Revista de la
Unión Matemática Argentina

Volume 60, number 1 (2019)

June 2019
Primary decomposition and secondary representation of modules. Masoumeh Hasanzad, Jafar A'zami
In this paper we study the notions of primary decomposition and secondary representation of modules over a commutative ring with identity. Also we review these concepts over injective and projective modules.
Lie $n$-multiplicative mappings on triangular $n$-matrix rings. Bruno L. M. Ferreira, Henrique Guzzo Jr.
We extend to triangular $n$-matrix rings and Lie $n$-multiplicative maps a result about Lie multiplicative maps on triangular algebras due to Xiaofei Qi and Jinchuan Hou.
Higher order mean curvatures of SAC half-lightlike submanifolds of indefinite almost contact manifolds. Fortuné Massamba, Samuel Ssekajja
We introduce higher order mean curvatures of screen almost conformal (SAC) half-lightlike submanifolds of indefinite almost contact manifolds, admitting a semi-symmetric non-metric connection. We use them to generalize some known results by Duggal and Sahin on totally umbilical half-lightlike submanifolds [Int. J. Math. Math. Sci. 2004, no. 68, 3737-3753]. Also, we derive a new integration formula via the divergence of some special vector fields tangent to these submanifolds, which we later use to characterize minimal and maximal submanifolds. Several examples, where possible, are also included to illustrate the main concepts.
Branching laws: some results and new examples. Oscar Márquez, Sebastián Simondi, Jorge A. Vargas
For a connected, noncompact simple matrix Lie group $G$ so that a maximal compact subgroup $K$ has a three dimensional simple ideal, in this note we analyze the admissibility of the restriction of irreducible square integrable representations for the ambient group when they are restricted to certain subgroups that contain the three dimensional ideal. In this setting we provide a formula for the multiplicity of the irreducible factors. Also, for general $G$ such that $G/K$ is an Hermitian $G$-manifold we give a necessary and sufficient condition so that an arbitrary square integrable representation of the ambient group is admissible over the semisimple factor of $K$.
On the structure of split involutive Hom-Lie color algebras. Valiollah Khalili
In this paper we study the structure of arbitrary split involutive regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular Hom-Lie color algebra $\mathcal{L}$ is of the form $\mathcal{L}=\mathcal{U}\oplus\sum_{[\alpha]\in\Pi/\sim} I_{[\alpha]}$, with $\mathcal{U}$ a subspace of the involutive abelian subalgebra $\mathcal{H}$ and any $I_{[\alpha]}$, a well-described involutive ideal of $\mathcal{L}$, satisfying $[I_{[\alpha]}, I_{[\beta]}]=0$ if $[\alpha]\neq[\beta]$. Under certain conditions, in the case of $\mathcal{L}$ being of maximal length, the simplicity of the algebra is characterized and it is shown that $\mathcal{L}$ is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular Hom-Lie color algebra. Finally, an example will be provided to characterise the inner structure of split involutive Hom-Lie color algebras.
The multivariate bisection algorithm. Manuel López Galván
The aim of this paper is to study the bisection method in $\mathbb{R}^n$. We propose a multivariate bisection method supported by the Poincaré–Miranda theorem in order to solve non-linear systems of equations. Given an initial cube satisfying the hypothesis of the Poincaré–Miranda theorem, the algorithm performs congruent refinements through its center by generating a root approximation. Through preconditioning we will prove the local convergence of this new root finder methodology and moreover we will perform a numerical implementation for the two dimensional case.
Some new $Z$-eigenvalue localization sets for tensors and their applications. Zhengge Huang, Ligong Wang, Zhong Xu , Jingjing Cui
In this paper some new $Z$-eigenvalue localization sets for general tensors are established, which are proved to be tighter than those newly derived by Wang et al. [Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 187–198]. Also, some relationships between the $Z$-eigenvalue inclusion sets presented by Wang et al. and the new $Z$-eigenvalue localization sets for tensors are given. Besides, we discuss the effects of orthonormal transformations for the proposed sets. As applications of the proposed sets, some improved upper bounds for the $Z$-spectral radius of weakly symmetric nonnegative tensors are given. Numerical examples are also given to verify the advantages of our proposed results over some known ones.
Direct theorems of trigonometric approximation for variable exponent Lebesgue spaces. Ramazan Akgün
Jackson type direct theorems are considered in variable exponent Lebesgue spaces $L^{p(x)}$ with exponent $p(x)$ satisfying $1\leq \operatorname{ess\,inf}_{x\in [0,2\pi ]}p(x)$, $\operatorname{ess\,sup}_{x\in [0,2\pi]}p(x) < \infty$, and the Dini–Lipschitz condition. Jackson type direct inequalities of trigonometric approximation are obtained for the modulus of smoothness based on one sided Steklov averages \[ \mathfrak{Z}_{v}f(\cdot) := \frac{1}{v}\int\nolimits_{0}^{v}f (\cdot+t) \,dt \] in these spaces. We give the main properties of the modulus of smoothness \[ \Omega_{r}(f,v)_{p(\cdot)} := \left\Vert (\mathbf{I}-\mathfrak{Z}_{v})^{r}f\right\Vert_{p(\cdot)}\quad (r\in \mathbb{N}) \] in $L^{p(x)}$, where $\mathbf{I}$ is the identity operator. An equivalence of the modulus of smoothness and Peetre's $K$-functional is established.
Pricing American put options under stochastic volatility using the Malliavin derivative. Mohamed Kharrat
The aim of this paper is to develop a methodology based on Malliavin calculus, in order to price American options under stochastic volatility. This leads to compute the conditional expectation $\mathbb{E}(P_{t}(X_{t}, V_{t})\mid(X_{l},V_{l}))$ for any $ 0\leq l < t$, where $V_{t}$ is generated by the Cox-Ingersoll-Ross (CIR) process. Some simulations and comparisons are given.
Trajectorial market models: arbitrage and pricing intervals. Sebastián E. Ferrando, Alfredo L. González, Iván L. Degano, Massoomeh Rahsepar
The paper develops general, non-probabilistic market models based on trajectory sets and minmax price bounds leading to price intervals for European options. The approach provides the trajectory based analogue of a martingale process as well as a generalization that allows a limited notion of arbitrage in the market while still providing coherent option prices. An illustrative example is described in detail. Several properties of the price bounds are obtained, in particular a connection with risk neutral pricing is established for trajectory markets associated to a continuous-time martingale model.
Relative modular uniform approximation by means of the power series method with applications. Kuldip Raj, Anu Choudhary
We introduce the notion of relative convergence by means of a four dimensional matrix in the sense of the power series method, which includes Abel's as well as Borel's methods, to prove a Korovkin type approximation theorem by using the test functions $\{1,y,z,y^{2}+z^{2}\}$ and a double sequence of positive linear operators defined on modular spaces. We also endeavor to examine some applications related to this new type of approximation.
Top local cohomology modules over local rings and the weak going-up property. Asghar Farokhi, Alireza Nazari
Let $(R,\mathfrak{m})$ be a Noetherian local ring and let $\widehat{R}$ denote the $\mathfrak{m}$-adic completion of $R$. In this paper, we introduce the concept of the weak going-up property for the extension $R\subseteq \widehat{R}$ and we give some characterizations of this property. In particular, we show that this property is equivalent to the strong form of the Lichtenbaum–Hartshorne Vanishing Theorem. Also, when $R$ satisfies the weak going-up property, we show that for a finitely generated $R$-module $M$ of dimension $d$, and ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, we have $\operatorname{Att}_{R}(\operatorname{H}^{d}_{\mathfrak{a}}(M)) = \operatorname{Att}_{R}(\operatorname{H}^{d}_{\mathfrak{b}}(M))$ if and only if $\operatorname{H}^{d}_{\mathfrak{a}}(M)\cong \operatorname{H}^{d}_{\mathfrak{b}}(M)$, and we find a criterion for the cofiniteness of Artinian top local cohomology modules.
Formal torsors under reductive group schemes. Benedictus Margaux
We consider the algebraization problem for torsors over a proper formal scheme under a reductive group scheme. Our results apply to the case of semisimple group schemes (which is addressed in detail).
Hörmander conditions for vector-valued kernels of singular integrals and their commutators. Andrea L. Gallo, Gonzalo H. Ibañez Firnkorn, María Silvina Riveros
We study Coifman type estimates and weighted norm inequalities for singular integral operators and their commutators, given by the convolution with a vector-valued kernel. We define a weaker Hörmander type condition associated with Young functions for the vector-valued kernels. With this general framework we obtain as an example the result for the square operator and its commutator given in [M. Lorente, M. S. Riveros, and A. de la Torre, J. Math. Anal. Appl. 336 (2007), no. 1, 577–592].
On the self-conjugateness of differential forms on bounded domains. Ricardo Abreu Blaya, Juan Bory Reyes, Efrén Morales Amaya, José María Sigarreta Almira
Suppose $\Omega$ is a bounded domain in $\mathbb{R}^n$ with boundary $\Gamma$ and let $\mathcal{W}$ be a non-homogeneous differential form harmonic in $\Omega$ and Hölder-continuous in $\Omega\cup\Gamma$. In this paper we study and obtain some necessary and sufficient conditions for the self-conjugateness of $\mathcal{W}$ in terms of its boundary value $\mathcal{W}|_\Gamma=\omega$.
A remark on trans-Sasakian 3-manifolds. Yaning Wang, Wenjie Wang
Let $M$ be a trans-Sasakian $3$-manifold of type $(\alpha,\beta)$. In this paper, we give a negative answer to the question proposed by S. Deshmukh [Mediterr. J. Math. 13 (2016), no. 5, 2951–2958], namely we prove that the differential equation $\nabla\beta=\xi(\beta)\xi$ on $M$ does not necessarily imply that $M$ is homothetic to either a Sasakian or cosymplectic manifold even when $M$ is compact. Many examples are constructed to illustrate this result.
Finite dimensional Hopf algebras over the Kac-Paljutkin algebra $H_8$. Yuxing Shi
Let $H_8$ be the Kac–Paljutkin algebra [Trudy Moskov. Mat. Obšč. 15 (1966), 224–261], which is the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra. All simple Yetter–Drinfel'd modules over $H_8$ are given, and finite-dimensional Nichols algebras over $H_8$ are determined completely. It turns out that they are all of diagonal type. In fact, they are of Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$, respectively. By the way, we calculate Gelfand–Kirillov dimensions for some Nichols algebras. As an application, we complete the classification of the finite-dimensional Hopf algebras over $H_8$ according to the lifting method.