Counter-examples for refinement of conjectures and proofs in primary school mathematics

The purpose of this study is to explore how primary school students reexamine their conjectures and proofs when they confront counter-examples to the conjectures they have proved. In the case study, a pair of Japanese fifth graders thought that they had proved their primitive conjecture with manipulative objects (that is, they constructed an action proof), and then the author presented a counter-example to them. Confronting the counter-example functioned as a driving force for them to refine their conjectures and proofs. They understood the reason why their conjecture was false through their analysis of its proof and therefore could modify their primitive conjecture. They also identified the part of the proof which was applicable to the counter-example. This identification and their action proof were essential for their invention of a more comprehensive conjecture.     

Rethinking the discovery function of proof within the context of proofs and refutations

Proof and proving are important components of school mathematics and have multiple functions in mathematical practice. Among these functions of proof, this paper focuses on the discovery function that refers to invention of a new statement or conjecture by reflecting on or utilizing a constructed proof. Based on two cases in which eighth and ninth graders engaged in proofs and refutations, we demonstrate that facing a counterexample of a primitive statement can become a starting point of students’ activity for discovery, and that a proof of the primitive statement can function as a useful tool for inventing a new conjecture that holds for the counterexample. An implication for developing tasks by which students can experience this discovery function is mentioned.     

Influence of the vacuum level upon the growth of carbon nanotubes on silicon carbide surface

We have investigated the influence of the vacuum level upon the growth of carbon nanotubes (CNTs) on 6H-SiC () surface.CNTs of about 160 nm in length were formed densely and uniformly on the 6H-SiC surface during annealing at 1700 °C in a high vacuum (∼10⁻² Pa). CNTs of about 1 μm in length were formed during annealing at 1700 °C in an ultra-high vacuum (∼10⁻⁷ Pa). However, CNTs were not formed and SiO₂ layers were formed on the SiC surface at 1700 °C in air. It is found that longer CNTs can grow up in an ultra-high vacuum, moreover, a little aligned and low-density graphite layers, or carbon nanofibers can also grow up.     

Direct through anionic,cationic, and radical active species: Terminal carbon–halogen bond for “controlled”/living polymerizations of styrene

In this work, we examined the synthesis of novel block (co)polymers by mechanistic transformation through anionic, cationic, and radical living polymerizations using terminal carbon–halogen bond as the dormant species. First, the direct halogenation of growing species in the living anionic polymerization of styrene was examined with CCl₄ to form a carbon–halogen terminal, which can be employed as the dormant species for either living cationic or radical polymerization. The mechanistic transformation was then performed from living anionic polymerization into living cationic or radical polymerization using the obtained polymers as the macroinitiator with the SnCl₄/n‐Bu₄NCl or RuCp^*Cl(PPh₃)/Et₃N initiating system, respectively. Finally, the combination of all the polymerizations allowed the synthesis block copolymers including unprecedented gradient block copolymers composed of styrene and p‐methylstyrene. © 2018 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019 , 57, 465–473     

Formal total synthesis of fostriecin via 1,4-asymmetric induction using cobalt-alkyne complex

The synthesis of a protected dephosphofostriecin, and thereby a formal synthesis of fostriecin, has been accomplished. Two of the four chiral centers are controlled by an external chiral auxiliary and the other two are synthesized stereoselectively, one by a novel 1,4-asymmetric induction using cobalt-alkyne complex, and the other by 1,3-asymmetric induction.     

Direct construction of bicyclic heterocycles by palladium-catalyzed domino cyclization of propargyl bromides

The palladium-catalyzed domino cyclization of propargyl bromides having two nucleophilic functional groups is described. Treatment of 1,7-diamino-5-bromohept-3-yne derivatives with catalytic Pd(PPh3)4 in the presence of NaH in MeOH gives the 2,7-diazabicyclo[4.3.0]non-5-enes in good yields. Interestingly, the regioselectivity of the reaction is completely controlled by the relative reactivity of the amine functional groups, irrespective of the position of the nucleophiles. The malonate derivative also undergoes domino cyclization to produce a hexahydroindole derivative.     

Single-crystal-to-single-crystal photocyclization of 4-(2,4,6-triisopropylbenzoyl)benzoic acid in the salt crystal with (S)-phenylethylamine

Ultraviolet irradiation of a salt crystal of 4-(2,4,6-triisopropylbenzoyl)benzoic acid with (S)-phenylethylamine promoted single-crystal-to-single-crystal photocyclization to give an enantiopure (R,R)-cyclopentenol and almost racemic cyclobutenol. The reaction paths were elucidated by X-ray crystallographic analysis of the crystal before and after irradiation.     

Fluorescence determination of N-acetylaspartic acid in the rat cerebrum homogenate using high-performance liquid chromatography with pre-column fluorescence derivatization

N-acetyl-L-aspartic acid (NAA) is an endogenous compound, and its brain concentration is suggested to be altered in neurological disorders. In the present study, a fluorescence determination method for NAA was developed by employing reversed-phase high-performance liquid chromatography (HPLC) with pre-column fluorescence derivatization using 4-N,N-dimethylaminosulfonyl-7-N-(2-aminoethyl)amino-2,1,3-benzoxadiazole (DBD-ED). Using methylsuccinic acid as the internal standard, a linear calibration curve for NAA was constructed in the range 125-1000 microM (n=3). The detection limit on the column was approximately 5.0 fmol (signal-to-noise ratio 3). The proposed HPLC method was applied to determine NAA in the rat cerebrum homogenate. Cerebrum NAA was successfully determined using 10 microL of the homogenate, and the validation data for the proposed HPLC method demonstrated satisfactory results. Intra- and inter-day precision and accuracy were within 1.1-7.0 and -8.1-6.3%, respectively. The concentration of NAA in the male rat cerebrum (13 weeks old) was 84+/-4.6 micromol/mg protein (n=3).     

Diffusion of metal complexes inside of silica-surfactant nanochannels within a porous alumina membrane

The present paper describes diffusivities of a series of metal complexes inside of silica-surfactant nanochannels (channel diameter = 3.4 nm), which were formed within a porous alumina membrane by a surfactant-templated method using cetyltrimethylammonium bromide (CTAB) as a template surfactant. The metal complexes used in this study were Fe(CN)6(3-), Ru(NH3)6(3-), ferrocenecarboxylic acid (Fc-COO-), (ferrocenylmethyl)-trimethylammonium (Fc-NMe3+), N,N-(dimethylamminomethyl)ferrocene (Fc-NMe2), and ferrocene methanol (Fc-OH). Apparent diffusion coefficients of these metal complexes were estimated by measuring their mass transports through the silica-surfactant nanochannels. The estimated apparent diffusion coefficients were on the order of 10(-11) cm2 s(-1) for Fe(CN)6(3-) and Ru(NH3)6(3-), and these values were five orders of magnitude smaller than those in a bulk aqueous solution. For the ferrocene derivatives, the apparent diffusion coefficients of charged ferrocene derivatives are almost the same (5.3 x 10(-11) cm2 s(-1) for Fc-COO- and 5.4 x 10(-11) cm2 s(-1) for Fc-NMe3+), whereas neutral ferrocene derivatives (Fc-NMe2 and Fc-OH) show faster diffusion than the charged species. In addition, the apparent diffusion coefficient of Fc-NMe2 (27 x 10(-11) cm2 s(-1)) was about three times larger than that of Fc-OH (10 x 10-11 cm2 s(-1)). The difference in these diffusion coefficients is discussed by considering the mesostructure of the silica-surfactant nanochannels, that is, an ionic interface with cationic head groups of CTA and their counteranions, a hydrophobic interior of the micellar phase, and a silica framework. As a result, it is inferred that the slow diffusivities of the charged metal complexes are due to the electrostatic interaction between the charged species and the ionic interface, whereas less interaction between neutral ferrocenes and the ionic interface causes distribution of metal complexes into the hydrophobic micellar phase, which is a less viscous medium compared to the ionic interface, resulting in the faster diffusivities of the neutral species.