Tag Archives: mathematics

Category Theory for the Sciences

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Category Theory for the Sciences by David I. Spivak
2014 | ISBN: 0262028131 | English | 496 pages | PDF | 7 MB

Category Theory for the Sciences
Category theory was invented in the 1940s to unify and synthesize different areas in mathematics, and it has proven remarkably successful in enabling powerful communication between disparate fields and subfields within mathematics. This book shows that category theory can be useful outside of mathematics as a rigorous, flexible, and coherent modeling language throughout the sciences. Information is inherently dynamic; the same ideas can be organized and reorganized in countless ways, and the ability to translate between such organizational structures is becoming increasingly important in the sciences. Category theory offers a unifying framework for information modeling that can facilitate the translation of knowledge between disciplines. Written in an engaging and straightforward style, and assuming little background in mathematics, the book is rigorous but accessible to non-mathematicians. Using databases as an entry to category theory, it begins with sets and functions, then introduces the reader to notions that are fundamental in mathematics: monoids, groups, orders, and graphs — categories in disguise. After explaining the "big three" concepts of category theory — categories, functors, and natural transformations — the book covers other topics, including limits, colimits, functor categories, sheaves, monads, and operads. The book explains category theory by examples and exercises rather than focusing on theorems and proofs. It includes more than 300 exercises, with selected solutions. is intended to create a bridge between the vast array of mathematical concepts used by mathematicians and the models and frameworks of such scientific disciplines as computation, neuroscience, and physics.
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Non-Linear Instabilities in Plasmas and Hydrodynamics

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Non-Linear Instabilities in Plasmas and Hydrodynamics
English | Physics, Mathematics | November 1999 | ISBN-10: 0750304839 | 180 pages | Djvu | 1 mb
For the first time in a single book, Non-Linear Instabilities in Plasmas and Hydrodynamics presents the underlying physics of fast secondary instabilities.

Non-Linear Instabilities in Plasmas and Hydrodynamics
This exceptionally well-written, introductory book discusses the basic ideas of the physics of secondary or induced, nonlinear instabilities in wave-sustaining media. The authors, world-renowned experts in the field, have brought together the results of papers scattered throughout the literature to explain subjects as diverse as fluctuation chaos, wave-turbulent instabilities, vortex dynamos, beam-plasma interactions, plasma confinement, and the origins of typhoons in the Earth's atmosphere and magnetic fields in galaxies. Paving the way for new and exciting research in the future, this broad, interdisciplinary book enables a wide range of physicists to apply the concepts discussed to obtain new results in plasma physics, space physics, hydrodynamics, and geophysics.
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Nonlinear Waves in Elastic Crystals

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Nonlinear Waves in Elastic Crystals
English | Physics, Mathematics | January 2000 | ISBN-10: 0198534841 | 320 pages | Djvu | 3.2 mb
The mathematical modelling of changing structures in materials is of increasing importance to industry where applications of the theory are found in subjects as diverse as aerospace and medicine.

Nonlinear Waves in Elastic Crystals
This book deals with aspects of the nonlinear dynamics of deformable ordered solids (known as elastic crystals) where the nonlinear effects combine or compete with each other. Physical and mathematical models are discused and computational aspects are also included. Different models are considered – on discrete as well as continuum scales – applying heat, electricity, or magnetism to the crystal structure and these are analysed using the equations of rational mechanics. In this way the student is introduced to the important equations of nonlinear science that describe shock waves, solitons and chaos and also the non-exactly integrable systems or partial differential equations. A large number of problems and examples are included, many taken from recent research and involving both one-dimensional and two-dimensional problems as well as some coupled degress of freedom.
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The Weyl Operator and its Generalization

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The Weyl Operator and its Generalization (Pseudo-Differential Operators) by Leon Cohen
English | 2012-12-12 | ISBN: 3034802935 | PDF | 171 pages | 0,9 MB

The Weyl Operator and its Generalization
The discovery of quantum mechanics in the years 1925-1930 necessitated the consideration of associating ordinary functions with non-commuting operators. Methods were proposed by Born/Jordan, Kirkwood, and Weyl. Sometime later, Moyal saw the connection between the Weyl rule and the Wigner distribution, which had been proposed by Wigner in 1932 as a way of doing quantum statistical mechanics. The basic idea of associating functions with operators has since been generalized and developed to a high degree. It has found several application fields, including quantum mechanics, pseudo-differential operators, time-frequency analysis, quantum optics, wave propagation, differential equations, image processing, radar, and sonar. This book aims at bringing together the results from the above mentioned fields in a unified manner and showing the reader how the methods have been applied. A wide audience is addressed, particularly students and researchers who want to obtain an up-to-date working knowledge of the field. The mathematics is accessible to the uninitiated reader and is presented in a straightforward manner.
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A Course on Borel Sets by S. M. Srivastava

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A Course on Borel Sets by S. M. Srivastava
English | April 13, 1998 | ISBN: 0387984127 | 271 Pages | PDF | 1 MB
A thorough introduction to Borel sets and measurable selections, acting as a stepping stone to descriptive set theory by presenting such important techniques as universal sets, prewellordering, scales, etc.

A Course on Borel Sets by S. M. Srivastava
It contains significant applications to other branches of mathematics and serves as a self-contained reference accessible by mathematicians in many different disciplines.
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Abstract Harmonic Analysis of Continuous Wavelet Transforms (Lecture Notes in Mathematics)

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Abstract Harmonic Analysis of Continuous Wavelet Transforms (Lecture Notes in Mathematics) by Hartmut Führ
English | Apr 6, 2005 | ISBN: 3540242597 | 193 Pages | PDF | 1 MB
This volume contains a systematic discussion of wavelet-type inversion formulae based on group representations, and their close connection to the Plancherel formula for locally compact groups.

Abstract Harmonic Analysis of Continuous Wavelet Transforms (Lecture Notes in Mathematics)
The connection is demonstrated by the discussion of a toy example, and then employed for two purposes: Mathematically, it serves as a powerful tool, yielding existence results and criteria for inversion formulae which generalize many of the known results. Moreover, the connection provides the starting point for a – reasonably self-contained – exposition of Plancherel theory. Therefore, the volume can also be read as a problem-driven introduction to the Plancherel formula.
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Precisely Predictable Dirac Observables

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Precisely Predictable Dirac Observables
English | Physics, Mechanics, Mathematics | 18. October 2006 | ISBN-10: 1402051689 | 269 pages | pdf | 3 mb
This work presents a Clean Quantum Theory of the Electron, based on Dirac's equation.

Precisely Predictable Dirac Observables
"Clean" in the sense of a complete mathematical explanation of the well known paradoxes of Dirac's theory and a connection to classical theory. It discusses the existence of an accurate split between physical states belonging to the electron and to the positron as well as the fact that precisely predictable observables must preserve this split.
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TTC How We Learn (Audiobook)

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TTC How We Learn (Audiobook)
English | MP3 128 kbps | 24 Lectures | 361 MB
Learning is a lifelong adventure. It starts in your mother's womb, accelerates to high speed in infancy and childhood, and continues through every age, whether you're actively engaged in mastering a new skill, intuitively discovering an unfamiliar place, or just sleeping, which is fundamental to helping you consolidate and hold on to what you've learned. You are truly born to learn around the clock.

TTC How We Learn (Audiobook)
But few of us know how we learn, which is the key to learning and studying more effectively. For example, you may be surprised by the following:

People tend to misjudge what they have learned well, what they don't yet know, and what they do and do not need to practice.
Moments of confusion, frustration, uncertainty, and lack of confidence are part of the process of acquiring new skills and new knowledge.
Humans and animals explore their worlds for the sake of learning, regardless of rewards and punishment connected with success.
You can teach an old dog new tricks. In fact, older learners have the benefit of prior knowledge and critical skills—two advantages in learning.
Shedding light on what's going on when we learn and dispelling common myths about the subject, How We Learn introduces you to this practical and accessible science in 24 half-hour lectures presented by Professor Monisha Pasupathi of the University of Utah, an award-winning psychology teacher and expert on how people of all ages learn.

A Course about You

Customers of The Great Courses are already devoted to lifelong learning and may be surprised at how complicated the process of learning is. We have a single word for it—learn—but it occurs in a fascinating variety of ways, which Professor Pasupathi recounts in detail. She describes a wide range of experiments that may strike a familiar chord as you recognize something about yourself or others:

scripts: We have trouble recalling specific events until we have first learned scripts for those events. Young children are prodigious learners of scripts, but so are first-time parents, college freshmen, foreign travelers, and new employees.
Variable ratio reinforcement: Children whining for candy are usually refused, but the few occasions when parents give in encourage maximal display of the behavior. The same principle is behind the success of slot machines and other unpredictable rewards.
Storytelling: Telling stories is fundamentally an act of learning about ourselves. The way we recount experiences, usually shortly after the event, has lasting effects on the way we remember those experiences and what we learn from them.
Sleeper effect: Have you ever heard something from an unreliable source and later found yourself believing it? Over time, we tend to remember information but forget the source. Paradoxically, this effect is stronger when the source is less credible.
Dr. Pasupathi's many examples cover the modern history of research on learning—from behaviorist theory in the early 20th century to the most recent debates about whether IQ can be separated from achievement, or whether a spectrum of different learning styles and multiple intelligences really exist.

What You Will Learn

You start by examining 10 myths about learning. These can get in the way of making the fullest use of the extraordinary capacity for learning and include widespread beliefs, such as that college-educated people already know how to maximize learning or that a person must be interested in a subject in order to learn it.

Professor Pasupathi then covers mistaken theories of learning, such as that lab animals and humans learn in the same way or that the brain is a tabula rasa, a blank slate that can absorb information without preparation. Babies might seem to be a counterexample, showing that you can learn from scratch. However, you examine what newborns must know at birth in order for them to learn so much, so quickly.

Next you explore in depth how humans master different tasks, from learning a native language or a second language, to becoming adept at a sport or a musical instrument, to learning a new city or a problem-solving strategy, to grasping the distinctive style of thinking required in mathematics and science. Then you look inside the learning process itself, where many factors come into play, including what is being learned and the context, along with the emotions, motivations, and goals of the learner. You close by considering individual differences. Some people seem to learn without effort. How do they do it?

Tips on Learning

Along the way, Professor Pasupathi offers frequent advice on how to excel in many different learning situations:

Mastering material: Testing yourself is a very effective strategy for mastering difficult material. Try taking a blank sheet of paper and writing down everything you can recall about the subject. Then go back and review the material. Next, try another blank sheet of paper.
Second-language learning: Becoming fluent in a second language in adulthood is difficult because your brain is tuned to your native language and misses important clues in the new language. To overcome this obstacle, immerse yourself among native speakers of the new language.
Motivating a child: When trying to motivate a schoolchild to learn, avoid controlling language, create opportunities to give the child a sense of choice, and be careful about excessive praise and other forms of rewards, which can actually undermine learning.
Maintaining a learning edge: Middle-aged and older adults can preserve their learning aptitude by exercising to maintain cardiovascular health, staying mentally active, and periodically trying a new challenge, such as learning to draw or studying new dance steps.
Adventures in Learning

Winner of prestigious teaching awards from her university's chapter of Psi Chi, the National Honor Society in Psychology, Dr. Pasupathi brings today's exciting field of learning research alive. Her descriptions of ongoing work in her field, in which she is a prominent participant, are vivid and insightful, allowing you to put yourself into a given experiment and ask, "How would I react under these circumstances? What does this tell me about my own approach to learning?"

By the time How We Learn ends, you will appreciate the incredible breadth of what we learn in our lifetimes, understand the commonality and diversity of human learning experiences, and come away with strategies for enhancing your own adventures in learning.

"Learning is a human birthright," says Professor Pasupathi. "Everything about us is built for lifelong learning—from our unusually long childhood and our large prefrontal cortex to our interest in novelty and challenge." And she finds reason for optimism about the future of humanity due to our almost miraculous capacity to learn.

About Your Professor

Dr. Monisha Pasupathi is Associate Professor of Psychology at the University of Utah. She holds a Ph.D. in Psychology from Stanford University. She joined the faculty at Utah in 1999 after completing a postdoctoral fellowship at the Max Planck Institute for Human Development in Germany.

Professor Pasupathi has been honored multiple times for her teaching. She was named Best Psychology Professor by her university's chapter of Psi Chi, the National Honor Society in Psychology. Psi Chi also awarded her the Outstanding Educator Award and Favorite Professor Award.

Professor Pasupathi's research focuses on how people of all ages learn from their experiences, particularly through storytelling. She is coeditor of Narrative Development in Adolescence: Creating the Storied Self, and her work has been published widely in scholarly journals.

Directory of TTC Teaching Company – How We Learn 2012

01 Myths about Learning.mp3
02 Why No Single Learning Theory Works.mp3
03 Learning as Information Processing.mp3
04 Creating Representations.mp3
05 Categories, Rules, and scripts.mp3
06 What Babies Know.mp3
07 Learning Your Native Tongue.mp3
08 Learning a Second Language.mp3
09 Learning How to Move.mp3
10 Learning Our Way Around.mp3
11 Learning to Tell Stories.mp3
12 Learning Approaches in Math and Science.mp3
13 Learning as Theory Testing.mp3
14 Integrating Different Domains of Learning.mp3
15 Cognitive Constraints on Learning.mp3
16 Choosing Learning Strategies.mp3
17 Source Knowledge and Learning.mp3
18 The Role of Emotion in Learning.mp3
19 Cultivating a Desire to Learn.mp3
20 Intelligence and Learning.mp3
21 Are Learning Styles Real.mp3
22 Different People, Different Interests.mp3
23 Learning across the Lifespan.mp3
24 Making the Most of How We Learn.mp3
How We Learn.txt

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