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 Psicologia: Reflex？o e Crítica , 2004, DOI: 10.1590/S0102-79722004000200013 Abstract: the paper concerns the learning of additive structures on its way to the multiplicative ones according to the equilibration model and vergnaud's propositions about conceptual fields. it describes the nature and the changes of children's notations produced on tasks of equalization and partition of quantities, and verifies the meaning of those notations in the study of the psychogenetic relationship between those structures. the 12 participants (6,4 to 9,5 years old), attending two state elementary schools from suburbs of two different metropolitan areas, performed in triads the tasks, which alternated moments of practical compositions, notations production and their interpretation. the qualitative microgenetic analysis of videotaped data resulted in notation types and in tendencies of the conceptual transformations shown by the notations. the discussion underlines the nature and the progression of the described notations, and the relevant role of equalization and repartition as inherent schemata to the psychogenetic relationship between the focused structures.
 Psicologia: Teoria e Pesquisa , 2005, DOI: 10.1590/S0102-37722005000200012 Abstract: according to vergnaud's and piaget's proposals, the study aims to describe children's conceptions of partitive division expressed on learning tasks of sharing numerical collections, and to identify the grasp of consciousness levels of the typical relations which define that kind of division. this process is supposed to be enhanced by tasks, which alternate moments of practical sharing of numerical collections, notation production and their interpretation. the microgenetic analysis of videotaped data describes the nature and the changes of six students' solutions (7 to 8 years old) to those tasks. these students were attending a state elementary school and they performed the tasks in triads, during two sessions. results show children's conceptions of partitive division disposed on two hierarchies, one for the practical executions and other for the notations. both hierarchies give support to the identification of the grasp of consciousness levels of schemata and relations inherent to that type of division. the status of the sharing activities on the comprehension of partitive division and the role of the grasp of consciousness of actions in this conceptual learning are discussed.
 Revista chilena de literatura , 2011, DOI: 10.4067/S0718-22952011000100005 Abstract: descripción de las grandezas de la ciudad de santiago de chile is one of the few texts included in the satirical and comic textual corpus of colonial chilean poetry. the authorship is still controversial. recently, some critical editions have been made, although they are not fully acceptable taking into consideration the textual fixation. this paper intends to point out the weaknesses in those modern editions, and consequently the need of a new one. the prospective new edition should follow rigorous philological criteria and a complete annotation to clarify the prolific dialogical games and witticisms present all along the descripción verses.
 Carlos Mata Induráin Revista Chilena de Literatura , 2011, Abstract: La Descripción de las grandezas de la ciudad de Santiago de Chile, uno de los pocos textos del corpus de la poesía satírico-burlesca del Chile colonial, de autoría discutida, ha contado con algunas ediciones recientes que no resultan del todo satisfactorias, especialmente desde el punto de vista de la fijación textual. Este trabajo quiere poner de relieve las carencias de esas ediciones modernas y, por consiguiente, la necesidad de una edición crítica, hecha con criterios filológicos rigurosos, y completada con una anotación amplia que clarifique los abundantes juegos dialógicos y las agudezas varias que recorren los versos de la Descripción. Descripción de las grandezas de la ciudad de Santiago de Chile is one of the few texts included in the satirical and comic textual corpus of colonial Chilean poetry. The authorship is still controversial. Recently, some critical editions have been made, although they are not fully acceptable taking into consideration the textual fixation. This paper intends to point out the weaknesses in those modern editions, and consequently the need of a new one. The prospective new edition should follow rigorous philological criteria and a complete annotation to clarify the prolific dialogical games and witticisms present all along the Descripción verses.
 Fedor Pakhomov Mathematics , 2015, Abstract: Caucal hierarchy is a well-known class of graphs with decidable monadic theories. It were proved by L. Braud and A. Carayol that well-orderings in the hierarchy are the well-orderings with order types less than $\varepsilon_0$. Naturally, every well-ordering from the hierarchy could be considered as a constructive system of ordinal notations. In proof theory constructive systems of ordinal notations with fixed systems of cofinal sequences are used for the purposes of classification of provable recursive functions of theories. We show that any well-ordering from the hierarchy could be extended by a monadically definable system of cofinal sequences with Bachmann property. We show that the growth speed of functions from fast-growing hierarchy based on constructive ordinal notations from Caucal hierarchy may be only slightly influenced by the choice of monadically definable systems of cofinal sequences. We show that for ordinals less than $\omega^\omega$ a fast-growing hierarchy based on any system of ordinal notations from Caucal hierarchy coincides with L\"ob-Wainer hierarchy.
 Computer Science , 2009, Abstract: This paper concerns instruction sequences that contain probabilistic instructions, i.e. instructions that are themselves probabilistic by nature. We propose several kinds of probabilistic instructions, provide an informal operational meaning for each of them, and discuss related work. On purpose, we refrain from providing an ad hoc formal meaning for the proposed kinds of instructions. We also discuss the approach of projection semantics, which was introduced in earlier work on instruction sequences, in the light of probabilistic instruction sequences.
 Jeremy E. Cohen Computer Science , 2015, Abstract: This paper gives an overview of notations used in multiway array processing. We redefine the vectorization and matricization operators to comply with some properties of the Kronecker product. The tensor product and Kronecker product are also represented with two different symbols, and it is shown how these notations lead to clearer expressions for multiway array operations. Finally, the paper recalls the useful yet widely unknown properties of the array normal law with suggested notations.
 Paul Libbrecht Computer Science , 2010, Abstract: Mathematical notations around the world are diverse. Not as much as requiring computing machines' makers to adapt to each culture, but as much as to disorient a person landing on a web-page with a text in mathematics. In order to understand better this diversity, we are building a census of notations: it should allow any content creator or mathematician to grasp which mathematical notation is used in which language and culture. The census is built collaboratively, collected in pages with a given semantic and presenting observations of the widespread notations being used in existing materials by a graphical extract. We contend that our approach should dissipate the fallacies found here and there about the notations in "other cultures" so that a better understanding of the cultures can be realized. The exploitation of the census in the math-bridge project is also presented: this project aims at taking learners "where they are in their math-knowledge" and bring them to a level ready to start engineering studies. The census serves as definitive reference for the transformation elements that generate the rendering of formul{\ae} in web-browsers.
 Lecturas de Economía , 2005, Abstract: decisions on allocation of scarce resources constitute one of the elements that define the nature of economy. these decisions related to the scarcity problem, appear in individual and social contexts and generate rationing problems which are faced by the modern economic analysis from two perspectives: the first one is given by the cooperative theory of the games and the second one is the axiomatic study of the various proposed solutions. the latter is used in this article with the purpose of finding solutions to bankruptcy situations, which are simple rationing problems with a remarkable variety of interpretations and a precise mathematical formulation. to accomplish this, we take into account a comparative analysis of four classic rules (proportional, equality, thalmud and equal loss) that enables finding a solution by means of applying different equality criteria. the election of one or another criterion shall depend, generally, on the specific type of problem.
 Antonio Villar Lecturas de Economía , 2005, Abstract: Las decisiones de asignación de recursos escasos constituyen uno de los elementos que definen la naturaleza de la Economía. Estas decisiones, asociadas al problema de la escasez, aparecen tanto en contextos individuales como sociales y generan problemas de racionamiento, los cuales han sido abordados por el análisis económico moderno desde dos enfoques: el primero, es proporcionado por la teoría cooperativa de los juegos y el segundo, es el estudio axiomático de las diversas soluciones propuestas. El último enfoque se emplea en este artículo con el objetivo de dar solución a situaciones de bancarrota, las cuales son un tipo de problemas de racionamiento sencillo con una notable variedad de interpretaciones y una formulación matemática precisa. Para esto, se aborda el análisis comparativo de cuatro reglas clásicas (regla proporcional, igualitaria, del Talmud y de igual pérdida) que permiten encontrar una solución aplicando diferentes criterios de equidad. La elección de uno u otro criterio dependerá, por lo general, del tipo concreto de problema de que se trate.
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