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.     

Heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle

We investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction-advection-diffusion equations with periodic boundary conditions. Using zero number properties of the solutions and the phase shift equivariance of the equation, we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves.     

Notiz über Parachloraldehyd

Ohne Zusammenfassung     

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.     

Synthesis and chiroptical properties of two new planar-chiral macrocycles

Could simple intraannular-arm macrocyclic systems exist in enantiopure stable forms? The effective synthesis of two representative compounds of such a class, their resolution into enantiomers, and experiments justifying their stability toward racemization are presented.     

Selective Propargylation of Carbonyl Compounds and Imines with Barium Reagents

A Barbier‐type regioselective propargylation of aldehydes and ketones with (3‐bromobut‐1‐ynyl)trimethylsilane has been achieved using reactive barium as a low‐valent metal in THF. Especially in the case of ketones, the corresponding homopropargylic alcohols form almost exclusively. In the reaction of α,β‐unsaturated carbonyl compounds, only 1,2‐adducts have been observed. This method is also applicable to propargylation of imines, and the corresponding homopropargylic amines are obtained regiospecifically in good yields with diastereomeric ratios of up to 87:13.     

Synthesis and Characterization of Silica@Copper Core–Shell Nanoparticles: Application for Conjugate Addition Reactions

Silica@copper (SiO₂@Cu) core–shell nanoparticles were synthesized and well characterized by XRD, TEM, AFM, XPS, UV/Vis, TGA–MS, and ICP–AES techniques. The synthesized SiO₂@Cu core–shell nanoparticles were employed as catalysts for the conjugate addition of amines to α,β‐unsaturated compounds in water to obtain β‐amino carbonyl compounds in excellent yields in shorter reaction times. Furthermore, the catalyst works well for hetero‐Michael addition reactions of heteroatom nucleophiles such as thiols to α,β‐unsaturated compounds. As the reaction is performed in water, it allows for easy recycling of the catalyst with consistent activity.