Errors in algebraic statements translation during the creation of an algebraic domino

We present a research study which main objective is to inquire into secondary school students´ ability to translate and relate algebraic statements which are presented in the symbolic and verbal representation systems. Data collection was performed with 26 14-15 years old students to whom we proposed the creation of an algebraic domino, designed for this research, and its subsequent use in a tournament. Here we present an analysis of the errors made in such translations. Among the obtained results, we note that the students found easier to translate statements from the symbolic to the verbal representation and that most errors in translating from verbal to symbolic expressions where derived from the particular characteristics of algebraic language. Other types of errors are also identified. KEYWORDS: Algebraic language, domino, errors, translation between representation systems, verbal representation

Discrete harmonic analysis associated with ultraspherical expansions

We study discrete harmonic analysis associated with ultraspherical orthogonal functions. We establish weighted l^p-boundedness properties of maximal operators and Littlewood-Paley g-functions defined by Poisson and heat semigroups generated by certain difference operator. We also prove weighted l^p-boundedness properties of transplantation operators associated to the system of ultraspherical functions. In order to show our results we previously establish a vector-valued local Calder\'on-Zygmund theorem in our discrete setting

Conical square functions associated with Bessel, Laguerre and Schr\"odinger operators in UMD Banach spaces

In this paper we consider conical square functions in the Bessel, Laguerre and Schr\"odinger settings where the functions take values in UMD Banach spaces. Following a recent paper of Hyt\"onen, van Neerven and Portal, in order to define our conical square functions, we use $\gamma$-radonifying operators. We obtain new equivalent norms in the Lebesgue-Bochner spaces $L^p((0,\infty ),\mathbb{B})$ and $L^p(\mathbb{R}^n,\mathbb{B})$, $1<p<\infty$, in terms of our square functions, provided that $\mathbb{B}$ is a UMD Banach space. Our results can be seen as Banach valued versions of known scalar results for square functions

UMD Banach spaces and square functions associated with heat semigroups for Schr\"odinger and Laguerre operators

In this paper we define square functions (also called Littlewood-Paley-Stein functions) associated with heat semigroups for Schr\"odinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) L^p-boundedness properties for the square functions to our Banach valued setting by using \gamma-radonifying operators. We also prove that these L^p-boundedness properties of the square functions actually characterize the Banach spaces having the UMD property

Solutions of Weinstein equations representable by Bessel Poisson integrals of BMO functions

We consider the Weinstein type equation $\mathcal{L}_\lambda u=0$ on $(0,\infty )\times (0,\infty )$, where $\mathcal{L}_\lambda=\partial _t^2+\partial _x^2-\frac{\lambda (\lambda -1)}{x^2}$, with $\lambda >1$. In this paper we characterize the solutions of $\mathcal{L}_\lambda u=0$ on $(0,\infty )\times(0,\infty )$ representable by Bessel-Poisson integrals of BMO-functions as those ones satisfying certain Carleson properties

BMO functions and Balayage of Carleson measures in the Bessel setting

By $BMO_o(R)$ we denote the space consisting of all those odd and bounded mean oscillation functions on R. In this paper we characterize the functions in $BMO_o(R)$ with bounded support as those ones that can be written as a sum of a bounded function on $(0,\infty )$ plus the balayage of a Carleson measure on $(0,\infty )\times (0,\infty )$ with respect to the Poisson semigroup associated with the Bessel operator $B_\lambda =-x^{-\lambda }Dx^{2\lambda }Dx^{-\lambda}$, $\lambda >0$. This result can be seen as an extension to Bessel setting of a classical result due to Carleson

Multiple Paleozoic magmatic-orogenic events in the Central Extremadura batholith (Iberian Variscan belt, Spain)

Background The Central Extremadura batholith located in the southeast part of the Central Iberian Zone (e.g. Iberian Autochthonous domain of the Iberian Variscan belt) was originally thought to have been generated entirely during Carboniferous igneous activity. However, some recent geochronological work has shown the existence of Ordovician plutonic rocks. Purpose The aim of this study is to re-examine the age of granitic rocks in the Central Extremadura batholith and complement this information with new field and geochemical data. This data set is used: to constrain the relative timing of plutons emplacement, as well as deformation and metamorphism preserved in the host rocks; to track deep crustal rocks and granitic magma sources; and to discuss prevailing tectonic evolutionary models for the Paleozoic evolution of the Iberian Variscan belt. Methods We use geochemical and SHRIMP U-Pb zircon geochronology data of three granitic plutons (Ruanes, Plasenzuela and Albalá) from the Central Extremadura batholith to track magmatic sources and provide a better understanding of temporal and spatial relationships between deformation and magmatism in the Iberian Variscan belt. Results Ruanes tonalite dated at 464 ± 2 Ma is peraluminous, magnesian and calc-alkaline, as typical of a magmatic arc setting. We report, for the first time, the occurrence of a Middle Ordovician intrusion spatially and temporally related to host deformed rocks of the Central Iberian Zone (e.g. the Iberian Autochthonous domain), which reached high-grade metamorphic conditions. Plasenzuela two-mica leucogranite is strongly peraluminous and of anatectic origin and includes a Neoproterozoic and Ordovician population of inherited zircon grains. This granite possibly derived from the partial melting of a crustal source composed of Neoproterozoic metapelites and metagreywackes (Schist-Greywacke Complex) and/or Lower Ordovician gneisses (Ollo de Sapo Formation), both including greywackes of volcano-sedimentary origin and peraluminous composition. The crystallization age of 330 ± 7 Ma obtained for the syn-kinematic Plasenzuela two-mica leucogranite constrains the functioning of D2 dextral strike-slip shear zones within the Iberian Autochthonous domain. The age of 309 ± 2 Ma obtained for the Albalá cordierite-bearing monzogranite matches the age interval of the calc-alkaline magmatic suite post-dating the main Variscan D1–D3 structures in the Iberian Autochthonous domain. Conclusion The new data presented in this study make it possible to recognize multiple Paleozoic magmatic-orogenic events (e.g. Caledonian, Variscan and Cimmerian) in the Central Extremadura batholith. During the Ordovician, the emplacement of intermediate magmas at shallow depths gave rise to extensive metamorphism due to heat transfer to the host rocks. The onset of this Ordovician plutonic–metamorphic complex in the Iberian Autochthonous domain is contemporaneous with the development of an active continental margin probably related to the subduction of the Iapetus–Tornquist Ocean (i.e. the Caledonian orogeny). During the Lower Carboniferous, these D2 strike-slip domains acted as lateral margins of largescale gravitational collapses associated with the SE-direct transport of low-angle extensional shear zones (i.e. the Variscan cycle). The emplacement of Upper Carboniferous arc type granitic rocks is interpreted in the context of the amalgamation of Pangaea and the spatial proximity of Iberia relative to the Eurasian active margin in the course of Paleotethys subduction (i.e. the Cimmerian orogeny)

Square functions in the Hermite setting for functions with values in UMD spaces

In this paper we characterize the Lebesgue Bochner spaces $L^p(\mathbb{R}^n,B)$, $1<p<\infty$, by using Littlewood-Paley $g$-functions in the Hermite setting, provided that $B$ is a UMD Banach space. We use $\gamma$-radonifying operators $\gamma (H,B)$ where $H=L^2((0,\infty),\frac{dt}{t})$. We also characterize the UMD Banach spaces in terms of $L^p(\mathbb{R}^n,B)$-$L^p(\mathbb{R}^n,\gamma (H,B))$ boundedness of Hermite Littlewood-Paley $g$-functions

γ\gamma-radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces

In this paper we study Littlewood-Paley-Stein functions associated with the Poisson semigroup for the Hermite operator on functions with values in a UMD Banach space \B. If we denote by $H$ the Hilbert space L^2((0,\infty),dt/t),\gamma(H,\B) represents the space of $\gamma$-radonifying operators from $H$ into \B. We prove that the Hermite square function defines bounded operators from BMO_\mathcal{L}(\R,\B) (respectively, H^1_\mathcal{L}(\R, \B)) into BMO_\mathcal{L}(\R,\gamma(H,\B)) (respectively, H^1_\mathcal{L}(\R, \gamma(H,\B))), where $BMO_\mathcal{L}$ and $H^1_\mathcal{L}$ denote $BMO$ and Hardy spaces in the Hermite setting. Also, we obtain equivalent norms in BMO_\mathcal{L}(\R, \B) and H^1_\mathcal{L}(\R,\B) by using Littlewood-Paley-Stein functions. As a consequence of our results, we establish new characterizations of the UMD Banach spaces.Comment: 31 page

LpL^p-boundedness properties for the maximal operators for semigroups associated with Bessel and Laguerre operators

In this paper we prove that the generalized (in the sense of Caffarelli and Calder\'on) maximal operators associated with heat semigroups for Bessel and Laguerre operators are weak type (1,1). Our results include other known ones and our proofs are simpler than the ones for the known special cases.Comment: 8 page