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| doughnut-like mathematical surface: Mathematical Tools for Neuroscience Richard A. Clement, 2022-04-21 This book provides a brief but accessible introduction to a set of related, mathematical ideas that have proved useful in understanding the brain and behaviour. If you record the eye movements of a group of people watching a riverside scene then some will look at the river, some will look at the barge by the side of the river, some will look at the people on the bridge, and so on, but if a duck takes off then everybody will look at it. How come the brain is so adept at processing such biological objects? In this book it is shown that brains are especially suited to exploiting the geometric properties of such objects. Central to the geometric approach is the concept of a manifold, which extends the idea of a surface to many dimensions. The manifold can be specified by collections of n-dimensional data points or by the paths of a system through state space. Just as tangent planes can be used to analyse the local linear behaviour of points on a surface, so the extension to tangent spaces can be used to investigate the local linear behaviour of manifolds. The majority of the geometric techniques introduced are all about how to do things with tangent spaces. Examples of the geometric approach to neuroscience include the analysis of colour and spatial vision measurements and the control of eye and arm movements. Additional examples are used to extend the applications of the approach and to show that it leads to new techniques for investigating neural systems. An advantage of following a geometric approach is that it is often possible to illustrate the concepts visually and all the descriptions of the examples are complemented by comprehensively captioned diagrams. The book is intended for a reader with an interest in neuroscience who may have been introduced to calculus in the past but is not aware of the many insights obtained by a geometric approach to the brain. Appendices contain brief reviews of the required background knowledge in neuroscience and calculus. |
| doughnut-like mathematical surface: Mathematics Galore! Christopher J. Budd, Christopher Sangwin, 2001-05-17 Provides materials for eight Saturday workshops to excite teenagers about the possibilities and fun of mathematics. Each chapter begins with detailed historical and mathematical information on the subject for delivering a talk, then lists exercises for small group work. Topics include network theory for mazes, trigonometry for sundials, the design of castles, and code breaking. Annotation copyrighted by Book News, Inc., Portland, OR |
| doughnut-like mathematical surface: Mathematics: A Very Short Introduction Timothy Gowers, 2002-08-22 The aim of this book is to explain, carefully but not technically, the differences between advanced, research-level mathematics, and the sort of mathematics we learn at school. The most fundamental differences are philosophical, and readers of this book will emerge with a clearer understanding of paradoxical-sounding concepts such as infinity, curved space, and imaginary numbers. The first few chapters are about general aspects of mathematical thought. These are followed by discussions of more specific topics, and the book closes with a chapter answering common sociological questions about the mathematical community (such as Is it true that mathematicians burn out at the age of 25?) ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable. |
| doughnut-like mathematical surface: A Mathematical Gift, I Kenji Ueno, Kōji Shiga, Shigeyuki Morita, 2003 Three volumes originating from a series of lectures in mathematics given by professors of Kyoto University in Japan for high school students. |
| doughnut-like mathematical surface: Do Not Erase Jessica Wynne, 2021-06-22 A photographic exploration of mathematicians’ chalkboards “A mathematician, like a painter or poet, is a maker of patterns,” wrote the British mathematician G. H. Hardy. In Do Not Erase, photographer Jessica Wynne presents remarkable examples of this idea through images of mathematicians’ chalkboards. While other fields have replaced chalkboards with whiteboards and digital presentations, mathematicians remain loyal to chalk for puzzling out their ideas and communicating their research. Wynne offers more than one hundred stunning photographs of these chalkboards, gathered from a diverse group of mathematicians around the world. The photographs are accompanied by essays from each mathematician, reflecting on their work and processes. Together, pictures and words provide an illuminating meditation on the unique relationships among mathematics, art, and creativity. The mathematicians featured in this collection comprise exciting new voices alongside established figures, including Sun-Yung Alice Chang, Alain Connes, Misha Gromov, Andre Neves, Kasso Okoudjou, Peter Shor, Christina Sormani, Terence Tao, Claire Voisin, and many others. The companion essays give insights into how the chalkboard serves as a special medium for mathematical expression. The volume also includes an introduction by the author, an afterword by New Yorker writer Alec Wilkinson, and biographical information for each contributor. Do Not Erase is a testament to the myriad ways that mathematicians use their chalkboards to reveal the conceptual and visual beauty of their discipline—shapes, figures, formulas, and conjectures created through imagination, argument, and speculation. |
| doughnut-like mathematical surface: Mathematical Ideas Charles David Miller, Vern E. Heeren, 1993 A textbook designed with a variety of students in mind and suited for several types of courses, including mathematics for liberal arts students, survey courses in mathematics, and mathematics for prospective and in-service elementary and middle-school teachers. Some 80% of the exercises are new to this edition, which also sports extensive use of color and changes in format to create a fresh look. Annotation copyright by Book News, Inc., Portland, OR |
| doughnut-like mathematical surface: Introduction to the Mathematics of Subdivision Surfaces Lars-Erik Andersson, Neil F. Stewart, 2010-05-13 This is an introduction to the mathematical theory which underlies subdivision surfaces, as it is used in computer graphics and animation. Subdivision surfaces enable a designer to specify the approximate form of a surface that defines an object and then to refine it to get a more useful or attractive version. A considerable amount of mathematical theory is needed to understand the characteristics of the resulting surfaces, and this book explains the material carefully and rigorously. The text is highly accessible, organising subdivision methods in a unique and unambiguous hierarchy which builds insight and understanding. The material is not restricted to questions related to regularity of subdivision surfaces at so-called extraordinary points, but gives a broad discussion of the various methods. It is therefore an excellent preparation for more advanced texts that delve more deeply into special questions of regularity. |
| doughnut-like mathematical surface: Knotted Doughnuts and Other Mathematical Entertainments Martin Gardner, 2020-10-06 Martin Gardner's Mathematical Games columns in Scientific American inspired and entertained several generations of mathematicians and scientists. Gardner in his crystal-clear prose illuminated corners of mathematics, especially recreational mathematics, that most people had no idea existed. His playful spirit and inquisitive nature invite the reader into an exploration of beautiful mathematical ideas along with him. These columns were both a revelation and a gift when he wrote them; no one--before Gardner--had written about mathematics like this. They continue to be a marvel. This is the original 1986 edition and contains columns published from 1972-1974. |
| doughnut-like mathematical surface: My Best Mathematical and Logic Puzzles Martin Gardner, 2013-04-10 The noted expert selects 70 of his favorite short puzzles, including such mind-bogglers as The Returning Explorer, The Mutilated Chessboard, Scrambled Box Tops, and dozens more involving logic and basic math. Solutions included. |
| doughnut-like mathematical surface: Think of a Number M. E. Lines, 2021-12-17 How many colors are needed to color a map? Must hailstones numbers always fall to the ground? Can statistics prove anything? What is a perfect square, and who has found the ultimate one? How do numbers affect national security? What kinds of problems confront the traveling salesman? Does anyone know how best to pack balls together? What is life like in 4 (or 3 1/2) dimensions? How does a clock count, and why should we care? What number secrets do sunflowers and pine cones conceal? What is a monster doing in mathematics? These and many other fascinating questions about familiar numbers like 1, 2, and 3 are explored in Malcolm Line's second adventure into the world of numbers. Written in a lively and readable style, Think of a Number relates the story of some of the most famous problems that have confronted the world's experts over the centuries, from the earliest interests of the ancient Greeks to the very cutting-edge of modern research involving today's most powerful computers. The book explores the relationship between numbers and nature in its broadest sense and discovers the beauty of fractals and chaos. Requiring little or no prior knowledge of mathematics, this resource will be fascinating reading for anyone with an interest in numbers and their role in the natural world. |
| doughnut-like mathematical surface: Modeling Biological Systems: James W. Haefner, 2005-12-05 I Principles 1 1 Models of Systems 3 1. 1 Systems. Models. and Modeling . . . . . . . . . . . . . . . . . . . . 3 1. 2 Uses of Scientific Models . . . . . . . . . . . . . . . . . . . . . . . . 4 1. 3 Example: Island Biogeography . . . . . . . . . . . . . . . . . . . . . 6 1. 4 Classifications of Models . . . . . . . . . . . . . . . . . . . . . . . . 10 1. 5 Constraints on Model Structure . . . . . . . . . . . . . . . . . . . . . 12 1. 6 Some Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1. 7 Misuses of Models: The Dark Side . . . . . . . . . . . . . . . . . . . 13 1. 8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 The Modeling Process 17 2. 1 Models Are Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2. 2 Two Alternative Approaches . . . . . . . . . . . . . . . . . . . . . . 18 2. 3 An Example: Population Doubling Time . . . . . . . . . . . . . . . . 24 2. 4 Model Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2. 5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3 Qualitative Model Formulation 32 3. 1 How to Eat an Elephant . . . . . . . . . . . . . . . . . . . . . . . . . 32 3. 2 Forrester Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3. 3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3. 4 Errors in Forrester Diagrams . . . . . . . . . . . . . . . . . . . . . . 44 3. 5 Advantages and Disadvantages of Forrester Diagrams . . . . . . . . . 44 3. 6 Principles of Qualitative Formulation . . . . . . . . . . . . . . . . . . 45 3. 7 Model Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3. 8 Other Modeling Problems . . . . . . . . . . . . . . . . . . . . . . . . 49 viii Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. 9 Exercises 53 4 Quantitative Model Formulation: I 4. 1 From Qualitative to Quantitative . . . . . . . . . . . . . . . . . Finite Difference Equations and Differential Equations 4. 2 . . . . . . . . . . . . . . . . 4. 3 Biological Feedback in Quantitative Models . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 4 Example Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. 5 Exercises 5 Quantitative Model Formulation: I1 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 1 Physical Processes 81 . . . . . . . . . . . . . . . 5. 2 Using the Toolbox of Biological Processes 89 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 3 Useful Functions 96 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 4 Examples 102 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. 5 Exercises 104 6 Numerical Techniques 107 . . . . . . . . . . . . . . . . . . . . . . . 6. 1 Mistakes Computers Make 107 . . . . . . . . . . . . . . . . . . . . . . . . . . 6. 2 Numerical Integration 110 . . . . . . . . . . . . . . . . 6. 3 Numerical Instability and Stiff Equations 115 . . . . . . . . . . . . . . |
| doughnut-like mathematical surface: The Knot Book Colin Conrad Adams, 2004 Knots are familiar objects. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. This work offers an introduction to this theory, starting with our understanding of knots. It presents the applications of knot theory to modern chemistry, biology and physics. |
| doughnut-like mathematical surface: The Best Writing on Mathematics 2016 Mircea Pitici, 2017-03-07 The year's finest mathematics writing from around the world This annual anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, The Best Writing on Mathematics 2016 makes available to a wide audience many articles not easily found anywhere else—and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here Burkard Polster shows how to invent your own variants of the Spot It! card game, Steven Strogatz presents young Albert Einstein's proof of the Pythagorean Theorem, Joseph Dauben and Marjorie Senechal find a treasure trove of math in New York's Metropolitan Museum of Art, and Andrew Gelman explains why much scientific research based on statistical testing is spurious. In other essays, Brian Greene discusses the evolving assumptions of the physicists who developed the mathematical underpinnings of string theory, Jorge Almeida examines the misperceptions of people who attempt to predict lottery results, and Ian Stewart offers advice to authors who aspire to write successful math books for general readers. And there's much, much more. In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a bibliography of other notable writings and an introduction by the editor, Mircea Pitici. This book belongs on the shelf of anyone interested in where math has taken us—and where it is headed. |
| doughnut-like mathematical surface: Principles of Knowledge Representation and Reasoning Jon Doyle, Erik Sandewall, Pietro Torasso, 1994 The proceedings of KR '94 comprise 55 papers on topics including deduction an search, description logics, theories of knowledge and belief, nonmonotonic reasoning and belief revision, action and time, planning and decision-making and reasoning about the physical world, and the relations between KR |
| doughnut-like mathematical surface: Diasporic Agencies: Mapping the City Otherwise Nishat Awan, 2017-05-15 Diasporic Agencies addresses the neglected subject of how architecture and urban design can respond to the consequences of increasing migration. Arguing that diasporic inhabitations can only be understood as the co-production of space, subjectivity and politics, the book explores questions of difference, belonging and movement in the city. Through focusing on a series of examples, it reveals how diasporas produce new types of spaces and develop new subjectivities in the contemporary European metropolis. It explores the way in which geo-politics affects individual lives and how national and regional borders inscribe themselves onto diasporic bodies. The book claims that the multiple belongings of diasporic citizens, half-here and half-there, provoke a crisis in the standard modes of architectural representation that tend to homogenise and flatten experience. Instead Diasporic Agencies makes a case for a non-representational approach, where the displacement of the diasporic subject and their consequent reterritorialisation of space are developed as modes of thinking and doing. In parallel, mapping otherwise is proposed as a tool for spatial practitioners to work with these multi-layered spaces. The book is aimed at spatial practitioners and theorists of all sorts - architects, artists, geographers, urban designers - anyone with a general interest in mapping or those interested in working through issues related to migration and the contemporary city. |
| doughnut-like mathematical surface: The Mathematical Tourist Ivars Peterson, 1998-04-15 In the first edition of The Mathematical Tourist, renowned science journalist Ivars Peterson took readers on an unforgettable tour through the sometimes bizarre, but always fascinating, landscape of modern mathematics. Now the journey continues in a new, updated edition that includes all the latest information on mathematical proofs, fractals, prime numbers, and chaos, as well as new material on * the relationship between mathematical knots and DNA * how computers based on quantum logic can significantly speed up the factoring of large composite numbers * the relationship between four-dimensional geometry and physical theories of the nature of matter * the application of cellular automata models to social questions and the peregrinations of virtual ants * a novel mathematical model of quasicrystals based on decagon-shaped tiles Blazing a trail through rows of austere symbols and dense lines of formulae, Peterson explores the central ideas behind the work of professional mathematicians-- how and where their pieces of the mathematical puzzle fit in, the sources of their ideas, their fountains of inspiration, and the images that carry them from one discovery to another. |
| doughnut-like mathematical surface: Language and Mathematics Marcel Danesi, 2016-06-06 This book explores the many disciplinary and theoretical links between language, linguistics, and mathematics. It examines trends in linguistics, such as structuralism, conceptual metaphor theory, and other relevant theories, to show that language and mathematics have a similar structure, but differential functions, even though one without the other would not exist. |
| doughnut-like mathematical surface: The Certainty of Uncertainty Mark Schaefer, 2018-08-23 The world is full of people who are very certain—in politics, in religion, in all manner of things. In addition, political, religious, and social organizations are marketing certainty as a cure all to all life’s problems. But is such certainty possible? Or even good? The Certainty of Uncertainty explores the question of certainty by looking at the reasons human beings crave certainty and the religious responses we frequently fashion to help meet that need. The book takes an in-depth view of religion, language, our senses, our science, and our world to explore the inescapable uncertainties they reveal. We find that the certainty we crave does not exist. As we reflect on the unavoidable uncertainties in our world, we come to understand that letting go of certainty is not only necessary, it’s beneficial. For, in embracing doubt and uncertainty, we find a more meaningful and courageous religious faith, a deeper encounter with mystery, and a way to build strong relationships across religious and philosophical lines. In The Certainty of Uncertainty, we see that embracing our belief systems with humility and uncertainty can be transformative for ourselves and for our world. |
| doughnut-like mathematical surface: Calculating the Cosmos Ian Stewart, 2016-10-25 A prize-winning popular science writer uses mathematical modeling to explain the cosmos. In Calculating the Cosmos, Ian Stewart presents an exhilarating guide to the cosmos, from our solar system to the entire universe. He describes the architecture of space and time, dark matter and dark energy, how galaxies form, why stars implode, how everything began, and how it's all going to end. He considers parallel universes, the fine-tuning of the cosmos for life, what forms extraterrestrial life might take, and the likelihood of life on Earth being snuffed out by an asteroid. Beginning with the Babylonian integration of mathematics into the study of astronomy and cosmology, Stewart traces the evolution of our understanding of the cosmos: How Kepler's laws of planetary motion led Newton to formulate his theory of gravity. How, two centuries later, tiny irregularities in the motion of Mars inspired Einstein to devise his general theory of relativity. How, eighty years ago, the discovery that the universe is expanding led to the development of the Big Bang theory of its origins. How single-point origin and expansion led cosmologists to theorize new components of the universe, such as inflation, dark matter, and dark energy. But does inflation explain the structure of today's universe? Does dark matter actually exist? Could a scientific revolution that will challenge the long-held scientific orthodoxy and once again transform our understanding of the universe be on the way? In an exciting and engaging style, Calculating the Cosmos is a mathematical quest through the intricate realms of astronomy and cosmology. |
| doughnut-like mathematical surface: Topology Now! Robert Messer, Philip Straffin, 2018-10-10 Topology is a branch of mathematics packed with intriguing concepts, fascinating geometrical objects, and ingenious methods for studying them. The authors have written this textbook to make the material accessible to undergraduate students without requiring extensive prerequisites in upper-level mathematics. The approach is to cultivate the intuitive ideas of continuity, convergence, and connectedness so students can quickly delve into knot theory, the topology of surfaces and three-dimensional manifolds, fixed points and elementary homotopy theory. The fundamental concepts of point-set topology appear at the end of the book when students can see how this level of abstraction provides a sound logical basis for the geometrical ideas that have come before. This organization exposes students to the exciting world of topology now(!) rather than later. Students using this textbook should have some exposure to the geometry of objects in higher-dimensional Euclidean spaces together with an appreciation of precise mathematical definitions and proofs. |
| doughnut-like mathematical surface: The Statistical Theory of Shape Christopher G. Small, 2012-12-06 In general terms, the shape of an object, data set, or image can be de fined as the total of all information that is invariant under translations, rotations, and isotropic rescalings. Thus two objects can be said to have the same shape if they are similar in the sense of Euclidean geometry. For example, all equilateral triangles have the same shape, and so do all cubes. In applications, bodies rarely have exactly the same shape within measure ment error. In such cases the variation in shape can often be the subject of statistical analysis. The last decade has seen a considerable growth in interest in the statis tical theory of shape. This has been the result of a synthesis of a number of different areas and a recognition that there is considerable common ground among these areas in their study of shape variation. Despite this synthesis of disciplines, there are several different schools of statistical shape analysis. One of these, the Kendall school of shape analysis, uses a variety of mathe matical tools from differential geometry and probability, and is the subject of this book. The book does not assume a particularly strong background by the reader in these subjects, and so a brief introduction is provided to each of these topics. Anyone who is unfamiliar with this material is advised to consult a more complete reference. As the literature on these subjects is vast, the introductory sections can be used as a brief guide to the literature. |
| doughnut-like mathematical surface: Teaching and Learning Mathematics Peter G. Dean, 2019-01-22 School mathematics is a complex subject and an ever-changing topic, but this book will help teachers, parents and employers to understand it better. |
| doughnut-like mathematical surface: Mosaic , 1988 |
| doughnut-like mathematical surface: Visual Differential Geometry and Forms Tristan Needham, 2021-07-13 An inviting, intuitive, and visual exploration of differential geometry and forms Visual Differential Geometry and Forms fulfills two principal goals. In the first four acts, Tristan Needham puts the geometry back into differential geometry. Using 235 hand-drawn diagrams, Needham deploys Newton’s geometrical methods to provide geometrical explanations of the classical results. In the fifth act, he offers the first undergraduate introduction to differential forms that treats advanced topics in an intuitive and geometrical manner. Unique features of the first four acts include: four distinct geometrical proofs of the fundamentally important Global Gauss-Bonnet theorem, providing a stunning link between local geometry and global topology; a simple, geometrical proof of Gauss’s famous Theorema Egregium; a complete geometrical treatment of the Riemann curvature tensor of an n-manifold; and a detailed geometrical treatment of Einstein’s field equation, describing gravity as curved spacetime (General Relativity), together with its implications for gravitational waves, black holes, and cosmology. The final act elucidates such topics as the unification of all the integral theorems of vector calculus; the elegant reformulation of Maxwell’s equations of electromagnetism in terms of 2-forms; de Rham cohomology; differential geometry via Cartan’s method of moving frames; and the calculation of the Riemann tensor using curvature 2-forms. Six of the seven chapters of Act V can be read completely independently from the rest of the book. Requiring only basic calculus and geometry, Visual Differential Geometry and Forms provocatively rethinks the way this important area of mathematics should be considered and taught. |
| doughnut-like mathematical surface: The Oxford Handbook of Philosophy of Physics Robert Batterman, 2013-03-14 This Oxford Handbook provides an overview of many of the topics that currently engage philosophers of physics. It surveys new issues and the problems that have become a focus of attention in recent years. It also provides up-to-date discussions of the still very important problems that dominated the field in the past. In the late 20th Century, the philosophy of physics was largely focused on orthodox Quantum Mechanics and Relativity Theory. The measurement problem, the question of the possibility of hidden variables, and the nature of quantum locality dominated the literature on the quantum mechanics, whereas questions about relationalism vs. substantivalism, and issues about underdetermination of theories dominated the literature on spacetime. These issues still receive considerable attention from philosophers, but many have shifted their attentions to other questions related to quantum mechanics and to spacetime theories. Quantum field theory has become a major focus, particularly from the point of view of algebraic foundations. Concurrent with these trends, there has been a focus on understanding gauge invariance and symmetries. The philosophy of physics has evolved even further in recent years with attention being paid to theories that, for the most part, were largely ignored in the past. For example, the relationship between thermodynamics and statistical mechanics—-once thought to be a paradigm instance of unproblematic theory reduction—-is now a hotly debated topic. The implicit, and sometimes explicit, reductionist methodology of both philosophers and physicists has been severely criticized and attention has now turned to the explanatory and descriptive roles of non-fundamental,'' phenomenological theories. This shift of attention includes old'' theories such as classical mechanics, once deemed to be of little philosophical interest. Furthermore, some philosophers have become more interested in less fundamental'' contemporary physics such as condensed matter theory. Questions abound with implications for the nature of models, idealizations, and explanation in physics. This Handbook showcases all these aspects of this complex and dynamic discipline. |
| doughnut-like mathematical surface: One Hundred Curious Mathematical Problems William Richard Ransom, 1955 |
| doughnut-like mathematical surface: Counting on Frameworks Jack Graver, 2001-09-06 Consider a scaffolding that is constructed by bolting together rods and beams. Is it strong enough to support the workers and their equipment? This is the basic problem in rigidity theory, an area of interest to a wide range of people including those studying graph theory or mathematical modelling. |
| doughnut-like mathematical surface: Teaching Maths D.M. Neal, 2013-10-23 School mathematics is a complex subject and an ever-changing topic, but this book will help teachers, parents and employers to understand it better. |
| doughnut-like mathematical surface: English for Mathematics TIM LC UMM, 2016-09-17 English for Mathematics is written to fulfill students’ needs to learn English as a preparatory for job communication. This book is designed to provide an opportunity to develop students’ English skills more communicatively and meaningfully. It consists of twenty eight units. Each unit presents reading, writing, and speaking section. Reading section consists of prereading, reading comprehension and vocabulary exercises related to the topic of the text. In writing section, some structures and sentence patterns are completed with guided writing exercises. Meanwhile, in speaking section, students are provided with models and examples followed by practical activities which are presented in various ways. In addition, students are also equipped with listening comprehension skill which is presented in a separate textbook. The materials have been arranged and graded in accordance with their language levels. Above of all, to improve the quality of this textbook, criticism and suggestions for better editions are highly appreciated |
| doughnut-like mathematical surface: The Five Dimension Space-Time Universe;A creation and grand unified field theory model K.W. Wong, Gisela A. M. Dreschhoff, Hogne Jungner, 2014-05-01 This book is intended to present to the readers familiar with the basic skills in physics and mathematics, the 5 Dimension Space-Time field theory and its projection into the 4 Dimension Space-Time Lorentz field theory. It is not a review on other 5D theories nor is it intended as a sophisticated mathematically complete presentation, although it is certainly possible to be so formulated. |
| doughnut-like mathematical surface: Leonardo , 1982 International journal of contemporary visual artists. |
| doughnut-like mathematical surface: Time and Space Barry Dainton, 2016-04-15 The first edition (2001) of this title quickly established itself on courses on the philosophy of time and space. This fully revised and expanded new edition sees the addition of chapters on Zeno's paradoxes, speculative contemporary developments in physics, and dynamic time, making the second edition, once again, unrivalled in its breadth of coverage. Surveying both historical debates and the ideas of modern physics, Barry Dainton evaluates the central arguments in a clear and unintimidating way and is careful to keep the conceptual issues throughout comprehensible to students with little scientific or mathematical training. The book makes the philosophy of space and time accessible for anyone trying to come to grips with the complexities of this challenging subject. With over 100 original line illustrations and a full glossary of terms, the book has the requirements of students firmly in sight and will continue to serve as an essential textbook for philosophy of time and space courses. |
| doughnut-like mathematical surface: Computational Support for Discrete Mathematics Nathaniel Dean, Gregory E. Shannon, With recent technological advances in workstations, graphics, graphical user interfaces, and object oriented programming languages, a significant number of researchers are developing general-purpose software and integrated software systems for domains in discrete mathematics, including graph theory, combinatorics, combinatorial optimization, and sets. This software aims to provide effective computational tools for research, applications prototyping, and teaching. In March 1992, DIMACS sponsored a workshop on Computational Support for Discrete Mathematics in order to facilitate interactions between the researchers, developers, and educators who work in these areas. Containing refereed papers based on talks presented at the workshop, this volume documents current and past research in these areas and should provide impetus for new interactions. |
| doughnut-like mathematical surface: Astronomy and Astrophysics - Volume I Oddbjørn Engvold Bozena Czerny, John Lattanzio and Rolf Stabell, 2012-11-30 Astronomy is the science of everything – with the exception of the Earth and everything on it and inside. Astronomy has a rich heritage dating back to the myths and legends of antiquity and the course of civilization has been greatly affected by mankind’s interpretation of what they saw in the starry sky and experienced through seasonal changes associated with the Sun and Moon. Early astronomy is associated with the definition of calendars which were needed to predict the dates of such as religious festivals and the numbers of months. A gradual shift of emphasis from astronomy to its sister, astrophysics, which took place through the 19th century, is generally attributed to the measurement of reliable stellar distances and the development of spectroscopy as a tool for understanding the physical nature of stars. Many paradigms in astronomy and its many subfields are continuously being shaken. New insights in the intricacy and elegance of the cosmos are steadily being obtained. Every few decennia, our concepts of the Universe are challenged and substantially modified. The reasons for this are the continuous development of new observing techniques and instruments for observatories both ground-based and in space, in addition to considerable progress in mathematics and physics, including computational ability. Our Universe harbors numerous phenomena and processes representing conditions that cannot be duplicated in terrestrial laboratories. Astronomy therefore frequently leads to fundamentally new insight and knowledge far beyond astronomy itself. Last but not least, it represents a first inspiring introduction to natural science, especially among young people, which is an extra motivation to many scientists to contribute to the Astronomy and Astrophysics Theme of this Encyclopedia. The book on Astronomy and Astrophysics with contributions from distinguished experts in the field, represents a first inspiring introduction to natural science, especially among young people, which is an extra motivation to many scientists to contribute to the Astronomy and Astrophysics Theme of this Encyclopedia. The first chapter which treats the development of astronomy and astrophysics in a historical perspective is followed by an account of the impact of astronomy on human culture and civilization. Observational astronomy is facing a number of environmental challenges. The nature and complexity of these and how the associated problems are met and overcome are described in the third article. Various aspects of our solar system are covered by authoritative articles on the Sun, planets including their satellites and smaller bodies, plus a review of the laws of motions and orbits of celestial bodies. The detection and studies of exo-solar planetary systems is rapidly developing field in astronomy which is treated in a separate chapter. Then follow fascinating up-to-date overviews on stars describing their formation, structure and life cycles. Stars are the building blocks of larger cosmic entities leading to the enigmatic galaxies composed of billions of stars, and gradually to clusters of galaxies. The final chapters cover the origin and evolution of galaxies and the large-scale structure of the Universe, including dark matter and dark energy which are among the most fascinating problems of physics today. These two volumes are aimed at the following five major target audiences: University and College students Educators, Professional practitioners, Research personnel and Policy analysts, managers, and decision makers and NGOs. |
| doughnut-like mathematical surface: Advances in Forming, Machining and Automation M. S. Shunmugam, M. Kanthababu, 2019-11-23 This volume comprises select proceedings of the 7th International and 28th All India Manufacturing Technology, Design and Research conference 2018 (AIMTDR 2018). The papers in this volume focus on forming and machining, and discuss both conventional technologies and the latest developments and innovations, including both experimental studies and simulations; while those on automation present the latest research on hardware as well as software aspects. This volume will be of interest to researchers, and practicing engineers alike. |
| doughnut-like mathematical surface: Lost in the Math Museum: A Survival Story Colin Adams, 2022-07-20 “But when I turned the handle on the door, suddenly the buzzing went crazy. I slapped my hands over my ears, when I should have jerked the door shut. It flew open, and I was face-to-face with the Weierstrass function. It was the ugliest function I could imagine, with kinks, and kinks on kinks and kinks on those. And it was shrieking in its buzz-like way, vibrating all over like a plucked string. I stood there, frozen for just a second, and then I was sprinting after the others, with the wild frantic buzzing right behind me.” From the twisted imagination of best-selling author Colin Adams (Zombies & Calculus, The Knot Book) comes this tale of sixteen-year-old Kallie trying to escape death at the hands of the exhibits in a mathematics museum. Kallie crosses paths with Carl Gauss, Bertrand Russell, Sophie Germain, G. H. Hardy, and John von Neumann, as she tries to save herself, her dad, and his colleague Maria from the deadly Hairy Ball theorem, the harrowing Hilbert Hotel, the bisecting Ham Sandwich machine, and a variety of other mathematical menaces. It's a wild romp through a mathematical bestiary featuring the bizarre, the exotic, and the counterintuitive. You'll never think of math the same way again. |
| doughnut-like mathematical surface: Topological Phases of Matter Roderich Moessner, Joel E. Moore, 2021-04-29 Topological Phases of Matter are an exceptionally dynamic field of research: several of the most exciting recent experimental discoveries and conceptual advances in modern physics have originated in this field. These have generated new, topological, notions of order, interactions and excitations. This text provides an accessible, unified and comprehensive introduction to the phenomena surrounding topological matter, with detailed expositions of the underlying theoretical tools and conceptual framework, alongside accounts of the central experimental breakthroughs. Among the systems covered are topological insulators, magnets, semimetals, and superconductors. The emergence of new particles with remarkable properties such as fractional charge and statistics is discussed alongside possible applications such as fault-tolerant topological quantum computing. Suitable as a textbook for graduate or advanced undergraduate students, or as a reference for more experienced researchers, the book assumes little prior background, providing self-contained introductions to topics as varied as phase transitions, superconductivity, and localisation. |
| doughnut-like mathematical surface: Principles of Topology Fred H. Croom, 2016-02-17 Originally published: Philadelphia: Saunders College Publishing, 1989; slightly corrected. |
| doughnut-like mathematical surface: The Words of Mathematics: An Etymological Dictionary of Mathematical Terms in English Steven Schwartzman, 1994-12-31 Explains the orgins of over 1500 mathematical terms used in English. This book concentrates on where those terms come from and what their literal meanings are. |
| doughnut-like mathematical surface: Brain and Perception Karl H. Pribram, 2013-09-05 Presented as a series of lectures, this important volume achieves four major goals: 1) It integrates the results of the author's research as applied to pattern perception -- reviewing current brain research and showing how several lines of inquiry have been converging to produce a paradigm shift in our understanding of the neural basis of figural perception. 2) It updates the holographic hypothesis of brain function in perception. 3) It emphasizes the fact that both distributed (holistic) and localized (structural) processes characterize brain function. 4) It portrays a neural systems analysis of brain organization in figural perception by computational models -- describing processing in terms of formalisms found useful in ordering data in 20th-century physical and engineering sciences. The lectures are divided into three parts: a Prolegomenon outlining a theoretical framework for the presentation; Part I dealing with the configural aspects of perception; and Part II presenting its cognitive aspects. The appendices were developed in a collaborative effort by the author, Kunio Yasue, and Mari Jibu (both of Notre Dame Seishin University of Okayama, Japan). |
The Best Old-Fashioned Doughnuts Recipe - Food Network
Fry the doughnut holes until deep golden brown, about 90 seconds per side. Transfer to the wire rack and let drain for 1 minute. Then transfer to the paper-towel lined baking sheet to cool for …
15 Fryer-Free Baked Doughnut Recipes - Food Network
Feb 26, 2025 · Chow down a hot fresh doughnut without having to fill a big pot full of fryer oil. Photo: Matt Scroll For More Photos. 1 / 15. How To Make Baked Doughnuts.
Doughnuts Recipes - Food Network
May 23, 2025 · Chow down a hot fresh doughnut without having to fill a big pot full of fryer oil. Vegan and Gluten-Free Lemon Doughnuts. 50 States of Doughnuts 51 Photos. No matter how …
Basic Doughnuts Recipe - Food Network Kitchen
In a large, deep heavy pot or an electric fryer heat the oil to 375 degrees. In a large bowl, beat together the egg and sugar. Stir in the milk and the shortening.
28 Decadent Doughnut Recipes You Can Make at Home - Food …
Nov 20, 2024 · Chocolate-glazed, jelly-filled, cinnamon-sugar and more—these doughnut recipes from Food Network are sure to make your day a little sweeter.
Doughnut Bread Pudding with Crispy Bacon Recipe - Food Network
Preheat the oven to 350 degrees F and butter a 3-quart oval baking dish. Put the doughnuts and bread into a very large bowl and mix to combine.
How to Make Doughnuts - Food Network
May 8, 2020 · Every doughnut lover knows that today's doughnuts are better than yesterday's, and freshly made doughnuts have a completely different flavor and texture than 2-hour-old …
Yeast Doughnuts Recipe | Alton Brown - Food Network
Cut out dough using a 2 1/2-inch doughnut cutter or pastry ring and using a 7/8-inch ring for the center whole. Set on floured baking sheet, cover lightly with a tea towel, and let rise for 30 ...
12 Recipes to Make with Store-Bought Doughnuts | Food Network
May 31, 2022 · Just sandwich a scoop of ice cream between two doughnut hole halves, roll in sprinkles (or chopped candies or nuts) and enjoy. Host Jamika Pessoa makes a Glazed Donut …
Long John Doughnuts Recipe - Food Network Kitchen
For the doughnuts: Place the warm water in the bowl of a stand mixer. Add a pinch of granulated sugar, then sprinkle the yeast on top and let stand until foamy, about 10 minutes.
The Best Old-Fashioned Doughnuts Recipe - Food Network
Fry the doughnut holes until deep golden brown, about 90 seconds per side. Transfer to the wire rack and let drain for 1 minute. Then transfer to the paper-towel lined baking sheet to cool for …
15 Fryer-Free Baked Doughnut Recipes - Food Network
Feb 26, 2025 · Chow down a hot fresh doughnut without having to fill a big pot full of fryer oil. Photo: Matt Scroll For More Photos. 1 / 15. How To Make Baked Doughnuts.
Doughnuts Recipes - Food Network
May 23, 2025 · Chow down a hot fresh doughnut without having to fill a big pot full of fryer oil. Vegan and Gluten-Free Lemon Doughnuts. 50 States of Doughnuts 51 Photos. No matter how …
Basic Doughnuts Recipe - Food Network Kitchen
In a large, deep heavy pot or an electric fryer heat the oil to 375 degrees. In a large bowl, beat together the egg and sugar. Stir in the milk and the shortening.
28 Decadent Doughnut Recipes You Can Make at Home - Food …
Nov 20, 2024 · Chocolate-glazed, jelly-filled, cinnamon-sugar and more—these doughnut recipes from Food Network are sure to make your day a little sweeter.
Doughnut Bread Pudding with Crispy Bacon Recipe - Food Network
Preheat the oven to 350 degrees F and butter a 3-quart oval baking dish. Put the doughnuts and bread into a very large bowl and mix to combine.
How to Make Doughnuts - Food Network
May 8, 2020 · Every doughnut lover knows that today's doughnuts are better than yesterday's, and freshly made doughnuts have a completely different flavor and texture than 2-hour-old …
Yeast Doughnuts Recipe | Alton Brown - Food Network
Cut out dough using a 2 1/2-inch doughnut cutter or pastry ring and using a 7/8-inch ring for the center whole. Set on floured baking sheet, cover lightly with a tea towel, and let rise for 30 ...
12 Recipes to Make with Store-Bought Doughnuts | Food Network
May 31, 2022 · Just sandwich a scoop of ice cream between two doughnut hole halves, roll in sprinkles (or chopped candies or nuts) and enjoy. Host Jamika Pessoa makes a Glazed Donut …
Long John Doughnuts Recipe - Food Network Kitchen
For the doughnuts: Place the warm water in the bowl of a stand mixer. Add a pinch of granulated sugar, then sprinkle the yeast on top and let stand until foamy, about 10 minutes.
