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Planar Maps, Random Walks and Circle Packing

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Book Series: Lecture Notes in Mathematics ISBN: 9783030279684 Year: Pages: 120 DOI: 10.1007/978-3-030-27968-4 Language: English
Publisher: Springer Nature
Subject: Mathematics
Added to DOAB on : 2020-01-29 11:21:10
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This open access book focuses on the interplay between random walks on planar maps and Koebe’s circle packing theorem. Further topics covered include electric networks, the He–Schramm theorem on infinite circle packings, uniform spanning trees of planar maps, local limits of finite planar maps and the almost sure recurrence of simple random walks on these limits. One of its main goals is to present a self-contained proof that the uniform infinite planar triangulation (UIPT) is almost surely recurrent. Full proofs of all statements are provided. A planar map is a graph that can be drawn in the plane without crossing edges, together with a specification of the cyclic ordering of the edges incident to each vertex. One widely applicable method of drawing planar graphs is given by Koebe’s circle packing theorem (1936). Various geometric properties of these drawings, such as existence of accumulation points and bounds on the radii, encode important probabilistic information, such as the recurrence/transience of simple random walks and connectivity of the uniform spanning forest. This deep connection is especially fruitful to the study of random planar maps. The book is aimed at researchers and graduate students in mathematics and is suitable for a single-semester course; only a basic knowledge of graduate level probability theory is assumed.

Measure, Integration & Real Analysis

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Book Series: Graduate Texts in Mathematics ISBN: 9783030331436 Year: Pages: 411 DOI: 10.1007/978-3-030-33143-6 Language: English
Publisher: Springer Nature
Subject: Mathematics
Added to DOAB on : 2020-02-04 11:21:16

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This open access textbook welcomes students into the fundamental theory of measure, integration, and real analysis. Focusing on an accessible approach, Axler lays the foundations for further study by promoting a deep understanding of key results. Content is carefully curated to suit a single course, or two-semester sequence of courses, creating a versatile entry point for graduate studies in all areas of pure and applied mathematics. Motivated by a brief review of Riemann integration and its deficiencies, the text begins by immersing students in the concepts of measure and integration. Lebesgue measure and abstract measures are developed together, with each providing key insight into the main ideas of the other approach. Lebesgue integration links into results such as the Lebesgue Differentiation Theorem. The development of products of abstract measures leads to Lebesgue measure on Rn. Chapters on Banach spaces, Lp spaces, and Hilbert spaces showcase major results such as the Hahn–Banach Theorem, Hölder’s Inequality, and the Riesz Representation Theorem. An in-depth study of linear maps on Hilbert spaces culminates in the Spectral Theorem and Singular Value Decomposition for compact operators, with an optional interlude in real and complex measures. Building on the Hilbert space material, a chapter on Fourier analysis provides an invaluable introduction to Fourier series and the Fourier transform. The final chapter offers a taste of probability. Extensively class tested at multiple universities and written by an award-winning mathematical expositor, Measure, Integration & Real Analysis is an ideal resource for students at the start of their journey into graduate mathematics. A prerequisite of elementary undergraduate real analysis is assumed; students and instructors looking to reinforce these ideas will appreciate the electronic Supplement for Measure, Integration & Real Analysis that is freely available online.

Programming for Computations - Python

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Book Series: Texts in Computational Science and Engineering ISBN: 9783030168773 Year: Pages: 332 DOI: 10.1007/978-3-030-16877-3 Language: English
Publisher: Springer Nature
Subject: Mathematics --- Science (General) --- Computer Science
Added to DOAB on : 2020-02-04 11:21:15
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Mathematics; Computer mathematics; Numerical analysis; Computer software; Numerical analysis

Forecasting and Assessing Risk of Individual Electricity Peaks

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Book Series: Mathematics of Planet Earth ISBN: 9783030286699 Year: Pages: 97 DOI: 10.1007/978-3-030-28669-9 Language: English
Publisher: Springer Nature
Subject: Mathematics --- Agriculture (General)
Added to DOAB on : 2020-02-04 11:21:18
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The overarching aim of this open access book is to present self-contained theory and algorithms for investigation and prediction of electric demand peaks. A cross-section of popular demand forecasting algorithms from statistics, machine learning and mathematics is presented, followed by extreme value theory techniques with examples. In order to achieve carbon targets, good forecasts of peaks are essential. For instance, shifting demand or charging battery depends on correct demand predictions in time. Majority of forecasting algorithms historically were focused on average load prediction. In order to model the peaks, methods from extreme value theory are applied. This allows us to study extremes without making any assumption on the central parts of demand distribution and to predict beyond the range of available data. While applied on individual loads, the techniques described in this book can be extended naturally to substations, or to commercial settings. Extreme value theory techniques presented can be also used across other disciplines, for example for predicting heavy rainfalls, wind speed, solar radiation and extreme weather events. The book is intended for students, academics, engineers and professionals that are interested in short term load prediction, energy data analytics, battery control, demand side response and data science in general.

National Reflections on the Netherlands Didactics of Mathematics

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Book Series: ICME-13 Monographs ISBN: 9783030338244 Year: Pages: 348 DOI: 10.1007/978-3-030-33824-4 Language: English
Publisher: Springer Nature
Subject: Education
Added to DOAB on : 2020-02-04 11:21:04
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This open access book, inspired by the ICME 13 Thematic Afternoon on “European Didactic Traditions”, consists of 17 chapters, in which educators from the Netherlands reflect on the teaching and learning of mathematics in their country and the role of the Dutch domain-specific instruction theory of Realistic Mathematics Education. Written by mathematics teachers, mathematics teacher educators, school advisors, and developers and researchers in the field of instructional material, textbooks, and examinations, the book offers a multitude of perspectives on important issues in Dutch mathematics education, both at primary and secondary school levels. Topics addressed include the theoretical underpinnings of the Dutch approach, the subject of mathematics in the Dutch educational system, teacher education and testing, the history of mathematics education and the use of history in teaching of mathematics, changes over time in subject matter domains and in the use of technology, and the process of innovation and how the Dutch and in particular one Dutch institute have worked on the reform.

International Reflections on the Netherlands Didactics of Mathematics

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Book Series: ICME-13 Monographs ISBN: 9783030202231 Year: Pages: 366 DOI: 10.1007/978-3-030-20223-1 Language: English
Publisher: Springer Nature
Subject: Education
Added to DOAB on : 2020-02-04 11:21:04
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This open access book, inspired by the ICME 13 Thematic Afternoon on “European Didactic Traditions”, takes readers on a journey with mathematics education researchers, developers and educators in eighteen countries, who reflect on their experiences with Realistic Mathematics Education (RME), the domain-specific instruction theory for mathematics education developed in the Netherlands since the late 1960s. Authors from outside the Netherlands discuss what aspects of RME appeal to them, their criticisms of RME and their past and current RME-based projects. It is clear that a particular approach to mathematics education cannot simply be transplanted to another country. As such, in eighteen chapters the authors describe how they have adapted RME to their individual circumstances and view on mathematics education, and tell their personal stories about how RME has influenced their thinking on mathematics education.

Boundary Value Problems, Weyl Functions, and Differential Operators

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Book Series: Monographs in Mathematics ISBN: 9783030367145 Year: Pages: 772 DOI: 10.1007/978-3-030-36714-5 Language: English
Publisher: Springer Nature
Subject: Science (General) --- Mathematics
Added to DOAB on : 2020-02-04 11:21:15
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This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory, employing abstract boundary mappings and Weyl functions. It includes self-contained treatments of the extension theory of symmetric operators and relations, spectral characterizations of selfadjoint operators in terms of the analytic properties of Weyl functions, form methods for semibounded operators, and functional analytic models for reproducing kernel Hilbert spaces. Further, it illustrates these abstract methods for various applications, including Sturm-Liouville operators, canonical systems of differential equations, and multidimensional Schrödinger operators, where the abstract Weyl function appears as either the classical Titchmarsh-Weyl coefficient or the Dirichlet-to-Neumann map. The book is a valuable reference text for researchers in the areas of differential equations, functional analysis, mathematical physics, and system theory. Moreover, thanks to its detailed exposition of the theory, it is also accessible and useful for advanced students and researchers in other branches of natural sciences and engineering.

Advances in Energy System Optimization

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Book Series: Trends in Mathematics ISBN: 9783030321574 Year: Pages: 178 DOI: 10.1007/978-3-030-32157-4 Language: English
Publisher: Springer Nature
Subject: Science (General) --- Business and Management --- Mathematics
Added to DOAB on : 2020-02-04 11:21:16
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The papers presented in this open access book address diverse challenges in decarbonizing energy systems, ranging from operational to investment planning problems, from market economics to technical and environmental considerations, from distribution grids to transmission grids, and from theoretical considerations to data provision concerns and applied case studies. While most papers have a clear methodological focus, they address policy-relevant questions at the same time. The target audience therefore includes academics and experts in industry as well as policy makers, who are interested in state-of-the-art quantitative modelling of policy relevant problems in energy systems. The 2nd International Symposium on Energy System Optimization (ISESO 2018) was held at the Karlsruhe Institute of Technology (KIT) under the symposium theme “Bridging the Gap Between Mathematical Modelling and Policy Support” on October 10th and 11th 2018. ISESO 2018 was organized by the KIT, the Heidelberg Institute for Theoretical Studies (HITS), the Heidelberg University, the German Aerospace Center and the University of Stuttgart.

De sphaera of Johannes de Sacrobosco in the Early Modern Period

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ISBN: 9783030308339 Year: Pages: 396 DOI: 10.1007/978-3-030-30833-9 Language: English
Publisher: Springer Nature
Subject: Languages and Literatures --- Science (General) --- History --- Mathematics
Added to DOAB on : 2020-02-05 11:21:05
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This open access book explores commentaries on an influential text of pre-Copernican astronomy in Europe. It features essays that take a close look at key intellectuals and how they engaged with the main ideas of this qualitative introduction to geocentric cosmology. Johannes de Sacrobosco compiled his Tractatus de sphaera during the thirteenth century in the frame of his teaching activities at the then recently founded University of Paris. It soon became a mandatory text all over Europe. As a result, a tradition of commentaries to the text was soon established and flourished until the second half of the 17th century. Here, readers will find an informative overview of these commentaries complete with a rich context. The essays explore the educational and social backgrounds of the writers. They also detail how their careers developed after the publication of their commentaries, the institutions and patrons they were affiliated with, what their agenda was, and whether and how they actually accomplished it. The editor of this collection considers these scientific commentaries as genuine scientific works. The contributors investigate them here not only in reference to the work on which it comments but also, and especially, as independent scientific contributions that are socially, institutionally, and intellectually contextualized around their authors.

Student Misconceptions and Errors in Physics and Mathematics

Authors: --- --- --- --- et al.
Book Series: IEA Research for Education ISBN: 9783030301880 Year: Pages: 165 DOI: 10.1007/978-3-030-30188-0 Language: English
Publisher: Springer Nature
Subject: Education --- Mathematics
Added to DOAB on : 2020-02-05 11:21:17

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This open access report explores the nature and extent of students’ misconceptions and misunderstandings related to core concepts in physics and mathematics and physics across grades four, eight and 12. Twenty years of data from the IEA’s Trends in International Mathematics and Science Study (TIMSS) and TIMSS Advanced assessments are analyzed, specifically for five countries (Italy, Norway, Russian Federation, Slovenia, and the United States) who participated in all or almost all TIMSS and TIMSS Advanced assessments between 1995 and 2015. The report focuses on students’ understandings related to gravitational force in physics and linear equations in mathematics. It identifies some specific misconceptions, errors, and misunderstandings demonstrated by the TIMSS Advanced grade 12 students for these core concepts, and shows how these can be traced back to poor foundational development of these concepts in earlier grades. Patterns in misconceptions and misunderstandings are reported by grade, country, and gender. In addition, specific misconceptions and misunderstandings are tracked over time, using trend items administered in multiple assessment cycles. The study and associated methodology may enable education systems to help identify specific needs in the curriculum, improve inform instruction across grades and also raise possibilities for future TIMSS assessment design and reporting that may provide more diagnostic outcomes.

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