Global Bounds for the Generalized Jensen Functional with Applications

In this article we give sharp global bounds for the generalized Jensen functional Jn(g,h;p,x). In particular, exact bounds are determined for the generalized power mean in terms from the class of Stolarsky means. As a consequence, we obtain the best possible global converses of quotients and differences of the generalized arithmetic, geometric and harmonic means.

As mudanças geradas pela Base Nacional Comum Curricular na abordagem do pensamento algébrico nos anos iniciais do Ensino Fundamental

ResumoEste artigo trata de uma pesquisa bibliográfica que tem por objetivo analisar dissertações e teses que investigaram o pensamento algébrico nos anos iniciais do Ensino Fundamental. O referencial teórico salienta um modelo dominante de ensino da Álgebra reduzido a uma Aritmética generalizada no Ensino Fundamental, que será bruscamente algebrizado no Ensino Médio. E na atual proposta da BNCC temos a algebrização do currículo desde os anos iniciais. Foram adotadas algumas palavras chave relacionadas ao pensamento algébrico para o levantamento dos trabalhos no banco de teses e dissertações da Coordenação de Aperfeiçoamento de Pessoal do Nível Superior (Capes). Em relação ao tema pensamento algébrico nos anos iniciais do Ensino Fundamental, foram encontrados onze trabalhos sobre sequências didáticas ou a produção escrita dos alunos e dois trabalhos sobre o professor dos anos iniciais. E podemos dizer também que nenhum dos estudos está relacionado à análise de materiais didáticos e existe, nesse levantamento, apenas um trabalho envolvendo a BNCC. Palavras-chave: Pensamento algébrico; Anos Iniciais; BNCC.AbstractThis article refers to a bibliographic research that aims to analyze dissertations and thesis that investigated algebraic reasoning in the early years of Elementary School. The theoretical framework highlights a dominant model of teaching Algebra reduced to a generalized Arithmetic in Elementary School, which will be sharply algebraized in High School. And the current proposal of the Base Nacional Comum Curricular (BNCC) (National Common Curricular Base) present an algebraized curriculum since the early years. Some keywords related to algebraic reasoning were adopted to survey the work in the thesis and dissertation database of the Coordenação de Aperfeiçoamento de Pessoal do Nível Superior (Capes) (Coordination for the Improvement of Higher Education Personnel). Regarding the theme algebraic reasoning in the early years of Elementary School, eleven works were found on didactic sequences or the student’s' written production and two works on the teacher of the early years. And we can also say that none of the studies is related to the analysis of teaching materials and there is, in this survey, only one work involving BNCC.Keywords: Algebraic reasoning; Elementary Education; BNCC.

Hydrotropic Pulping of Miscanthus to Obtain Pulp

The hydrotropic pulping of crushed straw of Miscanthus sacchariflorus Andersson to produce pulp was studied herein. A concentrated sodium benzoate solution was used as the reagent. Established regularities of the pulping temperature and time effects on delignification quality attributes such as pulp yield and residual lignin content in the pulp. The generalized arithmetic-mean optimization parameter enabled hydrotropic pulping optimum conditions to be identified to obtain hard pulp fit. Experimentally displayed the possibility of using cellulose for the manufacture of special paper grades

Forbidden arithmetic progressions in permutations of subsets of the integers

Permutations of the positive integers avoiding arithmetic progressions of length $5$ were constructed in (Davis et al, 1977), implying the existence of permutations of the integers avoiding arithmetic progressions of length $7$. We construct a permutation of the integers avoiding arithmetic progressions of length $6$. We also prove a lower bound of $\frac{1}{2}$ on the lower density of subsets of positive integers that can be permuted to avoid arithmetic progressions of length $4$, sharpening the lower bound of $\frac{1}{3}$ from (LeSaulnier and Vijay, 2011). In addition, we generalize several results about forbidden arithmetic progressions to construct permutations avoiding generalized arithmetic progressions.