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We present evidence of a secondary school teacher’s Knowledge of Mathematics Teaching when she teaches linear equations in a word problem setting. We focus on analyzing the examples and mathematics support that she provides her students with during two classes. When faced with contingencies, she takes decisions whereby she adapts or creates emerging action plans in order to enrich and help her students overcome their difficulties to learn linear equations. The decisions and actions observed during the classroom practices demonstrated that providing support and examples for her students and accomplishing the lesson goals—mediated by her teaching knowledge—were at odds with each other.

This book places thermodynamics on a system-theoretic foundation so as to harmonize it with classical mechanics. Using the highest standards of exposition and rigor, the authors develop a novel formulation of thermodynamics that can be viewed as a moderate-sized system theory as compared to statistical thermodynamics. This middle-ground theory involves deterministic large-scale dynamical system models that bridge the gap between classical and statistical thermodynamics. The authors' theory is motivated by the fact that a discipline as cardinal as thermodynamics--entrusted with some of the most perplexing secrets of our universe--demands far more than physical mathematics as its underpinning. Even though many great physicists, such as Archimedes, Newton, and Lagrange, have humbled us with their mathematically seamless eurekas over the centuries, this book suggests that a great many physicists and engineers who have developed the theory of thermodynamics seem to have forgotten that mathematics, when used rigorously, is the irrefutable pathway to truth. This book uses system theoretic ideas to bring coherence, clarity, and precision to an extremely important and poorly understood classical area of science.

This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis. It gives a thorough and modern approach to the classical theory and presents important and compelling applications across a spectrum of mathematics: dynamical systems, singular integral operators, inverse problems, the geometry of mappings, and the calculus of variations. It also gives an account of recent advances in harmonic analysis and their applications in the geometric theory of mappings. The book explains that the existence, regularity, and singular set structures for second-order divergence-type equations--the most important class of PDEs in applications--are determined by the mathematics underpinning the geometry, structure, and dimension of fractal sets; moduli spaces of Riemann surfaces; and conformal dynamical systems. These topics are inextricably linked by the theory of quasiconformal mappings. Further, the interplay between them allows the authors to extend classical results to more general settings for wider applicability, providing new and often optimal answers to questions of existence, regularity, and geometric properties of solutions to nonlinear systems in both elliptic and degenerate elliptic settings.

The provision of multidimensional holistic mathematics support for students transitioning from secondary school to university level is identified as a challenge currently facing global education systems including in South Africa. Ongoing attempts to select effective mathematics learning support reflect the enduring nature of a problem with multifaceted contributing factors. Despite a wealth of literature there has been little effort to consolidate diverse sources into a comprehensive and useful understanding. This study aims to close this gap by synthesising and integrating these disparate elements into a unified framework. The research employs a comprehensive rapid literature review, meticulously following the PRISMA guidelines. The review concentrates on papers which investigate the impact of specific types of mathematical support, on those conducting analyses of the efficacy of distinct mathematical support approaches, and papers proposing solutions towards improving first-year South African university students’ mathematics performance. To mitigate bias, the Rayyan AI-assisted literature review platform is utilised, ensuring a rigorous and unbiased selection process. This rapid literature review reveals a general hindrance to holistic mathematics learning support, in the form of an absence of a unified framework. We see this deficiency as potentially detrimentally impacting students, and as underscoring the need for a comprehensive approach. The findings suggest that mathematics support can be enhanced by integrating the core elements of a holistic approach, together with acknowledging the interconnectedness and mutual influence of the various elements.

This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line.

Mathematics and statistics support is essential in today’s higher education system. It gives students the guidance, expertise, skills and nurturing they need to be successful in their studies and beyond. It offers them opportunities to explore past their limits, self-motivate, self-assess, seek relevance, and ask questions. The Maths Hub at La Trobe University, Australia, recently changed, as we all did, with our face-to-face support in a physical space transitioned to face-to-face support via Zoom, almost overnight due to the COVID-19 pandemic. This is our tale – how we transitioned from the vibrant, busy, demanding physical space to a wholly online world. The advantages, the surprises, the incidents, the attendances, and the feedback, will be highlighted. Not only was the interaction and intimacy of one-to-one sessions successful, but also the increased engagement in the tailored cross-disciplinary online modules that we offer. It is the worst of times, but it is also the best of times for creating innovative online opportunities, and it is changing times. But the thirst for mathematics and statistics support remains strong.

This paper begins to build a theoretical framework that would enable the pharmaceutical industry to use network complexity measures as a way to identify drug targets. The variability of a betweenness measure for a network node is examined through different methods of network perturbation. Our results indicate a robustness of betweenness centrality in the identification of target genes.

Positron emission tomography, an imaging tool using radiolabeled tracers in humans and preclinical species, has been widely used in recent years in drug development, particularly in the central nervous system. One important goal of PET in drug development is assessing the occupancy of various molecular targets (e.g., receptors, transporters, enzymes) by exogenous drugs. The current linear mathematical approaches used to determine occupancy using PET imaging experiments are presented. These algorithms use results from multiple regions with different target content in two scans, a baseline (pre-drug) scan and a post-drug scan. New mathematical estimation approaches to determine target occupancy, using maximum likelihood, are presented. A major challenge in these methods is the proper definition of the covariance matrix of the regional binding measures, accounting for different variance of the individual regional measures and their nonzero covariance, factors that have been ignored by conventional methods. The novel methods are compared to standard methods using simulation and real human occupancy data. The simulation data showed the expected reduction in variance and bias using the proper maximum likelihood methods, when the assumptions of the estimation method matched those in simulation. Between-method differences for data from human occupancy studies were less obvious, in part due to small dataset sizes. These maximum likelihood methods form the basis for development of improved PET covariance models, in order to minimize bias and variance in PET occupancy studies.

This is the first book to present a complete characterization of Stein-Tomas type Fourier restriction estimates for large classes of smooth hypersurfaces in three dimensions, including all real-analytic hypersurfaces. The range of Lebesgue spaces for which these estimates are valid is described in terms of Newton polyhedra associated to the given surface.Isroil Ikromov and Detlef Müller begin with Elias M. Stein's concept of Fourier restriction and some relations between the decay of the Fourier transform of the surface measure and Stein-Tomas type restriction estimates. Varchenko's ideas relating Fourier decay to associated Newton polyhedra are briefly explained, particularly the concept of adapted coordinates and the notion of height. It turns out that these classical tools essentially suffice already to treat the case where there exist linear adapted coordinates, and thus Ikromov and Müller concentrate on the remaining case. Here the notion of r-height is introduced, which proves to be the right new concept. They then describe decomposition techniques and related stopping time algorithms that allow to partition the given surface into various pieces, which can eventually be handled by means of oscillatory integral estimates. Different interpolation techniques are presented and used, from complex to more recent real methods by Bak and Seeger.Fourier restriction plays an important role in several fields, in particular in real and harmonic analysis, number theory, and PDEs. This book will interest graduate students and researchers working in such fields.

The goals of this article and special issue are to highlight the value of mathematical biology approaches in industry, help foster future interactions, and suggest ways for mathematics Ph.D. students and postdocs to move into industry careers. We include a candid examination of the advantages and challenges of doing mathematics in the biopharma industry, a broad overview of the types of mathematics being applied, information about academic collaborations, and career advice for those looking to move from academia to industry (including graduating Ph.D. students).