector space and it has a $\mathbb{C}[t]$-module structure. The main results detect torsion classes in the Brieskorn module using explicit computations with differential forms. We compute the torsion of the Brieskorn module $B(f)$ for two variables in case of non-isolated singularities and show that torsion order is at most $1$. In addition, we find some interesting families in which $B(f)$ is torsion free even in case of non-isolated singularities. We exhibit several examples to compute the monomial basis for $B(f)$ and the construction of torsion elements for $n > 2$.
Some results on spectral distances of graphs. Irena M. Jovanović
Spectral distances of graphs related to some graph operations and modifications are considered. We show that the spectral distance between two graphs does not depend on their common eigenvalues, up to multiplicities. The spectral distance between some graphs with few distinct eigenvalues is computed and due to the results obtained one of the previously posed conjectures related to the A-spectral distances of graphs has been disproved. In order to investigate the cospectrality for special classes of graphs, we computed spectral distances and distance related graph parameters on several specially constructed sets of graphs.
Planar normal sections of focal manifolds of isoparametric hypersurfaces in spheres. Cristián U. Sánchez
The present paper contains some results about the algebraic sets of planar normal sections associated to the focal manifolds of homogeneous isoparametric hypersurfaces in spheres. With the usual identification of the tangent spaces to the focal manifold with subspaces of the tangent spaces to the isoparametric hypersurface, it is proven that the algebraic set of planar normal sections of the focal manifold is contained in that of the original hypersurface.