Support achievement in the latest syllabus (9709), for examination from 2020, with a stretching, practice-driven approach that builds the advanced skills required for Cambridge exam success and progression to further study. This new edition is fully aligned with the Pure Mathematics 2 & 3 part of the latest International AS & A Level syllabus, and contains a comprehensive mapping grid so you can be sure of complete support. Get students ready for higher education with a focus on real world application. From parabolic reflectors to technology in sport, up-to-date, international examples show how mathematics is used in real life. Students have plenty of opportunities to hone their skills with extensive graduated practice and thorough worked examples. Plus, give students realistic practice for their exams with exam-style questions covering every topic. Answers are included in the back of the book with full step-by-step solutions for all exercises and exam-style questions available on the accompanying support site. The online Student Book will be available on Oxford Education Bookshelf until 2028. Access is facilitated via a unique code, which is sent in the mail. The code must be linked to an email address, creating a user account. Access may be transferred once to a new user, once the initial user no longer requires access. You will need to contact your local Educational Consultant to arrange this.

This volume presents a unified approach to constructing iterative methods for solving irregular operator equations and provides rigorous theoretical analysis for several classes of these methods. The analysis of methods includes convergence theorems as well as necessary and sufficient conditions for their convergence at a given rate. The principal groups of methods studied in the book are iterative processes based on the technique of universal linear approximations, stable gradient-type processes, and methods of stable continuous approximations. Compared to existing monographs and textbooks on ill-posed problems, the main distinguishing feature of the presented approach is that it doesna (TM)t require any structural conditions on equations under consideration, except for standard smoothness conditions. This allows to obtain in a uniform style stable iterative methods applicable to wide classes of nonlinear inverse problems. Practical efficiency of suggested algorithms is illustrated in application to inverse problems of potential theory and acoustic scattering.

The volume can be read by anyone with a basic knowledge of functional analysis.

The book will be of interest to applied mathematicians and specialists in mathematical modeling and inverse problems.

Recent progress in research, teaching and communication has arisen from the use of new tools in visualization. To be fruitful, visualization needs precision and beauty. This book is a source of mathematical illustrations by mathematicians as well as artists. It offers examples in many basic mathematical fields including polyhedra theory, group theory, solving polynomial equations, dynamical systems and differential topology.For a long time, arts, architecture, music and painting have been the source of new developments in mathematics. And vice versa, artists have often found new techniques, themes and inspiration within mathematics. Here, while mathematicians provide mathematical tools for the analysis of musical creations, the contributions from sculptors emphasize the role of mathematics in their work.

The main topics reflect the fields of mathematics in which Professor O.A. Ladyzhenskaya obtained her most influential results.

One of the main topics considered in the volume is the Navier-Stokes equations. This subject is investigated in many different directions. In particular, the existence and uniqueness results are obtained for the Navier-Stokes equations in spaces of low regularity. A sufficient condition for the regularity of solutions to the evolution Navier-Stokes equations in the three-dimensional case is derived and the stabilization of a solution to the Navier-Stokes equations to the steady-state solution and the realization of stabilization by a feedback boundary control are discussed in detail. Connections between the regularity problem for the Navier-Stokes equations and a backward uniqueness problem for the heat operator are also clarified.

Generalizations and modified Navier-Stokes equations modeling various physical phenomena such as the mixture of fluids and isotropic turbulence are also considered. Numerical results for the Navier-Stokes equations, as well as for the porous medium equation and the heat equation, obtained by the diffusion velocity method are illustrated by computer graphs.

Some other models describing various processes in continuum mechanics are studied from the mathematical point of view. In particular, a structure theorem for divergence-free vector fields in the plane for a problem arising in a micromagnetics model is proved. The absolute continuity of the spectrum of the elasticity operator appearing in a problem for an isotropic periodic elastic medium with constant shear modulus (the Hill body) is established. Time-discretizationproblems for generalized Newtonian fluids are discussed, the unique solvability of the initial-value problem for the inelastic homogeneous Boltzmann equation for hard spheres, with a diffusive term representing a random background acceleration is proved and some qualitative properties of the solution are studied. An approach to mathematical statements based on the Maxwell model and illustrated by the Lavrent'ev problem on the wave formation caused by explosion welding is presented. The global existence and uniqueness of a solution to the initial boundary-value problem for the equations arising in the modelling of the tension-driven Marangoni convection and the existence of a minimal global attractor are established. The existence results, regularity properties, and pointwise estimates for solutions to the Cauchy problem for linear and nonlinear Kolmogorov-type operators arising in diffusion theory, probability, and finance, are proved. The existence of minimizers for the energy functional in the Skyrme model for the low-energy interaction of pions which describes elementary particles as spatially localized solutions of nonlinear partial differential equations is also proved.

Several papers are devoted to the study of nonlinear elliptic and parabolic operators. Versions of the mean value theorems and Harnack inequalities are studied for the heat equation, and connections with the so-called growth theorems for more general second-order elliptic and parabolic equations in the divergence or nondivergence form are investigated. Additionally, qualitative properties of viscosity solutions of fully nonlinear partial differential inequalities of elliptic and degenerate elliptic type areclarified. Some uniqueness results for identification of quasilinear elliptic and parabolic equations are presented and the existence of smooth solutions of a class of Hessian equations on a compact Riemannian manifold without imposing any curvature restrictions on the manifold is established.

Support achievement in the latest syllabus (9709), for examination from 2020, with a stretching, practice-driven approach that builds the advanced skills required for Cambridge exam success and progression to further study.

• This new edition is fully aligned with the Pure Mathematics 1 part of the latest International AS & A Level syllabus, and contains a comprehensive mapping grid so you can be sure of complete support.
• Get students ready for higher education with a focus on real-world application. From parabolic reflectors to technology in sport, up-to-date, international examples show how mathematics is used in real life.
• Students have plenty of opportunities to hone their skills with extensive graduated practice and thorough worked examples. Plus, give students realistic practice for their exams with exam-style questions covering every topic.
• Answers are included in the back of the book with full step-by-step solutions for all exercises and exam-style questions available on the accompanying support site.

Edexcel and A Level Modular Mathematics M2 features: Student-friendly worked examples and solutions, leading up to a wealth of practice questions. Sample exam papers for thorough exam preparation. Regular review sections consolidate learning. Opportunities for stretch and challenge presented throughout the course. 'Escalator section' to step up from GCSE. PLUS Free LiveText CD-ROM, containing Solutionbank and Exam Cafe to support, motivate and inspire students to reach their potential for exam success. Solutionbank contains fully worked solutions with hints and tips for every question in the Student Books. Exam Cafe includes a revision planner and checklist as well as a fully worked examination-style paper with examiner commentary.

This detailed textbook presents a great deal of material on ordered sets not previously published in the still rather limited textbook literature. It should be suitable as a text for a course on order theory.

MATHEMATICAL JOURNEYS-a unique book of math ideas for teachers, homeschoolers, and anyone who wants to learn about the world of mathematics. Unlock secrets and joy behind math ideas. Discover how math concepts evolve and connect. Stimulate curiosity and the awe behind mathematics. Develop creative and inspiring ways to introduce us to mathematical ideas. Important tool enhances and enriches mathematics distance learning Are you a teacher, parent, homeschooler, or just plain curious or perplexed by mathematics, if so Mathematical Journeys will open new vistas into why math is so important and essential in our lives while expanding your knowledge of math. Pappas' new book focuses on looking at math ideas not often covered in the traditional classroom...showing us that math is more than knowing how to add, subtract or solve equations. Explore such ideas as Discovering the golds of mathematics or Delving into non-Euclidean geometries. Pappas' design and writing style makes us wanting to learn more about math.

This book is about scientific inquiry. Designed for early and mid-career researchers, it is a practical manual for conducting and communicating high-quality research in (mathematics) education. Based on the authors' extensive experience as researchers, as mentors, and as members of the editorial team for the Journal for Research in Mathematics Education (JRME), this book directly speaks to researchers and their communities about each phase of the process for conceptualizing, conducting, and communicating high-quality research in (mathematics) education. In the late 2010s, both JRME and Educational Studies in Mathematics celebrated 50 years of publishing high-quality research in mathematics education. Many advances in the field have occurred since the establishment of these journals, and these anniversaries marked a milestone in research in mathematics education. Indeed, fifty years represents a small step for human history but a giant leap for mathematics education. The educational research community in general (and the mathematics education community in particular) has strongly advocated for original research, placing great emphasis on building knowledge and capacity in the field. Because it is an interdisciplinary field, mathematics education has integrated means and methods for scientific inquiry from multiple disciplines. Now that the field is gaining maturity, it is a good time to take a step back and systematically consider how mathematics education researchers can engage in significant, impactful scientific inquiry.

This book offers an elementary and engaging introduction to operator theory on the Hardy-Hilbert space. It provides a firm foundation for the study of all spaces of analytic functions and of the operators on them. Blending techniques from "soft" and "hard" analysis, the book contains clear and beautiful proofs. There are numerous exercises at the end of each chapter, along with a brief guide for further study which includes references to applications to topics in engineering.

This book provides an introduction to the computer language A Programming Language (APL), providing quick access to the powerful computational capabilities of the language for the newcomer to APL. It focuses on the mathematical nature of the language, and complements standard curricula.

This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of discrete series representations). The related concept of Dirac cohomology, which is defined using Dirac operators, is a far-reaching generalization that connects index theory in differential geometry to representation theory. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective. An excellent contribution to the mathematical literature of representation theory, this self-contained exposition offers a systematic examination and panoramic view of the subject. The material will be of interest to researchers and graduate students in representation theory, differential geometry, and physics.

Calculus Reform. Or, as many would prefer, calculus renewal. These are terms that, for better or worse, have become a part of the vocabulary in mathematics departments across the country. The movement to change the nature of the calculus course at the undergraduate and secondary levels has sparked discussion and controversy in ways as diverse as the actual changes. Such interactions range from "coffee pot conversations" to university curriculum committee agendas to special sessions on calculus renewal at regional and national conferences. But what is the significance of these activities? Where have we been and where are we going with calculus and, more importantly, the entire scope of undergraduate mathematics education? In April 1996, I received a fellowship from the American Educational Research Association (AERA) and the National Science Foundation (NSF). This fellowship afforded me the opportunity to work in residence at NSF on a number of evaluation projects, including the national impact of the calculus reform movement since 1988. That project resulted in countless communications with the mathematics community and others about the status of calculus as a course in isolation and as a significant player in the overall undergraduate mathematics and science experience for students (and faculty). While at NSF (and through a second NSF grant received while at the American Association for Higher Education), I also was part of an evaluation project for the Institution-wide Reform (IR) program.

Do your students believe that division "doesn't make sense" if the divisor is greater than the dividend? Explore rich, researched-based strategies and tasks that show how students are reasoning about and making sense of mulitplication and division. This book focuses on the specialised pedagogical content knowledge that you need to teach multiplication and division effectively in grades 3-5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with these computations - not only in their current work, but also in higher-level maths and a myriad of real-world contexts. Explore rich, research-based strategies and tasks that show how students are reasoning about and making sense of multiplication and division. Use the opportunities that these and similar tasks provide to build on their understanding while identifying and correcting misunderstandings that may be keeping them from taking the next steps in learning. About the Series: You have essential understanding. It's time to put it into practise in your teaching. The Putting Essential Understanding into Practice Series moves NCTM's Essential Understanding Series into the classroom. The new series details and explores best practises for teaching the essential ideas that students must grasp about fundamental topics in mathematics - topics that are challenging to learn and teach but are critical to the development of mathematical understanding. Classroom vignettes and samples of student work bring each topic to life and questions for reader reflection open it up for hands-on exploration. Each volume underscores connections with the Common Core State Standards for Mathematics while highlighting the knowledge of learners, curriculum, understanding into practise, instructional strategies and assessment that pedagogical content knowledge entails. Maximise the potential of student-centred learning and teaching by putting essential understanding into practise.

The book gives an up-to-date account of various approaches availablefor the analysis of infectious disease data. Most of the methods havebeen developed only recently, and for those based on particularlymodern mathematics, details of the computation are carefullyillustrated. Interpretation is discussed at some length and the emphasisthroughout is on making statistical inferences about epidemiologicallyimportant parameters.Niels G. Becker is Reader in Statistics at La Trobe University,Australia.