An Attempt to Provide Appropriate Mathematical Knowledge for Students of Economics

Mathematics courses for students of the social sciences have a vital role to play in providing relevant mathematical backgrounds for increasingly mathematical subjects.

The course described in this article has been designed for those first year degree students reading economics at University College, London, who have not passed G.C.E. A level? ? General Certificate of Education at Advanced level. mathematics. The aims and structure of the course are described, indicating those features which are thought to be of particular importance. The author reports her impression of the impact of the course on one group of students. The students’ own views about the course have also been collected and are summarized here, together with their more general views about mathematical education. Much of the analysis is thought to be relevant to all those involved in the mathematical education of social scientists.

     

An X-ray Crystallographic Study of the Related Triazenes, 1,4-Di[(<Emphasis Type="Italic">E</Emphasis>)-2-(2-nitrophenyl)-1-diazenyl]piperazine and Methyl 4-{(E)-2-(4-methylpiperazino)-1-diazenyl}benzoate

Abstract The crystal structures of methyl 4-{(E)-2-(4-methylpiperazino)-1-diazenyl}benzoate (2a) and 1,4-di[(E)-2-(2-nitrophenyl)-1-diazenyl]piperazine (3a) have been determined by single crystal X-ray diffraction analysis. The bis-triazene (3a) adopts an unusual pseudo-boat conformation in the piperazine ring, with a dihedral angle of 52.20(0.06)° between the two planes defined within the piperazine ring. The crystal structures of 2a and 3a are compared with the structure of the triazene (4) and the closely related bis-triazene (5). The piperazine ring of 2a adopts a typical chair conformation, whereas the piperazine ring of 3a adopts an unusual boat conformation. Crystal data: 2a C₁₃H₁₈N₄O₂, monoclinic, space group P2₁ /n, a = 13.849(3) ?, b = 6.577(1) ?, c = 14.904(3) ?, α = 90°, β = 96.098(3)°, γ = 90° and V = 1,349.8(4) ?³, for Z = 4. 3a C₁₆H₁₆N₈O₄, triclinic, space group P-1, a = 7.6066(6) ?, b = 8.3741(7) ?, c = 14.507(1) ?, α = 78.673(1)°, β = 81.877(1)°, γ = 73.445(1)° and V = 865.0(1) ?³, for Z = 2. Index abstract The crystal structures of methyl 4-{(E)-2-(4-methylpiperazino)-1-diazenyl}benzoate (2a) and 1,4-di[(E)-2-(2-nitrophenyl)-1-diazenyl]piperazine (3a) have been determined by single crystal X-ray diffraction analysis.     

X-ray crystal and molecular structure of cis-dichlorobis[1,2-ethanebis(diphenylphosphineoxide)] iron(III)tetrachloroferrate(III), [FeCl2(L-L)2] [FeCl4]

Reaction of iron(III) chloride with 1,2-ethanebis(diphenylphosphine oxide) in an ethanol/chloroform mixture formed a 11 adduct of stoichiometry FeCl₃(L-L) which has been characterized by X-ray crystallography as [FeCl₂(L-L)₂] [FeCl₄]. The structure was solved by the heavy atom method, using MoK diffractometer data and refined by full-matrix least squares toR=0.044 for 8506 observed reflections. The coordination is slightly distorted octahedral for the cation with acis-arrangement of the chlorine atoms and a seven-membered chelate ring formation by Ph₂P(O)CH₂CH₂P(O)Ph₂ with the donor O atoms also in acis arrangement. The Fe-Cl distances are equal at 2.306(1) Å but the Fe-O distances differ significantly in each of the two ligands, (avg. 2.076(2) Å; avg. 1.975(3) Å).     

Optical second harmonic generation studies of the nature of the GaAs(100) surface in air

We present in this paper a study of the non-linear optical response of the GaAs/air interface as measured by second harmonic generation. By careful choice of incidence and analysis electric vectors for the radiation the second harmonic response as a function of crystallographic orientation has been characterized for a GaAs(100) surface. The exact origin of the response remains unclear but comments on the surface specificity of the technique are made and the observed anisotropy in the data is explained in terms of the step density of the crystal surface.     

An innovative molybdenum column liner for oxygen and hydrogen stable isotope analysis by pyrolysis

The most widely used method for pyrolysing samples for hydrogen or oxygen isotopic analysis involves heating them to greater than 1300 degrees C in a helium stream passed through a glassy carbon tube in an alumina casing. There are a number of difficulties with this. Glassy carbon tubes are expensive and interaction between the carbon tube and the outer casing produces unwanted carbon monoxide by reduction of the alumina at high temperatures. The latter effect is overwhelming if temperatures of 1400 degrees C or greater are used for pyrolysis. We experimented with lining alumina casings with pure molybdenum sheet. It is relatively cheap, conforms well to the interior of the reactor tube (to avoid carrier and sample bypassing of the carbon pack), resists high temperatures and neither oxidises excessively nor absorbs the gases. The main disadvantages are that silver sample cups must be used and that the molybdenum degrades over time by formation of the carbide. We can maintain sharp peaks, high precision and good accuracy over more than 700 solid samples for both hydrogen and oxygen. The reactors last longer for water injections. The molybdenum in the columns does not contribute greatly to memory effects. The precision of analysis is dependent on other factors as well as the pyrolysis column, but for oxygen we typically achieve approximately <0.2 per thousand (sucrose), <0.25 per thousand (water) and <0.25 per thousand (leaf), sometimes using only a linear correction of drift, after dividing the run into 1 to 3 segments.     

Oxygen isotope analysis of carbonates in the calcite-dolomite-magnesite solid-solution by high-temperature pyrolysis: initial results

Accurate and efficient measurement of the oxygen isotope composition of carbonates (delta(C) (18)O) based on the mass spectrometric analysis of CO(2) produced by reacting carbonate samples with H(3)PO(4) is compromised by: (1) uncertainties associated with fractionation factors (alpha(CO)(2)C) used to correct measured oxygen isotope values of CO(2)(delta(CO(2)(18)O) to delta(C) (18)O; and (2) the slow reaction rates of many carbonates of geological and environmental interest with H(3)PO(4). In contrast, determination of delta(C) (18)O from analysis of CO produced by high-temperature (>1400 degrees C) pyrolytic reduction, using an elemental analyser coupled to continuous-flow isotope-ratio mass spectrometry (TC/EA CF-IRMS), offers a potentially efficient alternative that measures the isotopic composition of total carbonate oxygen and should, therefore, theoretically be free of fractionation effects. The utility of the TC/EA CF-IRMS technique was tested by analysis of carbonates in the calcite-dolomite-magnesite solid-solution and comparing the results with delta(C) (18)O measured by conventional thermal decomposition/fluorination (TDF) on the same materials. Initial results show that CO yields are dependent on both the chemical composition of the carbonate and the specific pyrolysis conditions. Low gas yields (<100% of predicted yield) are associated with positive (>+0.2 per thousand) deviations in delta(C(TC/EA) (18)O compared with delta(C(TDF) (18)O. At a pyrolysis temperature of 1420 degrees C the difference between delta(C) (18)O measured by TC/EA CF-IRMS and TDF (Delta(C(TC/EA,TDF) (18)O) was found to be negatively correlated with gas yield (r = -0.785) and this suggests that delta(C) (18)O values (with an estimated combined standard uncertainty of +/-0.38 per thousand) could be derived by applying a yield-dependent correction. Increasing the pyrolysis temperature to 1500 degrees C also resulted in a statistically significant correlation with gas yield (r = -0.601), indicating that delta(C) (18)O values (with an estimated uncertainty of +/-0.43 per thousand) could again be corrected using a yield-dependent procedure. Despite significant uncertainty associated with TC/EA CF-IRMS analysis, the magnitude of the uncertainty is similar to that associated with the application of poorly defined values of alpha(CO)(2), (C) used to derive delta(C) (18)O from delta(CO(2) (18)O measured by the H(3)PO(4) method for most common carbonate phases. Consequently, TC/EA CF-IRMS could provide a rapid alternative for the analysis of these phases without any effective deterioration in relative accuracy, while analytical precision could be improved by increasing the number of replicate analyses for both calibration standards and samples. Although automated gas preparation techniques based on the H(3)PO(4) method (ISOCARB, Kiel device, Gas-Bench systems) have the potential to measure delta(CO)(2) (18)O efficiently for specific, slowly reacting phases (e.g. dolomite), problems associated with poorly defined alpha(CO)(2), (C) remain. The application of the Principle of Identical Treatment is not a solution to the analysis of these phases because it assumes that a single fractionation factor may be defined for each phase within a solid-solution regardless of its precise chemical composition. This assumption has yet to be tested adequately.     

Evaluating the utility of ion mobility separation in combination with high-pressure liquid chromatography/mass spectrometry to facilitate detection of trace impurities in formulated drug products

Many formulated products contain complex polymeric excipients such as polyethylene glycols (PEGs). Such excipients can be readily ionized by electrospray and may be present at very high concentrations, thus making it very difficult to identify trace level impurities such as degradants in samples, even if hyphenated techniques such as liquid chromatography/mass spectrometry (LC/MS) are used. Ion mobility (IM) spectrometry is a very rapid gas-phase separation technique and offers additional separation capability within the LC timeframe. This work investigates the use of an IM separator in combination with high-pressure liquid chromatography (HPLC) and MS, to improve the separation of drug-related materials from excipients, thus aiding the identification of trace-level impurities in an anti-HIV medication, Combivir.     

Determination of the Pesticide Fenbutatin Oxide in Tomatoes,Cucumbers and Bananas by High Performance Liquid Chromatographic/Atmospheric Pressure Chemical Ionization-Mass Spectrometry

A rapid method, using high-performance liquid chromatography (HPLC)/atmospheric pressure chemical ionization-mass spectrometry, has been developed for the determination of fenbutatin oxide in tomatoes, cucumbers and bananas. Samples were homogenized with sodium carbonate and ethyl acetate, filtered through sodium sulphate, concentrated and solvent exchanged into acetonitrile prior to analysis. HPLC was performed on a Hypercarb column with 10:90 acetic acid (5% v/v glacial acetic acid in water)/acetonitrile at a flow rate of 1 mL/min. Positive ionization selected-ion monitoring was performed on the 7 isotopic cluster ions from the tris(2-methyl-2-phenylpropyl) tin fragment. A comparison of solvent-based and extract-based standards showed that tomato and cucumber matrices had a slight enhancement effect on the signal intensity, whereas the banana matrix exerted a signal suppression effect. Calibration was linear over the range 0.25–5.0 ng/μL. The mean spike recoveries (extracts spiked at 0.5 mg/kg) were 88% for tomatoes and 80% for both cucumbers and bananas with relative standard deviations of 6%, 7% and 8% respectively. Limits of detection were commodity dependent and ranged from 0.06–0.12 ng/μL (equivalent to 0.01–0.02 mg/kg in the crop). Ionization was stable for long analytical time periods. © 1997 Crown Copyright     

Effects of Curcumin Analogues DMC and EF24 in Combination with the Cytokine TRAIL against Kidney Cancer

The natural compound curcumin has been shown to have therapeutic potential against a wide range of diseases such as cancer. Curcumin reduces cell viability of renal cell carcinoma (RCC) cells when combined with TNF-related apoptosis-inducing ligand (TRAIL), a cytokine that specifically targets cancer cells, by helping overcome TRAIL resistance. However, the therapeutic effects of curcumin are limited by its low bioavailability. Similar compounds to curcumin with higher bioavailability, such as demethoxycurcumin (DMC) and 3,5-bis(2-fluorobenzylidene)-4-piperidone (EF24), can potentially have similar anticancer effects and show a similar synergy with TRAIL, thus reducing RCC viability. This study aims to show the effects of DMC and EF24 in combination with TRAIL at reducing ACHN cell viability and ACHN cell migration. It also shows the changes in death receptor 4 (DR4) expression after treatment with these compounds individually and in combination with TRAIL, which can play a role in their mechanism of action.     

Effect of additive length and chemistry on the morphology of blends of conjugated thiophenes and fullerene derivative acceptor molecules

Small molecule additives have been shown to increase the device efficiency of conjugated polymer (donor) and fullerene derivative (acceptor) based organic solar cells by modifying the morphology of the device active layer. In this paper we conduct a systematic study of how additives affect the donor‐acceptor morphology using molecular dynamics simulations of blends of thiophene‐based oligomers, mimicking poly(3‐dodecylthiophene) (P3DDT) or poly(2,2′:5′,2”‐3,3”‐didocyl‐terthiophene) (PTTT), and fullerene derivatives with additives of varying length and chemical functionalization, mimicking experimentally used additives like methyl ester additives, diiodooctane, and alkanedithiols. We find that functionalization of additives with end groups that are attracted to acceptor molecules are necessary to induce increased donor‐acceptor macrophase separation. In blends where acceptors intercalate between oligomer alkyl side chains, functionalized additives decrease acceptor intercalation. Functionalized additives with shorter alkyl segments increase acceptor macrophase separation more than additives with same chemical functionalization but longer alkyl segments. © 2015 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015 , 53, 1046–1057