Symmetry and the golden ratio in the analysis of a regular pentagon

The regular pentagon had a symbolic meaning in the Pythagorean and Platonic philosophies and a subsequent important role in Western thought, appearing also in arts and architecture. A property of regular pentagons, which was probably discovered by the Pythagoreans, is that the ratio between the diagonal and the side of these pentagons is equal to the golden ratio. Here, we will study some relations existing between a regular pentagon and this ratio. First, we will focus on the group of fivefold rotational symmetry, to find the position in the complex plane of the vertices of this geometric figure. Then, we will propose an analytic method to solve the same problem based on the Cartesian coordinates, a method where we find the golden ratio without any specific geometric consideration. This study shows a comparison of the use of complex numbers, symmetries and analytic methods, applied to a subject which can be interesting for general education in mathematics. In fact, the proposed approach can convey and link several concepts, requiring only a general pre-college education, showing at the same time the richness that mathematics can offer in solving geometric problems.     

Solving Internal Habit Formation Models Through Dynamic Programming in Infinite Dimension

In this paper, we study an economic model, where internal habits play a role. Their formation is described by a more general functional form than is usually assumed in the literature, because a finite memory effect is allowed. Indeed, the problem becomes the optimal control of a standard ordinary differential equation, with the past of the control entering both the objective function and an inequality constraint. Therefore, the problem is intrinsically infinite dimensional. To solve this model, we apply the dynamic programming approach and we find an explicit solution for the associated Hamilton–Jacobi–Bellman equation, which lets us write the optimal strategies in feedback form. Therefore, we contribute to the existing literature in two ways. Firstly, we fully develop the dynamic programming approach to a type of problem not studied in previous contributions. Secondly, we use this result to unveil the global dynamics of an economy characterized by generic internal habits.     

Exceptional ergodic directions in Eaton lenses

We construct examples of ergodic vortical flows in periodic configurations of Eaton lenses of fixed radius. We achieve this by studying a family of infinite translation surfaces that are ?²-covers of slit tori. We show that the Hausdorff dimension of lattices for which the vertical flow is ergodic is bigger than 3/2. Moreover, the lattices are explicitly constructed.     

A practical and expeditious method for the preparation of the potential anticancer agent 5,6,11,12,17,18,23,24-octahydrocyclododeca[1,2-b:4,5-b′:7,8-b″:10,11-b?]tetraindole (CTet)

A new synthetic method to obtain the potential anticancer agent 5,6,11,12,17,18,23,24-octahydrocyclododeca[1,2-b:4,5-b′:7,8-b′′:10,11-b′′′]tetraindole (CTet), starting from 1H-indole-3-carboxaldehyde and sulfamide, is described. Although a mixture of CTet and cyclic indole trimer (CTr) is formed, higher CTet/CTr ratio (4:6) and CTet yield (15%) make our protocol more favorable than those reported in the literature. A discussion on the possible reaction mechanism is reported.     

New monoterpene-derived phosphinopyridine ligands and their application in the enantioselective iridium-catalyzed hydrogenation

Pyridine derivatives with a phosphine or phosphinite pendant (1-11) have been synthesized from (+)-α-pinene, (−)-isopinocampheol, and/or (+)-camphor via Kröhnke annulation or another annulation method as the key step for the construction of the pyridine nucleus. The iridium complex of 6 proved to catalyze hydrogenation of the prochiral unfunctionalized alkene 44 with 94% ee, whereas the complex of 2 was most efficient in the hydrogenation of the cinnamyl-type ester 45 (83% ee).     

Synthesis, characterisation and cytotoxicity of polyoxometalate/carboxymethyl chitosan nanocomposites

Chitosan and its derivates continue to attract considerable research interest as effective drug carriers with good biocompatibility and high cellular uptake rates. We used these versatile features to tap the considerable biomedical potential of polyoxometalates (POMs) through their encapsulation into a carboxymethyl chitosan (CMC) matrix. The nanocapsules were prepared by ionic gelification with Ca(2+); their size distribution ranges from 60 to 150 nm. Because [Co(4)(H(2)O)(2)(PW(9)O(34))(2)](10-) is well known for its manifold properties, such as antiviral activity, it was selected as a model POM. The resulting composites were characterised with a wide range of analytical methods, which pointed to quantitative encapsulation of intact POMs within the CMC matrix. We studied the biocompatibility of the POM/CMC nanocomposites on HeLa cells through MTT and proliferation assays. Even after prolonged incubation times at high concentrations, the composites did not display cytotoxicity, thereby drastically reducing the side effects of the pristine POMs. This opens up new avenues for designing novel inorganic drug prototypes from bioactive POMs.     

A macrocyclic ligand as receptor and Zn(II)-complex receptor for anions in water: binding properties and crystal structures

Binding properties of 24,29-dimethyl-6,7,15,16-tetraoxotetracyclo[19.5.5.0(5,8).0(14,17)]-1,4,9,13,18,21,24,29-octaazaenatriaconta-Δ(5,8),Δ(14,17)-diene ligand L towards Zn(II) and anions, such as the halide series and inorganic oxoanions (phosphate (Pi), sulfate, pyrophosphate (PPi), and others), were investigated in aqueous solution; in addition, the Zn(II)/L system was tested as a metal-ion-based receptor for the halide series. Ligand L is a cryptand receptor incorporating two squaramide functions in an over-structured chain that connects two opposite nitrogen atoms of the Me(2)[12]aneN(4) polyaza macrocyclic base. It binds Zn(II) to form mononuclear species in which the metal ion, coordinated by the Me(2)[12]aneN(4) moiety, lodges inside the three-dimensional cavity. Zn(II)-containing species are able to bind chloride and fluoride at the physiologically important pH value of 7.4; the anion is coordinated to the metal center but the squaramide units play the key role in stabilizing the anion through a hydrogen-bonding network; two crystal structures reported here clearly show this aspect. Free L is able to bind fluoride, chloride, bromide, sulfate, Pi, and PPi in aqueous solution. The halides are bound at acidic pH, whereas the oxoanions are bound in a wide range of pH values ranging from acidic to basic. The cryptand cavity, abundant in hydrogen-bonding sites at all pH values, allows excellent selectivity towards Pi to be achieved mainly at physiological pH 7.4. By joining amine and squaramide moieties and using this preorganized topology, it was possible, with preservation of the solubility of the receptor, to achieve a very wide pH range in which oxoanions can be bound. The good selectivity towards Pi allows its discrimination in a manner not easily obtainable with nonmetallic systems in aqueous environment.