Mean and True Positions of Planets as Described in Gaṇitagannaḍi

The unpublished seventeenth-century Kannaḍa-language mathematical work Gaṇitagannaḍi is transmitted in a single palm-leaf manuscript. It was composed by Śaṅkaranārāyaṇa Jōisaru of Śṛṅgeri and is a karaṇa text cast as a commentary on the Vārṣikatantrasaṅgraha by Viddaṇācārya. Gaṇitagannaḍi's unique procedures for calculations wer introduced in an earlier paper in volume 8 (2020) of this journal. In the present paper the procedures for calculations of the mean and true positions of planets are described.

Relations between Modern Mathematics and Poetry: Czesław Miłosz; Zbigniew Herbert; Ion Barbu/Dan Barbilian

<p>This doctoral thesis is an examination of the relationship between poetry and mathematics, centred on three twentieth-century case studies: the Polish poets Czesław Miłosz (1911-2004) and Zbigniew Herbert (1924-1998), and the Romanian mathematician and poet Dan Barbilian/Ion Barbu (1895-1961). Part One of the thesis is a review of current scholarly literature, divided into two chapters. The first chapter looks at the nature of mathematics, outlining its historical developments and describing some major mathematical concepts as they pertain to the later case studies. This entails a focus on non-Euclidean geometries, modern algebra, and the foundations of mathematics in Europe; the nature of mathematical truth and language; and the modern historical evolution of mathematical schools in Poland and Romania. The second chapter examines some existing attempts to bring together mathematics and poetry, drawing on literature and science as an academic field; the role of the imagination and invention in the languages of both poetics and mathematics; the interest in mathematics among certain Symbolist poets, notably Mallarmé; and the experimental work of the French groups of mathematicians and mathematician-poets, Bourbaki and Oulipo. The role of metaphor is examined in particular. Part Two of the thesis is the case studies. The first presents the ethical and moral stance of Czesław Miłosz, investigating his attitudes towards classical and later relativistic science, in the light of the Nazi occupation and the Marxist regimes in Poland, and how these are reflected in his poetry. The study of Zbigniew Herbert is structured around a wide selection of his poetic oeuvre, and identifying his treatment of evolving and increasingly more complex mathematical concepts. The third case study, on Dan Barbilian, who published his poetry under the name Ion Barbu, begins with an examination of the mathematical school at Göttingen in the 1920s, tracing the influence of Gauss, Riemann, Klein, Hilbert and Noether in Barbilian’s own mathematical work, particularly in the areas of metric spaces and axiomatic geometry. In the discussion, the critical analysis of the mathematician and linguist Solomon Marcus is examined. This study finishes with a close reading of seven of Barbu’s poems. The relationship of mathematics and poetry has rarely been studied as a coherent academic field, and the relevant scholarship is often disconnected. A feature of this thesis is that it brings together a wide range of scholarly literature and discussion. Although primarily in English, a considerable amount of the academic literature collated here is in French, Romanian, Polish and some German. The poems themselves are presented in the original Polish and Romanian with both published and working translations appended in the footnotes. In the case of the two Polish poets, one a Nobel laureate and the other a multiple prize-winning figure highly regarded in Poland, this thesis is unusual in its concentration on mathematics as a feature of the poetry which is otherwise much-admired for its politically-engaged and lyrical qualities. In the case of the Romanian, Dan Barbilian, he is widely known in Romania as a mathematician, and most particularly as the published poet Ion Barbu, yet his work is little studied outside that country, and indeed much of it is not yet translated into English. This thesis suggests at an array of both theoretical and specific starting points for examining the multi-stranded and intricate relationship between mathematics and poetry, pointing to a number of continuing avenues of further research.</p>

Relations between Modern Mathematics and Poetry: Czesław Miłosz; Zbigniew Herbert; Ion Barbu/Dan Barbilian

<p>This doctoral thesis is an examination of the relationship between poetry and mathematics, centred on three twentieth-century case studies: the Polish poets Czesław Miłosz (1911-2004) and Zbigniew Herbert (1924-1998), and the Romanian mathematician and poet Dan Barbilian/Ion Barbu (1895-1961). Part One of the thesis is a review of current scholarly literature, divided into two chapters. The first chapter looks at the nature of mathematics, outlining its historical developments and describing some major mathematical concepts as they pertain to the later case studies. This entails a focus on non-Euclidean geometries, modern algebra, and the foundations of mathematics in Europe; the nature of mathematical truth and language; and the modern historical evolution of mathematical schools in Poland and Romania. The second chapter examines some existing attempts to bring together mathematics and poetry, drawing on literature and science as an academic field; the role of the imagination and invention in the languages of both poetics and mathematics; the interest in mathematics among certain Symbolist poets, notably Mallarmé; and the experimental work of the French groups of mathematicians and mathematician-poets, Bourbaki and Oulipo. The role of metaphor is examined in particular. Part Two of the thesis is the case studies. The first presents the ethical and moral stance of Czesław Miłosz, investigating his attitudes towards classical and later relativistic science, in the light of the Nazi occupation and the Marxist regimes in Poland, and how these are reflected in his poetry. The study of Zbigniew Herbert is structured around a wide selection of his poetic oeuvre, and identifying his treatment of evolving and increasingly more complex mathematical concepts. The third case study, on Dan Barbilian, who published his poetry under the name Ion Barbu, begins with an examination of the mathematical school at Göttingen in the 1920s, tracing the influence of Gauss, Riemann, Klein, Hilbert and Noether in Barbilian’s own mathematical work, particularly in the areas of metric spaces and axiomatic geometry. In the discussion, the critical analysis of the mathematician and linguist Solomon Marcus is examined. This study finishes with a close reading of seven of Barbu’s poems. The relationship of mathematics and poetry has rarely been studied as a coherent academic field, and the relevant scholarship is often disconnected. A feature of this thesis is that it brings together a wide range of scholarly literature and discussion. Although primarily in English, a considerable amount of the academic literature collated here is in French, Romanian, Polish and some German. The poems themselves are presented in the original Polish and Romanian with both published and working translations appended in the footnotes. In the case of the two Polish poets, one a Nobel laureate and the other a multiple prize-winning figure highly regarded in Poland, this thesis is unusual in its concentration on mathematics as a feature of the poetry which is otherwise much-admired for its politically-engaged and lyrical qualities. In the case of the Romanian, Dan Barbilian, he is widely known in Romania as a mathematician, and most particularly as the published poet Ion Barbu, yet his work is little studied outside that country, and indeed much of it is not yet translated into English. This thesis suggests at an array of both theoretical and specific starting points for examining the multi-stranded and intricate relationship between mathematics and poetry, pointing to a number of continuing avenues of further research.</p>

Mechanical and Dynamic Maps of Disc Brakes under Different Operating Conditions

The operating conditions during the braking process in an automobile affect the tribological contact between the pad and disc brake, thus, influencing the times and distances of braking and, in a more significant way, the safety of the braking process. This mathematical work aimed to provide a general visualization of the disc brake’s mechanical, dynamic, and thermal behavior under different operating conditions through 2D maps of the power dissipated, braking time, and braking distance of a disc brake with a ventilation blade N- 38 type. However, the dissipated energy on the disc brake in terms of temperature was analyzed considering Newton’s cooling law and mathematical calculations through classical theories of the dynamic and mechanical behavior of the disc brakes. For this purpose, the Response Surface Methodology (RSM) and Distance Weighted Least Squares (DWLS) fitting model considered different operating conditions of the disc brake. The results demonstrate that the disc brakes can be used effectively in severe operational requirements with a speed of 100 km/h and an ambient temperature of 27 °C, without affecting the occupant’s safety or the braking system and the pad. For the different conditions evaluated, the instantaneous temperature reaches values of 182.48 and 82.94 °C, where the high value was found for a total deceleration to 100 km/h to 0, which represent a total braking distance of around 44.20 to 114.96 m depending on the inclination angle (θ). Furthermore, the energy dissipation in the disc brakes depends strongly on the disc, blades and pad geometry, the type of material, parameters, and the vehicle operating conditions, as can be verified with mathematical calculation to validate the contribution of the effectiveness of the braking process during its real operation.

Correcting bias in log-linear instrument calibrations in the context of chemical ionization mass spectrometry

Quantitative calibration of analytes using chemical ionization mass spectrometers (CIMSs) has been hindered by the lack of commercially available standards of atmospheric oxidation products. To accurately calibrate analytes without standards, techniques have been recently developed to log-linearly correlate analyte sensitivity with instrument operating conditions. However, there is an inherent bias when applying log-linear calibration relationships that is typically ignored. In this study, we examine the bias in a log-linear-based calibration curve based on prior mathematical work. We quantify the potential bias within the context of a CIMS-relevant relationship between analyte sensitivity and instrument voltage differentials. Uncertainty in three parameters has the potential to contribute to the bias, specifically the inherent extent to which the nominal relationship can capture true sensitivity, the slope of the relationship, and the voltage differential below which maximum sensitivity is achieved. Using a prior published case study, we estimate an average bias of 30 %, with 1 order of magnitude for less sensitive compounds in some circumstances. A parameter-explicit solution is proposed in this work for completely removing the inherent bias generated in the log-linear calibration relationships. A simplified correction method is also suggested for cases where a comprehensive bias correction is not possible due to unknown uncertainties of calibration parameters, which is shown to eliminate the bias on average but not for each individual compound.

Using Rich Narratives to Engage Students in Worthwhile Mathematics: Children’s Literature, Movies and Short Films

Using children’s literature to support mathematics instruction has been connected to positive academic outcomes and learning dispositions; however, less is known about the use of audiovisual based narrative mediums to support student mathematical learning experiences. The current exploratory, qualitative study involved teaching three lessons based on challenging, problem solving tasks to two classes of Australian Year (Grade) 5 students (10 and 11 year olds). These tasks were developed from various narratives, each portrayed through a different medium (movie clip, short film, picture story book). Post lesson interviews were undertaken with 24 students inviting them to compare and contrast this lesson sequence with their usual mathematics instruction. Drawing on a self-determination theory lens, our analysis revealed that these lessons were experienced by students as both highly enjoyable and mathematically challenging. More specifically, it was found that presenting mathematics tasks based on rich and familiar contexts and providing meaningful choices about how to approach their mathematical work supported student autonomy. In addition, there was evidence that the narrative presentation supported student understanding of the mathematics through making the tasks clearer and more accessible, whilst the audiovisual mediums (movie clip, short film) in particular provided a dynamic representation of key mathematical ideas (e.g., transformation and scale). Students indicated an eclectic range of preferences in terms of their preferred narrative mediums for exploring mathematical ideas. Our findings support the conclusion that educators and researchers focused on the benefits of teaching mathematics through picture story books consider extending their definition of narrative to encompass other mediums, such as movie clips and short films.

Correcting Bias in Log-Linear Instrument Calibrations in the Context of Chemical Ionization Mass Spectrometry

Quantitative calibration of analytes using chemical ionization mass spectrometers (CIMS) has been hindered by the lack of commercially available standards of atmospheric oxidation products. To accurately calibrate analytes without standards, techniques have been recently developed to log-linearly correlate analyte sensitivity with instrument operating conditions. However, there is an inherent bias when applying log-linear calibration relationships that is typically ignored. In this study, we examine the bias in a log-linear based calibration curve based on prior mathematical work. We quantify the potential bias within the context of a CIMS-relevant relationship between analyte sensitivity and instrument voltage differentials. Uncertainty in three parameters has the potential to contribute to the bias, specifically the inherent extent to which the nominal relationship can capture true sensitivity, the slope of the relationship, and the voltage differential below which maximum sensitivity is achieved. Using a prior published case study, we estimate an average bias of 30%, with one order of magnitude for less sensitive compounds in some circumstances. A parameter-explicit solution is proposed in this work for completely removing the inherent bias generated in the log-linear calibration relationships. A simplified correction method is also suggested for cases where a comprehensive bias correction is not possible due to unknown uncertainties of calibration parameters, which is shown to eliminate the bias on average but not for each individual compound.

The Mathematical Proficiency Promoted by Mathematical Modelling

This study aims to investigate the mathematical proficiency promoted by mathematical modelling tasks that require students to get involved in the processes of developing mathematical models, instead of just using known or given models. The research methodology is grounded on design-based research, and the classroom design framework is supported by complexity science underpinnings. The research intervention consists of high-school students, from a grade 11 mathematics course, aiming to solve four different modelling tasks in four distinct moments. Data was collected during the intervention from students’ written mathematical work and audio and video recordings, and from recall interviews after the intervention. Data analysis was conducted based on a model of mathematical proficiency and assisted by interpretive diagrams created for this research purpose. This research study offers insight into mathematics teaching by portraying how mathematical modelling tasks can be integrated into mathematics classes to promote students’ mathematical proficiency. The study discusses observed expressions and behaviours in students’ development of mathematical proficiency and suggests a relationship between mathematical modelling processes and the promotion of mathematical proficiency. The study also reveals that students develop mathematical proficiency, even when they do not come to full resolutions of modelling tasks, which emphasizes the relevance of learning processes, and not only of the products of these processes.