An investigation of post-primary students' images of mathematics

This research study investigates the image of mathematics held by 5th-year post-primary students in Ireland. For this study, “image of mathematics” is conceptualized as a mental representation or view of mathematics, presumably constructed as a result of past experiences, mediated through school, parents, peers or society. It is also understood to include attitudes, beliefs, emotions, self-concept and motivation in relation to mathematics. This study explores the image of mathematics held by a sample of 356 5th-year students studying ordinary level mathematics. Students were aged between 15 and 18 years. In addition, this study examines the factors influencing students‟ images of mathematics and the possible reasons for students choosing not to study higher level mathematics for the Leaving Certificate. The design for this study is chiefly explorative. A questionnaire survey was created containing both quantitative and qualitative methods to investigate the research interest. The quantitative aspect incorporated eight pre-established scales to examine students‟ attitudes, beliefs, emotions, self-concept and motivation regarding mathematics. The qualitative element explored students‟ past experiences of mathematics, their causal attributions for success or failure in mathematics and their influences in mathematics. The quantitative and qualitative data was analysed for all students and also for students grouped by gender, prior achievement, type of post-primary school attending, co-educational status of the post-primary school and the attendance of a Project Maths pilot school. Students‟ images of mathematics were seen to be strongly indicated by their attitudes (enjoyment and value), beliefs, motivation, self-concept and anxiety, with each of these elements strongly correlated with each other, particularly self-concept and anxiety. Students‟ current images of mathematics were found to be influenced by their past experiences of mathematics, by their mathematics teachers, parents and peers, and by their prior mathematical achievement. Gender differences occur for students in their images of mathematics, with males having more positive images of mathematics than females and this is most noticeable with regards to anxiety about mathematics. Mathematics anxiety was identified as a possible reason for the low number of students continuing with higher level mathematics for the Leaving Certificate. Some students also expressed low mathematical self-concept with regards to higher level mathematics specifically. Students with low prior achievement in mathematics tended to believe that mathematics requires a natural ability which they do not possess. Rote-learning was found to be common among many students in the sample. The most positive image of mathematics held by students was the “problem-solving image”, with resulting implications for the new Project Maths syllabus in post-primary education. Findings from this research study provide important insights into the image of mathematics held by the sample of Irish post-primary students and make an innovative contribution to mathematics education research. In particular, findings contribute to the current national interest in Ireland in post-primary mathematics education, highlighting issues regarding the low uptake of higher level mathematics for the Leaving Certificate and also making a preliminary comparison between students who took part in the piloting of Project Maths and students who were more recently introduced to the new syllabus. This research study also holds implications for mathematics teachers, parents and the mathematics education community in Ireland, with some suggestions made on improving students‟ images of mathematics.

Summer Sprite for Orchestra

Summer Sprite for Orchestra was completed in December, 2004. The piece originated from a singular encounter with little angels at Chang-Kyung Palace, which is the oldest and the most beautiful palace in Korea, and where the kings of the Chosun Dynasty (1393-1897) lived. This encounter was in the summer of 2002. I certainly could not prove that those angels I met were real. Possibly they were the reflection of drops of water after a sudden shower on that summer day. However, I definitely remember that short, unforgettable, and mysterious moment and the angels' beautiful dance-like celebration. Summer Sprite is based on these special memories and the encounter with the little angels that summer. Summer Sprite consists of 3 movements: "Greeting," "Encounter," and "Celebration." These follow the course of my encounter with the little angels. In Summer Sprite, I wished to describe the image of the angels as well as the progression of greeting, encounter, and celebration with them. The moods that follow in Summer Sprite are by turns lyrical, poetic, fantastic, mysterious, and dream-like. In each movement, I describe the meeting of angels and composer through the use of the soloists -- violin (sometimes viola) and cello. As suggested by the subtitle of the first movement, "Greeting" portrays the moment when a surprised I met the angels. It begins with tam-tam, marimba, harp, and piano and sets a mysterious and dark mood. The second movement, "Encounter," is shorter than the first movement. This movement provides a more tranquil mood as well as more unique timbres resulting from the use of mutes and special instruments (English horn, harp, crotales, suspended cymbal, and celesta). The delicate expression of the percussion is particularly important in establishing the static mood of this movement . The last movement, °?Celebration,°± is bright and energetic. It is also the longest. Here, I require the most delicate changes of dynamics and tempo, the most vigorous harmonies, and the fastest rhythmic figures, as well as the most independent, lyrical, and poetic melodies. For bright orchestral tone color, I used various kinds of percussion such as timpani, xylophone, marimba, vibraphone, cymbals, side drum, tambourine, triangle, and bass drum. This last movement is divided rondo-like into five sections: The first (mm.1-3), second (mm.4 - rehearsal number 1), third (rehearsal numbers 2-4), fourth (rehearsal numbers 5-7), and fifth, (rehearsal numbers 8 -18). To sum up, Summer Sprite describes an unforgettable and mysterious moment in a my life. My intention was to portray this through a concerto-like framework. A model for this would be Brahms°Ø °?Double Concerto°± in A minor, op.102, in which the solo cello stands for my angel and the solo violin (sometimes solo viola) for me.

Design Optimization and Security For Communication Networks

In this work we introduce a new mathematical tool for optimization of routes, topology design, and energy efficiency in wireless sensor networks. We introduce a vector field formulation that models communication in the network, and routing is performed in the direction of this vector field at every location of the network. The magnitude of the vector field at every location represents the density of amount of data that is being transited through that location. We define the total communication cost in the network as the integral of a quadratic form of the vector field over the network area. With the above formulation, we introduce a mathematical machinery based on partial differential equations very similar to the Maxwell's equations in electrostatic theory. We show that in order to minimize the cost, the routes should be found based on the solution of these partial differential equations. In our formulation, the sensors are sources of information, and they are similar to the positive charges in electrostatics, the destinations are sinks of information and they are similar to negative charges, and the network is similar to a non-homogeneous dielectric media with variable dielectric constant (or permittivity coefficient). In one of the applications of our mathematical model based on the vector fields, we offer a scheme for energy efficient routing. Our routing scheme is based on changing the permittivity coefficient to a higher value in the places of the network where nodes have high residual energy, and setting it to a low value in the places of the network where the nodes do not have much energy left. Our simulations show that our method gives a significant increase in the network life compared to the shortest path and weighted shortest path schemes. Our initial focus is on the case where there is only one destination in the network, and later we extend our approach to the case where there are multiple destinations in the network. In the case of having multiple destinations, we need to partition the network into several areas known as regions of attraction of the destinations. Each destination is responsible for collecting all messages being generated in its region of attraction. The complexity of the optimization problem in this case is how to define regions of attraction for the destinations and how much communication load to assign to each destination to optimize the performance of the network. We use our vector field model to solve the optimization problem for this case. We define a vector field, which is conservative, and hence it can be written as the gradient of a scalar field (also known as a potential field). Then we show that in the optimal assignment of the communication load of the network to the destinations, the value of that potential field should be equal at the locations of all the destinations. Another application of our vector field model is to find the optimal locations of the destinations in the network. We show that the vector field gives the gradient of the cost function with respect to the locations of the destinations. Based on this fact, we suggest an algorithm to be applied during the design phase of a network to relocate the destinations for reducing the communication cost function. The performance of our proposed schemes is confirmed by several examples and simulation experiments. In another part of this work we focus on the notions of responsiveness and conformance of TCP traffic in communication networks. We introduce the notion of responsiveness for TCP aggregates and define it as the degree to which a TCP aggregate reduces its sending rate to the network as a response to packet drops. We define metrics that describe the responsiveness of TCP aggregates, and suggest two methods for determining the values of these quantities. The first method is based on a test in which we drop a few packets from the aggregate intentionally and measure the resulting rate decrease of that aggregate. This kind of test is not robust to multiple simultaneous tests performed at different routers. We make the test robust to multiple simultaneous tests by using ideas from the CDMA approach to multiple access channels in communication theory. Based on this approach, we introduce tests of responsiveness for aggregates, and call it CDMA based Aggregate Perturbation Method (CAPM). We use CAPM to perform congestion control. A distinguishing feature of our congestion control scheme is that it maintains a degree of fairness among different aggregates. In the next step we modify CAPM to offer methods for estimating the proportion of an aggregate of TCP traffic that does not conform to protocol specifications, and hence may belong to a DDoS attack. Our methods work by intentionally perturbing the aggregate by dropping a very small number of packets from it and observing the response of the aggregate. We offer two methods for conformance testing. In the first method, we apply the perturbation tests to SYN packets being sent at the start of the TCP 3-way handshake, and we use the fact that the rate of ACK packets being exchanged in the handshake should follow the rate of perturbations. In the second method, we apply the perturbation tests to the TCP data packets and use the fact that the rate of retransmitted data packets should follow the rate of perturbations. In both methods, we use signature based perturbations, which means packet drops are performed with a rate given by a function of time. We use analogy of our problem with multiple access communication to find signatures. Specifically, we assign orthogonal CDMA based signatures to different routers in a distributed implementation of our methods. As a result of orthogonality, the performance does not degrade because of cross interference made by simultaneously testing routers. We have shown efficacy of our methods through mathematical analysis and extensive simulation experiments.

Midlatitude daytime D region ionosphere variations measured from radio atmospherics

We measured the midlatitude daytime ionospheric D region electron density profile height variations in July and August 2005 near Duke University by using radio atmospherics (or sferics for short), which are the high-power, broadband very low frequency (VLF) signals launched by lightning discharges. As expected, the measured daytime D region electron density profile heights showed temporal variations quantitatively correlated with solar zenith angle changes. In the midlatitude geographical regions near Duke University, the observed quiet time heights decreased from ∼80 km near sunrise to ∼71 km near noon when the solar zenith angle was minimum. The measured height quantitative dependence on the solar zenith angle was slightly different from the low-latitude measurement given in a previous work. We also observed unexpected spatial variations not linked to the solar zenith angle on some days, with 15% of days exhibiting regional differences larger than 0.5 km. In these 2 months, 14 days had sudden height drops caused by solar flare X-rays, with a minimum height of 63.4 km observed. The induced height change during a solar flare event was approximately proportional to the logarithm of the X-ray flux. In the long waveband (wavelength, 1-8 Å), an increase in flux by a factor of 10 resulted in 6.3 km decrease of the height at the flux peak time, nearly a perfect agreement with the previous measurement. During the rising and decaying phases of the solar flare, the height changes correlated more consistently with the short, rather than the long, wavelength X-ray flux changes. © 2010 by the American Geophysical Union.

[Book review] Barry Mazur, Imagining Numbers: (Particularly the Square Root of Minus Fifteen)

Tony Mann provides a review of the book: Barry Mazur, Imagining Numbers: (Particularly the Square Root of Minus Fifteen), London: Allen Lane, 2003, ISBN: 0713996307

Returning to deposits...

Examines the Court of Appeal ruling in Midill (97PL) Ltd v Park Lane Estates Ltd on whether the court should exercise its discretion to return a deposit paid under a contract for the sale of a property where, despite the claimant having defaulted on its purchase, the owner obtained a higher price than would have been paid by the claimant when it subsequently sold the property to a third party. Considers the need for special or exceptional circumstances to be present to override the rule that deposits were to be forfeited upon the purchaser's default.

A unified 3D model of the VAR process

A 3D time-dependent model of the VAR process has been developed using CFD techniques. The model solves the coupled field equations for fluid flow, heat transfer (including phase change) and electromagnetic field, for both the electrode and the ingot. The motion of the electic arc 'preferred spot' can be specified based on observations. Correlations are sought between the local gap height, resulting from instantaneous liquid pool surface shape and electrode tip shape, and the arc motion. The detailed behaviour of the melting film on the electrode tip is studies using a spectral free surface technique, which allows investigation of the drops' detachment and drip shorts.

Investigation of the permeation of model formulations and a commercial ibuprofen formulation in Carbosil® and human skin using ATR-FTIR and multivariate spectral analysis

The purpose of the present study was to use attenuated total reflectance-Fourier transform infrared spectroscopy (ATR-FTIR) and target factor analysis (TFA) to investigate the permeation of model drugs and formulation components through Carbosil® membrane and human skin. Diffusion studies of saturated solutions in 50:50 water/ethanol of methyl paraben (MP), ibuprofen (IBU) and caffeine (CF) were performed on Carbosil® membrane. The spectroscopic data were analysed by target factor analysis, and evolution profiles of the signal for each component (i.e. the drug, water, ethanol and membrane) over time were obtained. Results showed that the data were successfully deconvoluted as correlations between factors from the data and reference spectra of the components, were above 0.8 in all cases. Good reproducibility over three runs for the evolution profiles was obtained. From the evolution profiles it was observed that water diffused better through the Carbosil® membrane than ethanol, confirming the hydrophilic properties of the Carbosil® membrane used. IBU diffused slower compared with MP and CF. The evolution profile of CF was very similar to that of water, probably because of the high solubility of CF in water, indicating that both compounds are diffusing concurrently. The second part of the work involved a study of the evolution profiles of the components of a commercial topical gel containing 5% (w/w) of ibuprofen as it permeated through human skin. Although the system was much more complex, data were still successfully deconvoluted and the different components of the formulation identified except for benzyl alcohol which might be attributed to the low concentrations of benzyl alcohol used in topical formulations. (C) 2009 Elsevier B.V. All rights reserved.

Local examination of skin diffusion using FTIR spectroscopic imaging and multivariate target factor analysis

In the context of trans-dermal drug delivery it is very important to have mechanistic insight into the barrier function of the skin's stratum corneum and the diffusion mechanisms of topically applied drugs. Currently spectroscopic imaging techniques are evolving which enable a spatial examination of various types of samples in a dynamic way. ATR-FTIR imaging opens up the possibility to monitor spatial diffusion profiles across the stratum corneum of a skin sample. Multivariate data analyses methods based on factor analysis are able to provide insight into the large amount of spectroscopically complex and highly overlapping signals generated. Multivariate target factor analysis was used for spectral resolution and local diffusion profiles with time through stratum corneum. A model drug, 4-cyanophenol in polyethylene glycol 600 and water was studied. Results indicate that the average diffusion profiles between spatially different locations show similar profiles despite the heterogeneous nature of the biological sample and the challenging experimental set-up.

Simultaneous monitoring of drug and solvent diffusion across a model membrane using ATR-FTIR spectroscopy

Attenuated total reflectance Fourier transform infrared (ATR-FTIR) spectroscopy has been used to simultaneously follow the diffusion of model drugs and solvent across polydimethylsiloxane (silicone) membrane. Three model drugs, cyanophenol (CNP), methyl nicotinate (MN) and butyl paraben (BP) were selected to cover a range of lipophilicities. Isostearyl isostearate (ISIS) was chosen as the solvent because its large molecular weight should facilitate observation of whether the drug molecules are able to diffuse through the membrane independently of the solvent. The diffusion of the three drugs and the solvent was successfully described by a Fickian model. The effects of parameters such as the absorption wavelength used to follow diffusion on the calculated diffusion coefficient were investigated. Absorption wavelength which affects the depth of penetration of the infrared radiation into the membrane did not significantly affect the calculated diffusion coefficient over the wavelength range tested. Each of the model drugs was observed to diffuse independently of the solvent across the membrane. The diffusion of a CNP-ISIS hydrogen bonded complex across the membrane was also monitored. The relative diffusion rates of the solute and solvent across the membrane can largely be accounted for by the molecular size of the permeant.

ATR-FTIR spectroscopy and spectroscopic imaging of solvent and permeant diffusion across model membranes

The uptake and diffusion of solvents across polymer membranes is important in controlled drug delivery, effects on drug uptake into, for example, infusion bags and containers, as well as transport across protective clothing. Attenuated Total Reflectance Fourier Transform Infrared (ATR-FTIR) spectroscopy has been used to monitor the effects of different solvents on the diffusion of a model compound, 4-cyanophenol (CNP) across silicone membrane and on the equilibrium concentration of CNP obtained in the membrane following diffusion. ATR-FTIR spectroscopic imaging of membrane diffusion was used to gain an understanding of when the boundary conditions applied to Fick's second law, used to model the diffusion of permeants across the silicone membrane do not hold. The imaging experiments indicated that when the solvent was not taken up appreciably into the membrane, the presence of discrete solvent pools between the ATR crystal and the silicone membrane can affect the diffusion profile of the permeant. This effect is more significant if the permeant has a high solubility in the solvent. In contrast, solvents that are taken up into the membrane to a greater extent, or those where the solubility of the permeant in the vehicle is relatively low, were found to show a good fit to the diffusion model. As such these systems allow the ATR-FTIR spectroscopic approach to give mechanistic insight into how the particular solvents enhance permeation. The solubility of CNP in the solvent and the uptake of the solvent into the membrane were found to be important influences on the equilibrium concentration of the permeant obtained in the membrane following diffusion. In general, solvents which were taken up to a significant extent into the membrane and which caused the membrane to swell increased the diffusion coefficient of the permeant in the membrane though other factors such as solvent viscosity may also be important.

Randomised controlled trial of effects of Helicobacter pylori infection and its eradication on heartburn and gastro-oesophageal reflux: the bristol helicobacter project

Objectives To investigate the effects of Helicobacter pylori infection and its eradication on heartburn and gastro-oesophageal reflux. Design Cross sectional study, followed by a randomised placebo controlled trial. Setting Seven general practices in Bristol, England. Participants 10 537 people, aged 20-59 years, with and without H pylori infection (determined by the 13C-urea breath test). Main outcome measures Prevalence of heartburn and gastro-oesophageal acid reflux at baseline and two years after treatment to eradicate H pylori infection. Results At baseline, H pylori infection was associated with increased prevalence of heartburn (odds ratio 1.14, 95% confidence interval 1.05 to 1.23) but not reflux (1.05, 0.97 to 1.14). In participants with H pylori infection, active treatment had no effect on the overall prevalence of heartburn (0.99, 0.88 to 1.12) or reflux (1.04, 0.91 to 1.19) and did not improve pre-existing symptoms of heartburn or reflux. Conclusions H pylori infection is associated with a slightly increased prevalence of heartburn but not reflux. Treatment to eradicate H pylori has no net benefit in patients with heartburn or gastro-oesophageal reflux