Revista de la
Unión Matemática Argentina

Volumen 52, número 1 (2011)

Eduardo H. Zarantonello (1918-2010). J. Tirao
De matemática y matemáticos. Eduardo H. Zarantonello

Conferencia originalmente ofrecida en septiembre de 1981 en Córdoba, Argentina, con motivo de la celebración del XXVº aniversario de la creación del Instituto de Matemática, Astronomía y Física de la Universidad Nacional de Córdoba.

Note: This text was not included in the print version of the Revista.

Factor congruences in semilattices. Pedro Sánchez Terraf
We characterize factor congruences in semilattices by using generalized notions of order ideal and of direct sum of ideals. When the semilattice has a minimum (maximum) element, these generalized ideals turn into ordinary (dual) ideals.
Properties of the bivariate confluent hypergeometric function kind 1 distribution. Daya K. Nagar and Fabio Humberto Sepúlveda-Murillo
The bivariate confluent hypergeometric function kind 1 distribution is defined by the probability density function proportional to x1ν1 − 1 x2ν2 − 11F1(α; β; −x1x2). In this article, we study several properties of this distribution and derive density functions of X1/X2, X1/(X1 + X2), X1 + X2 and 2 √(X1 X2). The density function of 2 √(X1 X2) is represented in terms of modified Bessel function of the second kind. We also show that for ν1 − ν2 = 1/2, 2 √(X1 X2) follows a confluent hypergeometric function kind 1 distribution.
Ergodic properties of linear operators. María Elena Becker
Let T be a bounded linear operator on a Banach space X. We prove some properties of X1 = { zX : limnk=1n (Tkz/k) exists} and we construct an operator T such that limn ||Tn/n|| = 0, but (IT)X is not included in X1.
A new application of power increasing sequences. Hüseyin Bor
In the present paper, we have proved a general summability factor theorem by using a general summability method. This theorem also includes several new results.
A note on integral C-parallel submanifolds in S7(c). D. Fetcu and C. Oniciuc
We find the explicit parametric equations of the flat 3-dimensional integral C-parallel submanifolds in the sphere S7 endowed with the deformed Sasakian structure defined by Tanno.
Weighted local BMO spaces and the local Hardy-Littlewood maximal operator. Aníbal Chicco Ruiz and Eleonor Harboure
We define a local type of a weighted BMO space on R+ and prove the boundedness of the local Hardy-Littlewood maximal function in that space, provided that the weight belongs to the local class A1loc.
Global Lp estimates for degenerate Ornstein-Uhlenbeck operators: a general approach. Ermanno Lanconelli
We present a new approach to prove global Lp estimates for degenerate Ornstein-Uhlenbeck operators in RN. We then show how to pave the way to extend such a technique to classes of general Hörmander operators. Several historical notes related to Calderón-Zygmund’s singular integrals theory in Euclidean and in non-Euclidean settings are also provided.
Triality and the normal sections of Cartan’s isoparametric hypersurfaces. Cristián U. Sánchez
The present paper is devoted to study the algebraic sets of normal sections of the so called Cartan’s isoparametric hypersurfaces MR, MC, MH and MO of complete flags in the projective planes RP2, CP2, HP2 and OP2. It presents a connection between “normed trialities” and the polynomial defining the algebraic sets of normal sections of these hypersurfaces. It contains also a geometric and topological description of these algebraic sets by means of the isotropy actions of the isoparametric hypersurfaces.
Approximation and shape preserving properties of the truncated Baskakov operator of max-product kind. Barnabás Bede, Lucian Coroianu and Sorin G. Gal
Starting from the study of the Shepard nonlinear operator of max-prod type in [2], [3], in the recent monograph [4], Open Problem 5.5.4, pp. 324-326, the Baskakov max-prod type operator is introduced and the question of the approximation order by this operator is raised. The aim of this note is to obtain the order of uniform approximation Cω1(f; 1/√n) (with the explicit constant C = 24) of another operator called the truncated max-prod Baskakov operator and to prove by a counterexample that in some sense, for arbitrary f this type of order of approximation with respect to ω1(f; √n) cannot be improved. However, for some subclasses of functions including for example the nondecreasing concave functions, the essentially better order of approximation ω1(f; 1/n) is obtained. Finally, some shape preserving properties are proved.
Some Slater type inequalities for convex functions of selfadjoint operators in Hilbert spaces. S. S. Dragomir
Some inequalities of the Slater type for convex functions of selfadjoint operators in Hilbert spaces $H$ under suitable assumptions for the involved operators are given. Amongst others, it is shown that if $A$ is a positive definite operator with $Sp(A) \subset [m,M]$ and $f$ is convex and has a continuous derivative on $[m,M]$, then for any $x\in H$ with $\left\Vert x\right\Vert =1$ the following inequality holds: \begin{multline*} 0\leq f\left( \frac{\left\langle Af^{\prime }\left( A\right) x,x\right\rangle }{\left\langle f^{\prime }\left( A\right) x,x\right\rangle }% \right) -\left\langle f\left( A\right) x,x\right\rangle \\ \leq \frac{1}{4}\cdot \sqrt{\frac{Mf^{\prime }\left( M\right) }{mf^{\prime }\left( m\right) }}\left( M-m\right) \left( f^{\prime }\left( M\right) -f^{\prime }\left( m\right) \right). \end{multline*}
Necessary and sufficient conditions for the Schur harmonic convexity or concavity of the extended mean values. Wei-Feng Xia, Yu-Ming Chu and Gen-Di Wang
In this paper, we prove that the extended values $E(r,s;x,y)$ are Schur harmonic convex (or concave, respectively) with respect to $(x,y)\in (0,\infty)\times(0,\infty)$ if and only if $(r,s)\in \{(r,s):s\geq-1, s\geq r,s+r+3\geq0\}\cup\{(r,s):r\geq-1, r\geq s,s+r+3\geq0\}$ (or $\{(r,s):s\leq-1, r\leq-1,s+r+3\leq0\},$ respectively).
Property (ω) and quasi-class (A, k) operators. M. H. M. Rashid
In this paper, we prove the following assertions: (i) If T is of quasi-class (A, k), then T is polaroid and reguloid; (ii) If T or T* is an algebraically of quasi-class (A, k) operator, then Weyls theorem holds for f(T) for every fHol(σ(T)); (iii) If T* is an algebraically of quasi-class (A, k) operator, then a-Weyls theorem holds for f(T) for every fHol(σ(T)); (iv) If T* is algebraically of quasi-class (A, k) then property (ω) holds for T.
Solution of Troesch’s problem using He’s polynomials. Syed Tauseef Mohyud-Din
In this paper, we apply He’s polynomials for finding the approximate solution of the Troesch’s problem which arises in the confinement of a plasma column by radiation pressure and applied physics. The proposed technique proved to be very effective and is easier to implement as compare to decomposition method.
Julio Rey Pastor, su posición en la escuela matemática argentina. Eduardo L. Ortiz
This paper was read at the Tandil Meeting of the UMA, in September 2010; in this presentation I keep to the original format of a lecture. I briefly consider three of the main attempts made in Argentina to establish a mathematics school, between 1817 and 1940, paying more attention to the third one, in which Julio Rey Pastor was the main character. Contrary to the earlier ones, in this last period mathematics began to be established as a distinct discipline, with its own problems and methods, while keeping close ties with other disciplines. In this lecture I’ll also consider some matters that emerged in parallel with these attempts, and make some remarks on the different approaches used to tackle them. At the same time I’ll try to relate the progress of mathematics in Argentina with the doctrines and ideas that, in different periods of its history, dominated its cultural life.

Volumen 52, número 2 (2011)

This volume contains the contributed papers from the Xth Encuentro Nacional de Analistas Alberto P. Calderón, which took place between August 25th and 28th, 2010, in La Falda, Córdoba.
Preface. Carlos Cabrelli and Eleonor Harboure
Hardy spaces associated with semigroups of operators. Jorge J. Betancor
This paper is a non exhaustive survey about Hardy spaces defined by semigroups of operators.
Sobolev spaces diversification. Bruno Bongioanni
This work attempts to be an overview of a variety of results concerning Sobolev spaces associated to some orthonormal systems, particularly the Hermite and Laguerre operators settings.
Five basic lemmas for symmetric tensor products of normed spaces. Daniel Carando and Daniel Galicer
We give the symmetric version of five lemmas which are essential for the theory of tensor products (and norms). These are: the approximation, extension, embedding, density and local technique lemma. Some applications of these tools to the metric theory of symmetric tensor products and to the theory of polynomials ideals are given.
Wiener’s lemma: pictures at an exhibition. Ilya A. Krishtal
In this expository paper we present various general extensions of the celebrated Wiener’s Tauberian Lemma, outline several ingredients that are often used in proofs and discuss a few applications to localization of frames.
Porosity, dimension, and local entropies: A survey. Pablo Shmerkin
Porosity and dimension are two useful, but different, concepts that quantify the size of fractal sets and measures. An active area of research concerns understanding the relationship between these two concepts. In this article we will survey the various notions of porosity of sets and measures that have been proposed, and how they relate to dimension. Along the way, we will introduce the idea of local entropy averages, which arose in a different context, and was then applied to obtain a bound for the dimension of mean porous measures.
Importance of Zak transforms for harmonic analysis. Edward N. Wilson

In engineering and applied mathematics, Zak transforms have been effectively used for over 50 years in various applied settings. As Gelfand observed in a 1950 paper, the variable coefficient Fourier series ideas articulated in Andre Weil’s famous book on integration lead to an exceedingly elementary proof of the Plancherel Theorem for LCA groups. The transform for functions on R appearing in Zak’s seminal 1967 paper is actually a special case of the LCA group transforms earlier introduced by Weil; Zak states this explicitly in his 1967 paper but the mathematical community nonetheless chose to name the transform for him.

In brief, the properties of Zak transforms are simply reflections of elementary Fourier series properties and the Plancherel Theorem for non-compact LCA groups is an immediate consequence of the fact that Fourier transforms are averages of Zak transforms. It is remarkable that only a small handful of mathematicians know this proof and that all textbooks continue to give much harder and less transparent proofs for even the case of the group R. Generalized Zak transforms arise naturally as intertwining operators for various representations of Abelian groups and allow formulation of many appealing theorems.