A q-linear analogue of the plane wave expansion

We obtain a q-linear analogue of Gegenbauer?s expansion of the plane wave. It is expanded in terms of the little q-Gegenbauer polynomials and the third Jackson q-Bessel function. The result is obtained by using a method based on bilinear biorthogonal expansions.     

An essential tension in mathematics education

There is a problem that goes through the history of calculus: the tension between the intuitive and the formal. Calculus continues to be taught as if it were natural to introduce the study of change and accumulation by means of the formalized ideas and concepts known as the mathematics of ε and δ. It is frequently considered as a failure that “students still seem to conceptualize limits via the imagination of motion.” These kinds of assertions show the tension, the rift created by traditional education between students’ intuitions and a misdirected formalization. In fact, I believe that the internal connections of the intuition of change and accumulation are not correctly translated into that arithmetical approach of ε and δ. There are other routes to formalization which cohere with these intuitions, and those are the ones discussed in this paper. My departing point is epistemic and once this discussion is put forward, I produce a narrative of classroom work, giving a special place to local conceptual organizations.     

An optimal replenishment policy for an EOQ model with partial backlogging

We study an inventory system where demand on the stockout period is partially backlogged. The backlogged demand ratio is a mixture of two exponential functions. The shortage cost has two significant costs: the unit backorder cost (which includes a fixed cost and a cost proportional to the length of time for which the backorder exists) and the cost of lost sales. A general procedure to determine the optimal policy and the minimum inventory cost for all the parameter values is developed. This model generalizes several inventory systems analyzed by different authors. Numerical examples are used to illustrate the theoretical results.     

An unexpected Pummerer rearrangement in the synthetic route to ethyl (2′-hydroxy-4′,5′-methylenedioxyphenyl)acetate: An alternative approach to 2,3-dimethylthio benzofurans

The synthesis of ethyl (2′-hydroxy-4′,5′-methylendioxophenyl)acetate, a fragment of the antihyperglycemic natural coumarin subcoriacin, is reported. We found an expeditious route to the title compound in five steps. Final metal catalyzed acid ethanolysis of the vinylic 1,1-methylthio methylsulfoxide derivative afforded the required aryl acetic ester, but in the absence of metal catalyst, an unexpected Pummerer rearrangement produced the 2,3-dimethylthiofuran derivative as the major product. This last result provides an alternative entry to 2,3-dimethlythiobenzofurans.     

Electrochemical and electron spin resonance studies of selected benzazolo[3,2-a]quinolinium salts

The electrochemical reduction of the chloride or perchlorate salts of benzazolo[3,2-a]quinolinium ion and several of its analogues is reported. The compounds studied are the perchlorate salt of 3-nitrobenzothiazolo-and 3-nitro-9-methoxybenzothiazolo[3,2-a]quinolinium, and the chloride salts of 7-ethyl-, 3-nitro-7-methyl-, 3-nitro-7-ethyl-, 3-nitro-7-isopropyl-, 3-nitro-7-butyl- and 3-nitro-7-benzylbenzimidazolo[3,2-a]-quinolinium, respectively. Cyclic voltammetry of the corresponding 3-nitrobenzothiazolo[3,2-a]quinolinium derivatives in N,N-dimethylformamide shows an irreversible peak potential at -0.6 and a quasi-reversible peak at -(1.2–1.3) volts, respectively, relative to the standard calomel electrode. In contrast, the corresponding 3-nitrobenzimidazolo[3,2-a]quinolinium derivatives show, in general, reversible peaks at near -0.8 and -(1.2–1.4) volts, respectively. Upon electrolytic reduction, only the nitro-substituted derivatives produced observable electron paramagnetic resonance electron spin resonance spectra. This observation is explained in terms of the stabilization of the radicals produced by the nitro group. Theoretical MM+/AM1/UHF calculations support the idea that the larger nitrogen splitting is caused by N-12 and the minor splittings by N-7 in the benzimidazolo[3,2-a]quinolinium ion series.     

Characterization of an all Sol-Gel Electrochromic Device WO3/Ormolyte/CeO2-TiO2

Electrochemical lithium intercalation in thin films of CeO₂-TiO₂ and WO₃, prepared by the sol-gel technique was investigated with cyclic voltammetry and spectroelectrochemical techniques in propylene carbonate solutions. A solid state system having the configuration WO₃/Ormolyte/CeO₂-TiO₂ has been assembled. The solid electrolyte, an organically modified electrolyte (ormolyte), was prepared with different [O]/[Li] ratios. The transmittance variation of this system during a potentiostatic step from –0.7 V to 0.8 V was about 35% at 550 nm.     

Cloud point extraction for cobalt preconcentration with on-line phase separation in a knotted reactor followed by ETAAS determination in drinking waters

A novel method for cobalt preconcentration by cloud point extraction with on-line phase separation in a PTFE knotted reactor and further determination by electrothermal atomic absorption spectrometry (ETAAS) is proposed. The cloud point system was formed in the presence of non-ionic micelles of polyethyleneglycolmono-p-nonylphenylether (PONPE 7.5) and it was retained on the inner walls of a knotted reactor (KR). The surfactant rich-phase was removed from the knotted reactor with 75 microL of methanol acidified with 0.8 mol L(-1) nitric acid, directly into the dosing hole of the L'Vov graphite tube. An enrichment factor of 15 was obtained with a preconcentration time of 60 s, with respect to the direct determination of cobalt by ETAAS in aqueous solutions. The value of the detection limit for the preconcentration of 5 mL of sample solution was 10 ng L(-1). The precision, expressed as the relative standard deviation (R.S.D.), for 10 replicate determinations at 0.5 microg L(-1) Co level was 4.5%. Verification of the accuracy was carried out by analysis of a standard reference material (NIST SRM 1640e "Trace elements in natural water"). The method was successfully applied to the determination of cobalt in drinking water samples.     

Continuum percolation of long lifespan clusters in a simple fluid

We present results on the percolation loci for chemical clusters and physical clusters of long lifespan. Chemical clusters are defined as sets of particles connected through particle-particle bonds that last for a given time tau. Physical clusters are sets of particles that remain close together at every instant for a given period of time tau. By using molecular dynamics simulations of a Lennard-Jones system we obtain the percolation loci at different values of tau as the lines in the temperature-density plane at which the system presents a spanning cluster in 50% of the configurations. We find that the percolation loci for chemical clusters shifts rapidly toward high densities as tau is increased. For moderate values of tau this line converges to the low-density branch of the liquid-solid coexistence curve. This implies that no stable chemical clusters can be found in the fluid phase. In contrast, the percolation loci for physical clusters tend to a limiting line, as tau tends to infinity, which is far from the liquid-solid transition line.     

An understanding of the electrophilic/nucleophilic behavior of electro-deficient 2,3-disubstituted 1,3-butadienes in polar diels-alder reactions. A density functional theory study

The electrophilic/nucleophilic behavior of dimethyl 2,3-dimethylenesuccinate 1, an electron-deficient 2,3-disubstituted 1,3-butadiene, in polar Diels-Alder reactions has been studied using DFT methods at the B3LYP/6-31G(d) level of theory. The electronic nature of bonding of the transition structures involved in the cycloaddition reactions of the diene 1 toward the nucleophilically activated dienophile 6 and the strong electrophilically activated dienophile 7 has been carefully examined within the natural bond orbital (NBO) and the topological analysis of the electron localization function (ELF) frameworks. Additionally, a study of the global electrophilicity pattern of the reagents at the ground state was performed. This evidence allows us to rationalize the participation of electron-deficient 2-susbtituted and 2,3-disubstituted 1,3-butadienes as nucleophiles in polar Diels-Alder reactions.     

Synthesis,Characterization and Photocleavage of Bis-decyl Pteroic Acid: A Folate Derivative with Affinity to Biomembranes†

Here, we provide mechanistic insight to the photocleavage of a compound in the folate family, namely pteroic acid. A bis-decyl chain derivative of pteroic acid was synthesized, structurally characterized and photochemically investigated. We showed that, like folic acid, pteroic acid and the decylated derivative undergo a photocleavage reaction in the presence of H₂O, while no reaction was observed in methanol solution. Furthermore, density functional theory calculations were carried out to predict relative stabilities of hypothetical mono-, bis- and tris-decylated pteroic acid derivatives to help rationalize the regioselectivity of the bis-decyl pteroic acid product. Additionally, the lipophilicity of the bis-decyl pteroic acid appears to confer a hydrophobic property enabling an interaction with biomembranes.