Patterns—a fundamental idea of mathematical thinking and learning

Taking advantage of patterns is typical of our everyday experience as well as our mathematical thinking and learning. For example a working day or a morning at school displays a certain structure, which can be described in terms of patterns. On the one hand regular structures give us the feeling of permanence and enable us to make predictions. On the other hand they also provide a chance to be creative and to vary common procedures. School students usually encounter patterns in math classes either as number patterns or geometric patterns. There are also patterns that teachers can find in analyzing the errors students make during their calculations (error patterns) as well as patterns that are inherent to mathematical problems. One could even go so far as to say that identifying and describing patterns is elementary for mathematics (cf. Devlin 2003). Practising good interacting with patterns supports not only the active learning of mathematics but also a deeper understanding of the world in general. Patterns can be explored, identified, extended, reproduced, compared, varied, represented, described and created. This paper provides some examples of pattern utilization and detailed analyses thereof. These ideas serve as “hooks” to encourage the good use of patterns to facilitate active learning processes in mathematics.     

Estimation consistante des paramètres d'un modèle non linéaire pour des données fonctionnelles discrétisées aléatoirement

We study a class of nonlinear regression models for scalar or vectorial response when the explanatory variable is a function. We introduce a consistent estimator of the parameters of models in this class when functions are evaluated at randomly chosen observation points. To cite this article: F. Rossi, B. Conan-Guez, C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

Studien über Quercetin und seine Derivate

Ohne Zusammenfassung     

Zur Darstellung von Methylendiaminderivaten

Ohne Zusammenfassung     

Notiz über Parachloraldehyd

Ohne Zusammenfassung     

Preparation of novel analgesics via diastereoselective nucleophilic addition to 1-dimethylamino-2-methylpentan-3-one

The diastereoselective synthesis of naphthyl amino alcohols via nucleophilic addition to racemic 1-dimethylamino-2-methylpentan-3-one was studied. The use of the appropriate experimental conditions allowed the synthesis of both diastereoisomers. The relative configurations were established via NOESY experiments.     

A New Two‐Dimensional Coordination Polymer of Mercury(II) with very High Thermal Stability

A 2D Hg^II coordination polymer containing ligands 1,2,4‐triazole (Htrz) and thiocyanate, [Hg(μ₃‐trz)(SCN)]_n ( 1 ) has been synthesized and characterized by elemental analysis and IR spectroscopy. The single‐crystal X‐ray data show the coordination number of Hg atoms is four and the ligand trz^? acts as a three‐fold donor. The thermal stability of compound 1 was studied by thermal gravimetric and differential thermal analyses. The composition and formation of the complex in methanol solution were found to be in support of its solid state structure.     

cos πρ theorems for δ-subharmonic functions

For ϕ a δ-subharmonic function, sharp results are obtained that connectA(r, ϕ), B(r, ϕ) andT(r, ϕ), whereA(r, ϕ)=inf_|z|=r ϕ(z),B(r, ϕ)=sup_|z|=r ϕ(z), andT(r, ϕ) is the Nevanlinna characteristics.