On the singular values of the Fox-Li operator

In this paper we study the singular values of the Fox-Li operator. Also, we study some interactions of the Fox-Li operator with its periodic pseudo-differential analogue in the spectral setting

Dynamical systems techniques in the analysis of neural systems

As we strive to understand the mechanisms underlying neural computation, mathematical models are increasingly being used as a counterpart to biological experimentation. Alongside building such models, there is a need for mathematical techniques to be developed to examine the often complex behaviour that can arise from even the simplest models. There are now a plethora of mathematical models to describe activity at the single neuron level, ranging from one-dimensional, phenomenological ones, to complex biophysical models with large numbers of state variables. Network models present even more of a challenge, as rich patterns of behaviour can arise due to the coupling alone. We first analyse a planar integrate-and-fire model in a piecewise-linear regime. We advocate using piecewise-linear models as caricatures of nonlinear models, owing to the fact that explicit solutions can be found in the former. Through the use of explicit solutions that are available to us, we categorise the model in terms of its bifurcation structure, noting that the non-smooth dynamics involving the reset mechanism give rise to mathematically interesting behaviour. We highlight the pitfalls in using techniques for smooth dynamical systems in the study of non-smooth models, and show how these can be overcome using non-smooth analysis. Following this, we shift our focus onto the use of phase reduction techniques in the analysis of neural oscillators. We begin by presenting concrete examples showcasing where these techniques fail to capture dynamics of the full system for both deterministic and stochastic forcing. To overcome these failures, we derive new coordinate systems which include some notion of distance from the underlying limit cycle. With these coordinates, we are able to capture the effect of phase space structures away from the limit cycle, and we go on to show how they can be used to explain complex behaviour in typical oscillatory neuron models

Dynamical systems techniques in the analysis of neural systems

As we strive to understand the mechanisms underlying neural computation, mathematical models are increasingly being used as a counterpart to biological experimentation. Alongside building such models, there is a need for mathematical techniques to be developed to examine the often complex behaviour that can arise from even the simplest models. There are now a plethora of mathematical models to describe activity at the single neuron level, ranging from one-dimensional, phenomenological ones, to complex biophysical models with large numbers of state variables. Network models present even more of a challenge, as rich patterns of behaviour can arise due to the coupling alone. We first analyse a planar integrate-and-fire model in a piecewise-linear regime. We advocate using piecewise-linear models as caricatures of nonlinear models, owing to the fact that explicit solutions can be found in the former. Through the use of explicit solutions that are available to us, we categorise the model in terms of its bifurcation structure, noting that the non-smooth dynamics involving the reset mechanism give rise to mathematically interesting behaviour. We highlight the pitfalls in using techniques for smooth dynamical systems in the study of non-smooth models, and show how these can be overcome using non-smooth analysis. Following this, we shift our focus onto the use of phase reduction techniques in the analysis of neural oscillators. We begin by presenting concrete examples showcasing where these techniques fail to capture dynamics of the full system for both deterministic and stochastic forcing. To overcome these failures, we derive new coordinate systems which include some notion of distance from the underlying limit cycle. With these coordinates, we are able to capture the effect of phase space structures away from the limit cycle, and we go on to show how they can be used to explain complex behaviour in typical oscillatory neuron models

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Mathematics meets physics: A contribution to their interaction in the 19th and the first half of the 20th century

Es gibt wohl kaum Wissenschaftsgebiete, in denen die wechselseitige Beeinflussung stärker ist als zwischen Mathematik und Physik. Eine wichtige Frage ist dabei die nach der konkreten Ausgestaltung dieser Wechselbeziehungen, etwa an einer Universität, oder die nach prägenden Merkmalen in der Entwicklung dieser Beziehungen in einem historischen Zeitabschnitt. Im Rahmen eines mehrjährigen Akademieprojekts wurden diese Beziehungen an den Universitäten in Leipzig, Halle und Jena für den Zeitraum vom Beginn des 19. bis zur Mitte des 20. Jahrhunderts untersucht und in fünf Bänden dargestellt. Der erste dieser Bände erschien in den Abhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig, die nachfolgenden als eigenständige Reihe unter dem Titel “Studien zur Entwicklung von Mathematik und Physik in ihren Wechselwirkungen“. Ein weiterer und abschließender Band dieser Reihe (der vorliegende) beinhaltet die Beiträge einer wissenschaftshistorischen Fachtagung im Jahr 2010, die das Thema in einem internationalen Kontext einbettet. Der vorliegende Band enthält die Beiträge der Tagung “Mathematics meets physics. A contribution to their interaction in the 19th and the first half of the 20th century”, die vom 22. bis 25. März 2010 in Leipzig stattfand. Die Konferenzbeiträge bestätigen die große Variabilität in der Gestaltung der Wechselbeziehungen zwischen Mathematik und Physik. In ihnen werden u.a. verschiedene Entwicklungsprozesse im 19. und 20. Jahrhundert (zur elektromagnetischen Feldtheorie, zur Quantenmechanik, zur Quantenfeldtheorie, zur Relativitätstheorie) aus unterschiedlichen Perspektiven analysiert. Weitere Beiträge stellen allgemeinere Fragestellungen der Entwicklung der Wechselbeziehungen in den Mittelpunkt und tragen zur Frage einer möglichen Unterscheidung unterschiedlicher Entwicklungsstufen im den Wechselverhältnis von Mathematik und Physik bei. Insgesamt ist einzuschätzen: Zum einen dokumentieren die in den Beiträgen vorgelegten Ergebnisse den Wert und die Notwendigkeit von Detailuntersuchungen, um die Entwicklung der Wechselbeziehungen zwischen Mathematik und Physik in ihrer Vielfalt und mit der nötigen Präzision zu erfassen, zum anderen lassen sie in ihrer Gesamtheit noch zu beantwortende Forschungsfragen erkennen.:Vorwort Karl-Heinz Schlote, Martina Schneider: Introduction Jesper Lützen: Examples and Reflections on the Interplay between Mathematics and Physics in the 19th and 20th Century Juraj Šebesta: Mathematics as one of the basic Pillars of physical Theory: a historical and epistemological Survey Karl-Heinz Schlote, Martina Schneider: The Interrelation between Mathematics and Physics at the Universities Jena, Halle-Wittenberg and Leipzig – a Comparison Karin Reich: Der erste Professor für Theoretische Physik an der Universität Hamburg: Wilhelm Lenz Jim Ritter: Geometry as Physics: Oswald Veblen and the Princeton School Erhard Scholz: Mathematische Physik bei Hermann Weyl – zwischen „Hegelscher Physik“ und „symbolischer Konstruktion der Wirklichkeit“ Scott Walter: Henri Poincaré, theoretical Physics, and Relativity Theory in Paris Reinhard Siegmund-Schultze: Indeterminismus vor der Quantenmechanik: Richard von Mises’ wahrscheinlichkeitstheoretischer Purismus in der Theorie physikalischer Prozesse Christoph Lehner: Mathematical Foundations and physical Visions: Pascual Jordan and the Field Theory Program Jan Lacki: From Matrices to Hilbert Spaces: The Interplay of Physics and Mathematics in the Rise of Quantum Mechanics Helge Kragh: Mathematics, Relativity, and Quantum Wave Equations Klaus-Heinrich Peters: Mathematische und phänomenologische Strenge: Distributionen in der Quantenmechanik und -feldtheorie Arianna Borrelli: Angular Momentum between Physics and Mathematics Friedrich Steinle: Die Entstehung der Feldtheorie: ein ungewöhnlicher Fall der Wechselwirkung von Physik und Mathematik? Vortragsprogramm Liste der Autoren Personenverzeichni

Framing Global Mathematics

This open access book is about the shaping of international relations in mathematics over the last two hundred years. It focusses on institutions and organizations that were created to frame the international dimension of mathematical research. Today, striking evidence of globalized mathematics is provided by countless international meetings and the worldwide repository ArXiv. The text follows the sinuous path that was taken to reach this state, from the long nineteenth century, through the two wars, to the present day. International cooperation in mathematics was well established by 1900, centered in Europe. The first International Mathematical Union, IMU, founded in 1920 and disbanded in 1932, reflected above all the trauma of WW I. Since 1950 the current IMU has played an increasing role in defining mathematical excellence, as is shown both in the historical narrative and by analyzing data about the International Congresses of Mathematicians. For each of the three periods discussed, interactions are explored between world politics, the advancement of scientific infrastructures, and the inner evolution of mathematics. Readers will thus take a new look at the place of mathematics in world culture, and how international organizations can make a difference. Aimed at mathematicians, historians of science, scientists, and the scientifically inclined general public, the book will be valuable to anyone interested in the history of science on an international level

The Politics of Medicine: Power, Actors, and Ideas in the Making of Health

The practice of medicine has become the prescribing of medicine. Reflecting a construct of health defined by Rationalism, individualism, and biomedical science, medicines (pharmaceuticals) are politically constructed to be the first – and sometimes only prescribed – line of defense against illness and disease. Pharmaceuticals also represent a highly desirable, ‘recession-proof’ component of many Nation-states’ (states’) export strategies, helping advanced economies, in particular, to maintain favorable trade balances and economic growth amidst the headwinds of deindustrialization. Higher use and the overreliance on pharmaceuticals promote an outsized role for certain actors and ideas in the making of global health, referring to the systems of medical practice, the norms defining health subconsciously and consciously, the politico-economic relations and decisions that prioritize certain qualities and determinants of health, and interventions relating to health. Concentrations of power deepened under globalization, reinforcing and internationalizing specific, hegemonic ideas about health that reflect the ideas and interests of dominant actors. These dynamics further privilege certain actors and ideas in political and economic processes, which have the practical effect of predetermining outcomes. In this way, power sustains the global normative and politico-economic conditions that comprise modern health—power makes health. This dissertation employs pharmaceuticals as a proxy to examine power asymmetries and market-oriented norms relating to health. The research examines the formative ideas and structuring role of power on the political salience, interests, values, and choices of the leading actors in global health. Rather than an exclusive focus on health’s visible outcomes, the research distinguishes the influence of power asymmetries expressed through norm formation and spread. It finds that health is a core issue of the 21st century global political economy and equitable scholarly focus and practical solutioning must be applied to the ideas, contexts, content, and processes that make health

Generative Bildarbeit : Zum transformativen Potential fotografischer Praxis.

Generative Picturing uses photography as a relational, ambivalent and undisciplined medium for education and research.Wir alle sind Fotografie! – Wir fotografieren, betrachten Fotos, sind darauf abgebildet und verwenden sie. Die Fotografie berührt und verstört, sie verbindet und trennt, sie beweist und ist vieldeutig. Das Beziehungshafte, das Ambivalente und die Undiszipliniertheit – als wesentliche Eigenschaften der Fotografie – werden in der Generativen Bildarbeit für Bildungs- und Forschungsprozesse genutzt. Die Fotografie wird dabei zum transdisziplinären Praxisfeld, in dem Menschen Fotos machen, diese in Gruppenprozessen zeigen und miteinander in Dialog treten. Auf prozesshafte und partizipative Weise gelangen dabei die Beteiligten und ihre generativen Themen in den Fokus. Anhand ihrer Bilder, beim Betrachten und Diskutieren darüber, erkunden sie die eigene Wahrnehmung und die „der Anderen“, hinterfragen die Kategorien „Eigen“ und „Fremd“ und die sozialen Grenzziehungen, die damit verbunden sind. Forschen und Lernen gehen dabei Hand in Hand und werden als Erkenntnis- und Transformationsprozesse wirksam. Durch Generative Bildarbeit wird die Fotografie als beziehungshaftes, ambivalentes und undiszipliniertes Medium für Bildungs- und Forschungsprozesse zum Einsatz gebracht