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2540 results in Computational Science

Page 46 of 127

  • 7 - Design Scenes of Online Code Learning Environments

  • from PART IV - ONLINE LEARNING
  • Edited by Michael Filimowicz, Simon Fraser University, British Columbia, Veronika Tzankova, Simon Fraser University, British Columbia
  • Book:
    Teaching Computational Creativity
    Published online:
    04 July 2017
    Print publication:
    02 May 2017, pp 163-190

  • Summary

    Abstract:

    An autoethnographic exploration of online code learning environments is situated within a context of multimedia cognition frameworks. This suggests new strategies for both the design of multimedia code learning content and facilities. What emerges from this analysis is the notion of a “three-screen scene” that articulates the spatially distributed cognitive tasks of programming workspace, note taking, and expanded context. The three-screen scene is considered as its own design frame for the development of both courseware and computer labs for self-learning coding skills.

Arithmetical Wonderland
  • Andrew C. F. Liu
  • Published by:
    Mathematical Association of America
    Published online:
    27 April 2017
    Print publication:
    30 October 2015
  • Arithmetical Wonderland is intended as an unorthodox mathematics textbook for students in elementary education, in a contents course offered by a mathematics department. The scope is deliberately restricted to cover only arithmetic, even though geometric elements are introduced whenever warranted. For example, what the Euclidean Algorithm for finding the greatest common divisors of two numbers has to do with Euclid is showcased.

    Many students find mathematics somewhat daunting. It is the author's belief that much of that is caused not by the subject itself, but by the language of mathematics. In this book, much of the discussion is in dialogues between Alice, of Wonderland fame, and the twins Tweedledum and Tweedledee who hailed from Through the Looking Glass. The boys are learning High Arithmetic or Elementary Number Theory from Alice, and the reader is carried along in this academic exploration. Thus many formal proofs are converted to soothing everyday language.

    Nevertheless, the book has considerable depth. It examines many arcane corners of the subject, and raises rather unorthodox questions. For instance, Alice tells the twins that six divided by three is two only because of an implicit assumption that division is supposed to be fair, whereas fairness does not come into addition, subtraction or multiplication. Some topics often not covered are introduced rather early, such as the concepts of divisibility and congruence.

  • Treatment for third-order nonlinear differential equations based on the Adomian decomposition method

  • Part of
  • Xueqin Lv, Jianfang Gao
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 20 / Issue 1 / 2017
    Published online by Cambridge University Press:
    01 April 2017, pp. 1-10
    • Article
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  • The Adomian decomposition method (ADM) is an efficient method for solving linear and nonlinear ordinary differential equations, differential algebraic equations, partial differential equations, stochastic differential equations, and integral equations. Based on the ADM, a new analytical and numerical treatment is introduced in this research for third-order boundary-value problems. The effectiveness of the proposed approach is verified by numerical examples.

  • Certain aspects of holomorphic function theory on some genus-zero arithmetic groups

  • Part of
  • Jay Jorgenson, Lejla Smajlović, Holger Then
  • Journal:
    LMS Journal of Computation and Mathematics / Volume 19 / Issue 2 / 2016
    Published online by Cambridge University Press:
    01 March 2017, pp. 360-381
    • Article
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  • There are a number of fundamental results in the study of holomorphic function theory associated to the discrete group $\operatorname{PSL}(2,\mathbb{Z})$ , including the following statements: the ring of holomorphic modular forms is generated by the holomorphic Eisenstein series of weights four and six, denoted by $E_{4}$ and $E_{6}$ ; the smallest-weight cusp form $\unicode[STIX]{x1D6E5}$ has weight twelve and can be written as a polynomial in $E_{4}$ and $E_{6}$ ; and the Hauptmodul $j$ can be written as a multiple of $E_{4}^{3}$ divided by $\unicode[STIX]{x1D6E5}$ . The goal of the present article is to seek generalizations of these results to some other genus-zero arithmetic groups $\unicode[STIX]{x1D6E4}_{0}(N)^{+}$ with square-free level $N$ , which are related to ‘Monstrous moonshine conjectures’. Certain aspects of our results are generated from extensive computer analysis; as a result, many of the space-consuming results are made available on a publicly accessible web site. However, we do present in this article specific results for certain low-level groups.

Volterra Integral Equations
  • An Introduction to Theory and Applications
  • Hermann Brunner
  • Published online:
    02 February 2017
    Print publication:
    20 January 2017
  • This book offers a comprehensive introduction to the theory of linear and nonlinear Volterra integral equations (VIEs), ranging from Volterra's fundamental contributions and the resulting classical theory to more recent developments that include Volterra functional integral equations with various kinds of delays, VIEs with highly oscillatory kernels, and VIEs with non-compact operators. It will act as a 'stepping stone' to the literature on the advanced theory of VIEs, bringing the reader to the current state of the art in the theory. Each chapter contains a large number of exercises, extending from routine problems illustrating or complementing the theory to challenging open research problems. The increasingly important role of VIEs in the mathematical modelling of phenomena where memory effects play a key role is illustrated with some 30 concrete examples, and the notes at the end of each chapter feature complementary references as a guide to further reading.

  • Preface

  • Hermann Brunner, Hong Kong Baptist University
  • Book:
    Volterra Integral Equations
    Published online:
    02 February 2017
    Print publication:
    20 January 2017, pp xi-xvi

  • Summary

    This monograph presents an introduction to the theory of linear and non-linear Volterra integral equations, ranging from Volterra's fundamental contributions and the resulting classical theory to more recent developments. The latter include Volterra functional integral equations with various kinds of delays, Volterra integral equations with highly oscillatory kernels, and Volterra integral equations with non-compact operators. One of the aims of the book is to introduce the reader to the current state of the art in the theory of Volterra integral equations. In addition, it illustrates – by means of a representative selection of examples – the increasingly important role Volterra integral equations play in the mathematical modelling of phenomena where memory effects play a key role.

    The book is intended also as a ‘stepping stone’ to the literature on the advanced theory of Volterra integral equations, as presented, for example, in the monumental and seminal monograph by Gripenberg (1990). The notes at the end of each chapter and the annotated references point the reader to such papers and books.

    We give a brief outline of the contents of the various chapters. As will be seen, Chapters 1, 2, 3 and 6 describe what might be called the classical theory of linear and non-linear Volterra integral equations, while the other chapters are concerned with more recent developments. Those chapters will also reveal that the theory of Volterra integral equations is by no means complete, and that many challenging problems (many of which are stated as Research Problems in the Exercise sections at the ends of the chapters) remain to be addressed.

    Chapter 1 contains an introduction to the basic theory of the existence and uniqueness of solutions of linear Volterra integral equations of the first and second kind, including equations with various types of integrable kernel singularities. It also gives a brief introduction to the ill-posed nature of first-kind equations (which will play a key role in Chapter 5 when analysing systems of integral-algebraic equations).

    The focus of Chapter 2 is on the regularity properties of the solutions of the linear Volterra integral equations discussed in Chapter 1. It also contains an introduction to linear functional Volterra integral equations with vanishing or non-vanishing delays: here, the regularity of the solution depends strongly on the type of delay.

Page 46 of 127