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Books > Science & Mathematics > Mathematics > Calculus & mathematical analysis
Hoermander operators are a class of linear second order partial
differential operators with nonnegative characteristic form and
smooth coefficients, which are usually degenerate
elliptic-parabolic, but nevertheless hypoelliptic, that is highly
regularizing. The study of these operators began with the 1967
fundamental paper by Lars Hoermander and is intimately connected to
the geometry of vector fields.Motivations for the study of
Hoermander operators come for instance from
Kolmogorov-Fokker-Planck equations arising from modeling physical
systems governed by stochastic equations and the geometric theory
of several complex variables. The aim of this book is to give a
systematic exposition of a relevant part of the theory of
Hoermander operators and vector fields, together with the necessary
background and prerequisites.The book is intended for self-study,
or as a reference book, and can be useful to both younger and
senior researchers, already working in this area or aiming to
approach it.
This book presents a collection of problems and solutions in
functional analysis with applications to quantum mechanics.
Emphasis is given to Banach spaces, Hilbert spaces and generalized
functions.The material of this volume is self-contained, whereby
each chapter comprises an introduction with the relevant notations,
definitions, and theorems. The approach in this volume is to
provide students with instructive problems along with
problem-solving strategies. Programming problems with solutions are
also included.
The Qualitative Theory of Ordinary Differential Equations (ODEs)
occupies a rather special position both in Applied and Theoretical
Mathematics. On the one hand, it is a continuation of the standard
course on ODEs. On the other hand, it is an introduction to
Dynamical Systems, one of the main mathematical disciplines in
recent decades. Moreover, it turns out to be very useful for
graduates when they encounter differential equations in their work;
usually those equations are very complicated and cannot be solved
by standard methods.The main idea of the qualitative analysis of
differential equations is to be able to say something about the
behavior of solutions of the equations, without solving them
explicitly. Therefore, in the first place such properties like the
stability of solutions stand out. It is the stability with respect
to changes in the initial conditions of the problem. Note that,
even with the numerical approach to differential equations, all
calculations are subject to a certain inevitable error. Therefore,
it is desirable that the asymptotic behavior of the solutions is
insensitive to perturbations of the initial state.Each chapter
contains a series of problems (with varying degrees of difficulty)
and a self-respecting student should solve them. This book is based
on Raul Murillo's translation of Henryk Zoladek's lecture notes,
which were in Polish and edited in the portal Matematyka Stosowana
(Applied Mathematics) in the University of Warsaw.
This book differs from traditional numerical analysis texts in that
it focuses on the motivation and ideas behind the algorithms
presented rather than on detailed analyses of them. It presents a
broad overview of methods and software for solving mathematical
problems arising in computational modeling and data analysis,
including proper problem formulation, selection of effective
solution algorithms, and interpretation of results. In the 20 years
since its original publication, the modern, fundamental perspective
of this book has aged well, and it continues to be used in the
classroom. This Classics edition has been updated to include
pointers to Python software and the Chebfun package, expansions on
barycentric formulation for Lagrange polynomial interpretation and
stochastic methods, and the availability of about 100 interactive
educational modules that dynamically illustrate the concepts and
algorithms in the book. Scientific Computing: An Introductory
Survey, Second Edition is intended as both a textbook and a
reference for computationally oriented disciplines that need to
solve mathematical problems.
This volume considers resistance networks: large graphs which are
connected, undirected, and weighted. Such networks provide a
discrete model for physical processes in inhomogeneous media,
including heat flow through perforated or porous media. These
graphs also arise in data science, e.g., considering
geometrizations of datasets, statistical inference, or the
propagation of memes through social networks. Indeed, network
analysis plays a crucial role in many other areas of data science
and engineering. In these models, the weights on the edges may be
understood as conductances, or as a measure of similarity.
Resistance networks also arise in probability, as they correspond
to a broad class of Markov chains.The present volume takes the
nonstandard approach of analyzing resistance networks from the
point of view of Hilbert space theory, where the inner product is
defined in terms of Dirichlet energy. The resulting viewpoint
emphasizes orthogonality over convexity and provides new insights
into the connections between harmonic functions, operators, and
boundary theory. Novel applications to mathematical physics are
given, especially in regard to the question of self-adjointness of
unbounded operators.New topics are covered in a host of areas
accessible to multiple audiences, at both beginning and more
advanced levels. This is accomplished by directly linking diverse
applied questions to such key areas of mathematics as functional
analysis, operator theory, harmonic analysis, optimization,
approximation theory, and probability theory.
Advances in techniques that reduce or eliminate the type of meshes
associated with finite elements or finite differences are reported
in the papers that form this volume. As design, analysis and
manufacture become more integrated, the chances are that software
users will be less aware of the capabilities of the analytical
techniques that are at the core of the process. This reinforces the
need to retain expertise in certain specialised areas of numerical
methods, such as BEM/MRM, to ensure that all new tools perform
satisfactorily within the aforementioned integrated process. The
maturity of BEM since 1978 has resulted in a substantial number of
industrial applications of the method; this demonstrates its
accuracy, robustness and ease of use. The range of applications
still needs to be widened, taking into account the potentialities
of the Mesh Reduction techniques in general. The included papers
originate from the 45th conference on Boundary Elements and other
Mesh Reduction Methods (BEM/MRM) and describe theoretical
developments and new formulations, helping to expand the range of
applications as well as the type of modelled materials in response
to the requirements of contemporary industrial and professional
environments.
Addresses computational methods that have proven efficient for the
solution of a large variety of nonlinear elliptic problems. These
methods can be applied to many problems in science and engineering,
but this book focuses on their application to problems in continuum
mechanics and physics. This book differs from others on the topic
by:* Presenting examples of the power and versatility of
operator-splitting methods.* Providing a detailed introduction to
alternating direction methods of multipliers and their
applicability to the solution of nonlinear (possibly non-smooth)
problems from science and engineering.* Showing that nonlinear
least-squares methods, combined with operator-splitting and
conjugate gradient algorithms, provide efficient tools for the
solution of highly nonlinear problems.
Extremum Seeking through Delays and PDEs, the first book on the
topic, expands the scope of applicability of the extremum seeking
method, from static and finite-dimensional systems to
infinite-dimensional systems. Readers will find: Numerous
algorithms for model-free real-time optimization are developed and
their convergence guaranteed. Extensions from single-player
optimization to noncooperative games, under delays and pdes, are
provided. The delays and pdes are compensated in the control
designs using the pde backstepping approach, and stability is
ensured using infinite-dimensional versions of averaging theory.
Accessible and powerful tools for analysis. This book is intended
for control engineers in all disciplines (electrical, mechanical,
aerospace, chemical), mathematicians, physicists, biologists, and
economists. It is appropriate for graduate students, researchers,
and industrial users.
This work is devoted to fixed point theory as well as the theory of
accretive operators in Banach spaces. The goal is to develop, in
self-contained way, the main results in both theories. Special
emphasis is given to the study how both theories can be used to
study the existence and uniqueness of solution of several types of
partial differential equations and integral equations.
This book is a general introduction to the statistical analysis of
networks, and can serve both as a research monograph and as a
textbook. Numerous fundamental tools and concepts needed for the
analysis of networks are presented, such as network modeling,
community detection, graph-based semi-supervised learning and
sampling in networks. The description of these concepts is
self-contained, with both theoretical justifications and
applications provided for the presented algorithms.Researchers,
including postgraduate students, working in the area of network
science, complex network analysis, or social network analysis, will
find up-to-date statistical methods relevant to their research
tasks. This book can also serve as textbook material for courses
related to thestatistical approach to the analysis of complex
networks.In general, the chapters are fairly independent and
self-supporting, and the book could be used for course composition
"a la carte". Nevertheless, Chapter 2 is needed to a certain degree
for all parts of the book. It is also recommended to read Chapter 4
before reading Chapters 5 and 6, but this is not absolutely
necessary. Reading Chapter 3 can also be helpful before reading
Chapters 5 and 7. As prerequisites for reading this book, a basic
knowledge in probability, linear algebra and elementary notions of
graph theory is advised. Appendices describing required notions
from the above mentioned disciplines have been added to help
readers gain further understanding.
Containing the proceedings from the 41st conference on Boundary
Elements and other Mesh Reduction Methods (BEM/MRM), this book is a
collection of high quality papers that report on advances in
techniques that reduce or eliminate the type of meshes associated
with such methods as finite elements or finite differences. As
design, analysis and manufacture become more integrated the chances
are that the users will be less aware of the capabilities of the
analytical techniques that are at the core of the process. This
reinforces the need to retain expertise in certain specialised
areas of numerical methods, such as BEM/MRM, to ensure that all new
tools perform satisfactorily in the integrated process. The
maturity of BEM since 1978 has resulted in a substantial number of
industrial applications that demonstrate the accuracy, robustness
and easy use of the technique. Their range still needs to be
widened, taking into account the potentialities of the Mesh
Reduction techniques in general. The papers in this volume help to
expand the range of applications as well as the type of materials
in response to industrial and professional requirements.
Study smarter and stay on top of your differential equations course
with the bestselling Schaum's Outline-now with the NEW Schaum's app
and website! Schaum's Outline of Differential Equations, Fifth
Edition is the go-to study guide for all students of science who
need to learn or refresh their knowledge of differential equations.
With an outline format that facilitates quick and easy review and
mirrors the course in scope and sequence, this book helps you
understand basic concepts and get the extra practice you need to
excel in the course. It supports the all major differential
equations textbooks and is useful for study in Calculus (I, II, and
III), Mathematical Modeling, Introductory Differential Equations
and Differential Equations. Chapters include an Introduction to
Modeling and Qualitative Methods, Classifications of First-Order
Differential Equations, Linear Differential Equations, Variation of
Parameters, Initial-Value Problems for Linear Differential
Equations, Graphical and Numerical Methods for Solving First-Order
Differential Equations, Solutions of Linear Differential Equations
with Constant Coefficients by Laplace Transforms, and more.
Features: NEW to this edition: the new Schaum's app and website!
NEW CHAPTERS include Autonomous Differential Equations and
Qualitative Methods; Eigenvalues and Eigenvectors; three chapters
dealing with Solutions of Systems of Autonomous Equations via
Eigenvalues and Eigenvectors (real and distinct, real and equal,
and complex conjugate Eigenvalues) 20 problem-solving videos online
563 solved problems Outline format provides a quick and easy review
of differential equations Clear, concise explanations of
differential equations concepts Hundreds of examples with
explanations of key concepts Supports all major textbooks for
differential equations courses Appropriate for the following
courses: Calculus (I, II, and III), Mathematical Modeling,
Introductory Differential Equations, and Differential Equations
This book provides a concrete introduction to a number of topics in
harmonic analysis, accessible at the early graduate level or, in
some cases, at an upper undergraduate level. Necessary
prerequisites to using the text are rudiments of the Lebesgue
measure and integration on the real line. It begins with a thorough
treatment of Fourier series on the circle and their applications to
approximation theory, probability, and plane geometry (the
isoperimetric theorem). Frequently, more than one proof is offered
for a given theorem to illustrate the multiplicity of approaches.
The second chapter treats the Fourier transform on Euclidean
spaces, especially the author's results in the three-dimensional
piecewise smooth case, which is distinct from the classical Gibbs -
Wilbraham phenomenon of one-dimensional Fourier analysis. The
Poisson summation formula treated in Chapter 3 provides an elegant
connection between Fourier series on the circle and Fourier
transforms on the real line, culminating in Landau's asymptotic
formulas for lattice points on a large sphere. Much of modern
harmonic analysis is concerned with the behavior of various linear
operators on the Lebesgue spaces Lp (Rn). Chapter 4 gives a gentle
introduction to these results, using the Riesz - Thorin theorem and
the Marcinkiewicz interpolation formula. One of the long-time users
of Fourier analysis is probability theory. In Chapter 5 the central
limit theorem, iterated log theorem, and Berry - Esseen theorems
are developed using the suitable Fourier-analytic tools. The final
chapter furnishes a gentle introduction to wavelet theory,
depending only on the L2 theory of the Fourier transform (the
Plancherel theorem). The basic notions of scale and location
parameters demonstrate the flexibility of the wavelet approach to
harmonic analysis. The text contains numerous examples and more
than 200 exercises, each located in close proximity to the related
theoretical material.
The introduction of cross diffusivity opens many questions in the
theory of reactiondiffusion systems. This book will be the first to
investigate such problems presenting new findings for researchers
interested in studying parabolic and elliptic systems where
classical methods are not applicable. In addition, The
Gagliardo-Nirenberg inequality involving BMO norms is improved and
new techniques are covered that will be of interest. This book also
provides many open problems suitable for interested Ph.D students.
In 1940 G. H. Hardy published A Mathematician's Apology, a
meditation on mathematics by a leading pure mathematician.
Eighty-two years later, An Applied Mathematician's Apology is a
meditation and also a personal memoir by a philosophically inclined
numerical analyst, one who has found great joy in his work but is
puzzled by its relationship to the rest of mathematics.
For a two-semester or three-semester course in Calculus for Life
Sciences. Calculus for Biology and Medicine, Third Edition,
addresses the needs of students in the biological sciences by
showing them how to use calculus to analyze natural
phenomena-without compromising the rigorous presentation of the
mathematics. While the table of contents aligns well with a
traditional calculus text, all the concepts are presented through
biological and medical applications. The text provides students
with the knowledge and skills necessary to analyze and interpret
mathematical models of a diverse array of phenomena in the living
world. Since this text is written for college freshmen, the
examples were chosen so that no formal training in biology is
needed.
This graduate-level introduction to ordinary differential equations
combines both qualitative and numerical analysis of solutions, in
line with Poincare's vision for the field over a century ago.
Taking into account the remarkable development of dynamical systems
since then, the authors present the core topics that every young
mathematician of our time--pure and applied alike--ought to learn.
The book features a dynamical perspective that drives the
motivating questions, the style of exposition, and the arguments
and proof techniques. The text is organized in six cycles. The
first cycle deals with the foundational questions of existence and
uniqueness of solutions. The second introduces the basic tools,
both theoretical and practical, for treating concrete problems. The
third cycle presents autonomous and non-autonomous linear theory.
Lyapunov stability theory forms the fourth cycle. The fifth one
deals with the local theory, including the Grobman-Hartman theorem
and the stable manifold theorem. The last cycle discusses global
issues in the broader setting of differential equations on
manifolds, culminating in the Poincare-Hopf index theorem. The book
is appropriate for use in a course or for self-study. The reader is
assumed to have a basic knowledge of general topology, linear
algebra, and analysis at the undergraduate level. Each chapter ends
with a computational experiment, a diverse list of exercises, and
detailed historical, biographical, and bibliographic notes seeking
to help the reader form a clearer view of how the ideas in this
field unfolded over time.
Jump-start your career as a data scientist--learn to develop
datasets for exploration, analysis, and machine learning SQL for
Data Scientists: A Beginner's Guide for Building Datasets for
Analysis is a resource that's dedicated to the Structured Query
Language (SQL) and dataset design skills that data scientists use
most. Aspiring data scientists will learn how to how to construct
datasets for exploration, analysis, and machine learning. You can
also discover how to approach query design and develop SQL code to
extract data insights while avoiding common pitfalls. You may be
one of many people who are entering the field of Data Science from
a range of professions and educational backgrounds, such as
business analytics, social science, physics, economics, and
computer science. Like many of them, you may have conducted
analyses using spreadsheets as data sources, but never retrieved
and engineered datasets from a relational database using SQL, which
is a programming language designed for managing databases and
extracting data. This guide for data scientists differs from other
instructional guides on the subject. It doesn't cover SQL broadly.
Instead, you'll learn the subset of SQL skills that data analysts
and data scientists use frequently. You'll also gain practical
advice and direction on "how to think about constructing your
dataset." Gain an understanding of relational database structure,
query design, and SQL syntax Develop queries to construct datasets
for use in applications like interactive reports and machine
learning algorithms Review strategies and approaches so you can
design analytical datasets Practice your techniques with the
provided database and SQL code In this book, author Renee Teate
shares knowledge gained during a 15-year career working with data,
in roles ranging from database developer to data analyst to data
scientist. She guides you through SQL code and dataset design
concepts from an industry practitioner's perspective, moving your
data scientist career forward!
For courses in Differential Equations and Linear Algebra. The right
balance between concepts, visualisation, applications, and skills
Differential Equations and Linear Algebra provides the conceptual
development and geometric visualisation of a modern differential
equations and linear algebra course that is essential to science
and engineering students. It balances traditional manual methods
with the new, computer-based methods that illuminate qualitative
phenomena - a comprehensive approach that makes accessible a wider
range of more realistic applications. The book combines core topics
in elementary differential equations with concepts and methods of
elementary linear algebra. It starts and ends with discussions of
mathematical modeling of real-world phenomena, evident in figures,
examples, problems, and applications throughout.
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