Two truncated identities of Gauss

Two new expansions for partial sums of Gauss' triangular and square numbers series are given. As a consequence, we derive a family of inequalities for the overpartition function $\bar{p}(n)$ and for the partition function $p_1(n)$ counting the partitions of $n$ with distinct odd parts. Some further inequalities for variations of partition function are proposed as conjectures.Comment: 9 pages, final versio

Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$\sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{\Phi_n(q)},$$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for $a,m\geq 1$, the first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to appear in Adv. Appl. Mat

Some Arithmetic Properties of the q-Euler Numbers and q-Sali\'e Numbers

For m>n\geq 0 and 1\leq d\leq m, it is shown that the q-Euler number E_{2m}(q) is congruent to q^{m-n}E_{2n}(q) mod (1+q^d) if and only if m\equiv n mod d. The q-Sali\'e number S_{2n}(q) is shown to be divisible by (1+q^{2r+1})^{\left\lfloor \frac{n}{2r+1}\right\rfloor} for any r\geq 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.Comment: 12 pages, see also http://math.univ-lyon1.fr/~gu

Multiple extensions of a finite Euler's pentagonal number theorem and the Lucas formulas

Motivated by the resemblance of a multivariate series identity and a finite analogue of Euler's pentagonal number theorem, we study multiple extensions of the latter formula. In a different direction we derive a common extension of this multivariate series identity and two formulas of Lucas. Finally we give a combinatorial proof of Lucas' formulas.Comment: 11 pages, to appear in Discrete Mathematics. See also http://math.univ-lyon1.fr/~gu

School-Related Stress and Depression in Adolescents With and Without Learning Disabilities: An Exploratory Study

This study examined school-related stress and depression in adolescents with and without learning disabilities. A total of 87 students (38 learning-disabled and 49 nondisabled) from secondary schools in Calgary completed questionnaires on depressive symptoms and on school-related stress. Results indicated that the adolescents with LD reported significantly higher levels of academic self-concept stress than their NonLD peers. However, the groups did not differ significantly on depression or on the other areas of school-related stress. Significant and positive correlations between school-related stress and depression were found, and the stress variables were found to be significant predictors of adolescent depression. Practical implications of the findings for parents and educators are discussed.Ce projet de recherche porte sur le stress et la dépression liés à l’école chez 87 adolescents, élèves au secondaire à Calgary. Parmi eux, 38 étaient atteints de troubles d’apprentissage et 49 ne l’étaient pas. Les élèves ont répondu à des questionnaires sur les symptômes de dépression et le stress lié à l’école. D’après les résultats, le niveau de stress lié à l’image de soi sur le plan académique était beaucoup plus élevé chez les adolescents atteints de troubles d’apprentissage que chez les adolescents qui ne l’étaient pas. Toutefois, il n’y avait pas de différence significative entre les groupes quant à la dépression ou d’autres stress liés à l’école. Les résultats ont révélé des corrélations significatives et positives entre le stress lié à l’école et la dépression. De plus, les chercheurs ont trouvé que les variables pour le stress étaient des prédicteurs significatifs de dépression chez les adolescents. Les auteurs discutent de conséquences pratiques pour les parents et les enseignants

Bijective Proofs of Gould's and Rothe's Identities

We first give a bijective proof of Gould's identity in the model of binary words. Then we deduce Rothe's identity from Gould's identity again by a bijection, which also leads to a double-sum extension of the $q$-Chu-Vandermonde formula.Comment: 4 page

Factors of binomial sums from the Catalan triangle

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers $n_1, ..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, the expression $$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer. Moreover, several related conjectures are proposed.Comment: 15 pages, final versio

Los dos saberes. La sinergia entre los saberes científicos y locales. Un diálogo entre técnicos agropecuarios y productores para mejorar la extensión e investigación en Guatemala

Este proyecto fue motivado por las ideas presentadas a continuación: En su afán de apoyar procesos de desarrollo, la Cooperación Internacional a menudo apoya programas simplistas que basan sus propuestas unilateralmente o en el conocimiento científico/tecnológico (exógeno) o en el conocimiento local (autóctono o endógeno). Estos programas no solamente tienen sesgos fuertes a favor de una de las fuentes de conocimiento sino, implícita o explícitamente, descalifica la otra fuente. En demasiadas instancias, se encuentra en las comunidades rurales proyectos respaldos por diferentes organizaciones (gubernamentales, privadas, ONGs) con agendas diferentes y mensajes profundamente contradictorios. Los perdedores son inevitablemente los supuestos beneficiarios. Esta situación es común en el desarrollo rural y agropecuario, en salud pública y medicina, y el desarrollo económico local. Se puede plantear la hipótesis que para el desarrollo agropecuario y rural es imprescindible usar creativa, eficiente y sinérgicamente las dos clases de conocimiento: los derivados de la ciencia moderna y los saberes locales, que incluyen los conocimientos tradicionales y de origen reciente de los grupos campesinos, indígenas y afrodescendientes. Los programas convencionales de extensión agrícola muchas veces presuponen que la ciencia y tecnología moderna son mejores que la no científica; mientras que otra tendencia creciente considera que lo tradicional es bueno y lo nuevo es indeseable. Estos dos modelos son encontrados en el sentido que muchas veces no admiten evidencias contrarias

Ex Situ and In Situ Ellipsometric Studies of the Thermal Oxide on InP

The thermally grown InP oxide as etched by an aqueous dilute HF solution has been studied by ellipsometric techniques. The ex-situ measurement reveals a two-layer structure for the oxide grown at 440°C. The refractive indices for both oxide layers have been determined using a two-layer optical model. The etching process has also been monitored ellipsometrically in the real etching environment, in-situ. A fused silica cell, which enables the windows to be aligned properly, has been specifically designed for the in-situ solution measurement. A liquid layer at the solutionoxide interface has been identified, and the layer is shown to contain P and In species resulting from the etching reactions. A theory based on the Lorentz-Lorenz relation results in a reasonable qualitative description of the liquid layer. During the etching of the oxide the liquid layer shrinks at a linear rate, and after removal of the outer oxide layer the liquid layer forms a dense electric double layer

Cropping system effects on soil quality in the Great Plains: Synthesis from a regional project

Soils perform a number of essential functions affecting management goals. Soil functions were assessed by measuring physical, chemical, and biological properties in a regional assessment of conventional (CON) and alternative (ALT) management practices at eight sites within the Great Plains. The results, reported in accompanying papers, provide excellent data for assessing how management practices collectively affect agronomic and environmental soil functions that benefit both farmers and society. Our objective was to use the regional data as an input for two new assessment tools to evaluate their potential and sensitivity for detecting differences (aggradation or degradation) in management systems. The soil management assessment framework (SMAF) and the agro-ecosystem performance assessment tool (AEPAT) were used to score individual soil properties at each location relative to expected conditions based on inherent soil-forming factors and to compute index values that provide an overall assessment of the agronomic and environmental impact of the CON and ALT practices. SMAF index values were positively correlated with grain yield (an agronomic function) and total organic matter (an agronomic and environmental function). They were negatively correlated with soil nitrate concentration at harvest (an indicator of environmental function). There was general agreement between the two assessment tools when used to compare management practices. Users can measure a small number of soil properties and use one of these tools to easily assess the effectiveness of soil management practices. A higher score in either tool identifies more environmentally and agronomically sustainable management. Temporal variability in measured indicators makes dynamic assessments of management practices essential. Water-filled pore space, aggregate stability, particulate organic matter, and microbial biomass were sensitive to management and should be included in studies aimed at improving soil management. Reductions in both tillage and fallow combined with crop rotation has resulted in improved soil function (e.g., nutrient cycling, organic C content, and productivity) throughout the Great Plains