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Basic Analysis: Volumes I-V is written with the aim of balancing
theory and abstraction with clear explanations and arguments, so
that students and researchers alike who are from a variety of
different areas can follow this text and use it profitably for
self-study. The first volume is designed for students who have
completed the usual calculus and ordinary differential equation
sequence and a basic course in linear algebra. This is a critical
course in the use of abstraction, but is just first volume in a
sequence of courses which prepare students to become practicing
scientists. The second volume focuses on differentiation in
n-dimensions and important concepts about mappings between finite
dimensional Euclidean spaces, such as the inverse and implicit
function theorem and change of variable formulae for
multidimensional integration. These important topics provide
background in important applied and theoretical areas which are no
longer covered in mathematical science curricula. Although it
follows on from the preceding volume, this is a self-contained
book, accessible to undergraduates with a standard course in
undergraduate analysis. The third volume is intended as a first
course in abstract linear analysis. This textbook covers metric
spaces, normed linear spaces and inner product spaces, along with
many other deeper abstract ideas such a completeness, operators and
dual spaces. These topics act as an important tool in the
development of a mathematically trained scientist. The fourth
volume introduces students to concepts from measure theory and
continues their training in the abstract way of looking at the
world. This is a most important skill to have when your life's work
will involve quantitative modeling to gain insight into the real
world. This text generalizes the notion of integration to a very
abstract setting in a variety of ways. We generalize the notion of
the length of an interval to the measure of a set and learn how to
construct the usual ideas from integration using measures. We
discuss carefully the many notions of convergence that measure
theory provides. The final volume introduces graduate students in
science with concepts from topology and functional analysis, both
linear and nonlinear. It is the fifth book in a series designed to
train interested readers how to think properly using mathematical
abstractions, and how to use the tools of mathematical analysis in
applications. It is important to realize that the most difficult
part of applying mathematical reasoning to a new problem domain is
choosing the underlying mathematical framework to use on the
problem. Once that choice is made, we have many tools we can use to
solve the problem. However, a different choice would open up
avenues of analysis from a different, perhaps more productive
perspective. In this volume, the nature of these critical choices
is discussed using applications involving the immune system and
cognition. Features: Can be used as a supplementary text for anyone
whose work requires that they begin to assimilate more abstract
mathematical concepts as part of their professional growth Function
as a traditional textbook as well as a resource for self-study
Suitable for mathematics students and for those in other
disciplines such as biology, physics, and economics and others
requiring a careful and solid grounding in the use of abstraction
in problem solving Emphasizes learning how to understand the
consequences of the underlying assumptions used in building a model
Regularly uses computation tools to help understand abstract
concepts.
Basic Analysis V: Functional Analysis and Topology introduces
graduate students in science to concepts from topology and
functional analysis, both linear and nonlinear. It is the fifth
book in a series designed to train interested readers how to think
properly using mathematical abstractions, and how to use the tools
of mathematical analysis in applications. It is important to
realize that the most difficult part of applying mathematical
reasoning to a new problem domain is choosing the underlying
mathematical framework to use on the problem. Once that choice is
made, we have many tools we can use to solve the problem. However,
a different choice would open up avenues of analysis from a
different, perhaps more productive, perspective. In this volume,
the nature of these critical choices is discussed using
applications involving the immune system and cognition. Features
Develops a proof of the Jordan Canonical form to show some basic
ideas in algebraic topology Provides a thorough treatment of
topological spaces, finishing with the Krein-Milman theorem
Discusses topological degree theory (Brouwer, Leray-Schauder, and
Coincidence) Carefully develops manifolds and functions on
manifolds ending with Riemannian metrics Suitable for advanced
students in mathematics and associated disciplines Can be used as a
traditional textbook as well as for self-study Author James K.
Peterson is an Emeritus Professor at the School of Mathematical and
Statistical Sciences, Clemson University. He tries hard to build
interesting models of complex phenomena using a blend of
mathematics, computation, and science. To this end, he has written
four books on how to teach such things to biologists and cognitive
scientists. These books grew out of his Calculus for Biologists
courses offered to the biology majors from 2007 to 2015. He has
taught the analysis courses since he started teaching both at
Clemson and at his previous post at Michigan Technological
University. In between, he spent time as a senior engineer in
various aerospace firms and even did a short stint in a software
development company. The problems he was exposed to were very hard,
and not amenable to solution using just one approach. Using tools
from many branches of mathematics, from many types of computational
languages, and from first-principles analysis of natural phenomena
was absolutely essential to make progress. In both mathematical and
applied areas, students often need to use advanced mathematics
tools they have not learned properly. So, he has recently written a
series of five books on mathematical analysis to help researchers
with the problem of learning new things after they have earned
their degrees and are practicing scientists. Along the way, he has
also written papers in immunology, cognitive science, and neural
network technology, in addition to having grants from the NSF,
NASA, and the US Army. He also likes to paint, build furniture, and
write stories.
Basic Analysis II: A Modern Calculus in Many Variables focuses on
differentiation in Rn and important concepts about mappings from Rn
to Rm, such as the inverse and implicit function theorem and change
of variable formulae for multidimensional integration. These topics
converge nicely with many other important applied and theoretical
areas which are no longer covered in mathematical science
curricula. Although it follows on from the preceding volume, this
is a self-contained book, accessible to undergraduates with a
minimal grounding in analysis. Features Can be used as a
traditional textbook as well as for self-study Suitable for
undergraduates in mathematics and associated disciplines Emphasises
learning how to understand the consequences of assumptions using a
variety of tools to provide the proofs of propositions
Basic Analysis III: Mappings on Infinite Dimensional Spaces is
intended as a first course in abstract linear analysis. This
textbook cover metric spaces, normed linear spaces and inner
product spaces, along with many other deeper abstract ideas such a
completeness, operators and dual spaces. These topics act as an
important tool in the development of a mathematically trained
scientist. Feature: Can be used as a traditional textbook as well
as for self-study Suitable for undergraduates in mathematics and
associated disciplines Emphasizes learning how to understand the
consequences of assumptions using a variety of tools to provide the
proofs of propositions
Basic Analysis I: Functions of a Real Variable is designed for
students who have completed the usual calculus and ordinary
differential equation sequence and a basic course in linear
algebra. This is a critical course in the use of abstraction, but
is just first volume in a sequence of courses which prepare
students to become practicing scientists. This book is written with
the aim of balancing the theory and abstraction with clear
explanations and arguments, so that students who are from a variety
of different areas can follow this text and use it profitably for
self-study. It can also be used as a supplementary text for anyone
whose work requires that they begin to assimilate more abstract
mathematical concepts as part of their professional growth.
Features Can be used as a traditional textbook as well as for
self-study Suitable for undergraduate mathematics students, or for
those in other disciplines requiring a solid grounding in
abstraction Emphasises learning how to understand the consequences
of assumptions using a variety of tools to provide the proofs of
propositions
This book shows how mathematics, computer science and science can
be usefully and seamlessly intertwined. It begins with a general
model of cognitive processes in a network of computational nodes,
such as neurons, using a variety of tools from mathematics,
computational science and neurobiology. It then moves on to solve
the diffusion model from a low-level random walk point of view. It
also demonstrates how this idea can be used in a new approach to
solving the cable equation, in order to better understand the
neural computation approximations. It introduces specialized data
for emotional content, which allows a brain model to be built using
MatLab tools, and also highlights a simple model of cognitive
dysfunction.
This book provides a self-study program on how mathematics,
computer science and science can be usefully and seamlessly
intertwined. Learning to use ideas from mathematics and computation
is essential for understanding approaches to cognitive and
biological science. As such the book covers calculus on one
variable and two variables and works through a number of
interesting first-order ODE models. It clearly uses MatLab in
computational exercises where the models cannot be solved by hand,
and also helps readers to understand that approximations cause
errors - a fact that must always be kept in mind.
This book offers a self-study program on how mathematics, computer
science and science can be profitably and seamlessly intertwined.
This book focuses on two variable ODE models, both linear and
nonlinear, and highlights theoretical and computational tools using
MATLAB to explain their solutions. It also shows how to solve cable
models using separation of variables and the Fourier Series.
This book offers a self-study program on how mathematics, computer
science and science can be profitably and seamlessly intertwined.
This book focuses on two variable ODE models, both linear and
nonlinear, and highlights theoretical and computational tools using
MATLAB to explain their solutions. It also shows how to solve cable
models using separation of variables and the Fourier Series.
This book shows how mathematics, computer science and science can
be usefully and seamlessly intertwined. It begins with a general
model of cognitive processes in a network of computational nodes,
such as neurons, using a variety of tools from mathematics,
computational science and neurobiology. It then moves on to solve
the diffusion model from a low-level random walk point of view. It
also demonstrates how this idea can be used in a new approach to
solving the cable equation, in order to better understand the
neural computation approximations. It introduces specialized data
for emotional content, which allows a brain model to be built using
MatLab tools, and also highlights a simple model of cognitive
dysfunction.
This book provides a self-study program on how mathematics,
computer science and science can be usefully and seamlessly
intertwined. Learning to use ideas from mathematics and computation
is essential for understanding approaches to cognitive and
biological science. As such the book covers calculus on one
variable and two variables and works through a number of
interesting first-order ODE models. It clearly uses MatLab in
computational exercises where the models cannot be solved by hand,
and also helps readers to understand that approximations cause
errors - a fact that must always be kept in mind.
Basic Analysis IV: Measure Theory and Integration introduces
students to concepts from measure theory and continues their
training in the abstract way of looking at the world. This is a
most important skill to have when your life's work will involve
quantitative modeling to gain insight into the real world. This
text generalizes the notion of integration to a very abstract
setting in a variety of ways. We generalize the notion of the
length of an interval to the measure of a set and learn how to
construct the usual ideas from integration using measures. We
discuss carefully the many notions of convergence that measure
theory provides. Features * Can be used as a traditional textbook
as well as for self-study * Suitable for advanced students in
mathematics and associated disciplines * Emphasises learning how to
understand the consequences of assumptions using a variety of tools
to provide the proofs of propositions
From America's favorite cooking teacher, multiple award-winner
James Peterson, an invaluable reference handbook.
Culinary students everywhere rely on the comprehensive and
authoritative cookbooks published by chef, instructor, and
award-winning author Jim Peterson. And now, for the first time,
this guru-to-the-professionals turns his prodigious knowledge into
a practical, chockablock, quick-reference, A-to-Z answer book for
the rest of us.
Look elsewhere for how to bone skate or trim out a saddle of lamb,
how to saute sweetbreads or flambe dessert. Look here instead for
how to zest a lemon, make the perfect hamburger, bread a chicken
breast, make (truly hot) coffee in a French press, make magic with
a Microplane. It's all here: how to season a castiron pan, bake a
perfect pie, keep shells from sticking to hardcooked eggs. How to
carve a turkey, roast a chicken, and chop, slice, beat, broil,
braise, or boil any ingredient you're likely to encounter.
Information on seasoning, saucing, and determining doneness (by
internal temperatures, timings, touch, and sight) guarantee that
you've eaten your last bland and overcooked meal.
Here are 500 invaluable techniques with nearly as many color
photographs, bundled into a handy, accessible format.
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