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Books > Science & Mathematics > Mathematics
This book provides a broad, interdisciplinary overview of
non-Archimedean analysis and its applications. Featuring new
techniques developed by leading experts in the field, it highlights
the relevance and depth of this important area of mathematics, in
particular its expanding reach into the physical, biological,
social, and computational sciences as well as engineering and
technology. In the last forty years the connections between
non-Archimedean mathematics and disciplines such as physics,
biology, economics and engineering, have received considerable
attention. Ultrametric spaces appear naturally in models where
hierarchy plays a central role - a phenomenon known as
ultrametricity. In the 80s, the idea of using ultrametric spaces to
describe the states of complex systems, with a natural hierarchical
structure, emerged in the works of Fraunfelder, Parisi, Stein and
others. A central paradigm in the physics of certain complex
systems - for instance, proteins - asserts that the dynamics of
such a system can be modeled as a random walk on the energy
landscape of the system. To construct mathematical models, the
energy landscape is approximated by an ultrametric space (a finite
rooted tree), and then the dynamics of the system is modeled as a
random walk on the leaves of a finite tree. In the same decade,
Volovich proposed using ultrametric spaces in physical models
dealing with very short distances. This conjecture has led to a
large body of research in quantum field theory and string theory.
In economics, the non-Archimedean utility theory uses probability
measures with values in ordered non-Archimedean fields. Ultrametric
spaces are also vital in classification and clustering techniques.
Currently, researchers are actively investigating the following
areas: p-adic dynamical systems, p-adic techniques in cryptography,
p-adic reaction-diffusion equations and biological models, p-adic
models in geophysics, stochastic processes in ultrametric spaces,
applications of ultrametric spaces in data processing, and more.
This contributed volume gathers the latest theoretical developments
as well as state-of-the art applications of non-Archimedean
analysis. It covers non-Archimedean and non-commutative geometry,
renormalization, p-adic quantum field theory and p-adic quantum
mechanics, as well as p-adic string theory and p-adic dynamics.
Further topics include ultrametric bioinformation, cryptography and
bioinformatics in p-adic settings, non-Archimedean spacetime,
gravity and cosmology, p-adic methods in spin glasses, and
non-Archimedean analysis of mental spaces. By doing so, it
highlights new avenues of research in the mathematical sciences,
biosciences and computational sciences.
This book analyses the models for major risks related to flight
safety in the aviation sector and presents risk estimation methods
through examples of several known aviation enterprises. The book
provides a comprehensive content for professionals engaged in the
development of flight safety regulatory framework as well as in the
design and operation of ground-based or on-board flight support
radio electronic systems. The book is also useful for senior
students and postgraduates in aviation specialties, especially
those related to air traffic management.
This book provides a comprehensive examination of preconditioners
for boundary element discretisations of first-kind integral
equations. Focusing on domain-decomposition-type and multilevel
methods, it allows readers to gain a good understanding of the
mechanisms and necessary techniques in the analysis of the
preconditioners. These techniques are unique for the discretisation
of first-kind integral equations since the resulting systems of
linear equations are not only large and ill-conditioned, but also
dense. The book showcases state-of-the-art preconditioning
techniques for boundary integral equations, presenting up-to-date
research. It also includes a detailed discussion of Sobolev spaces
of fractional orders to familiarise readers with important
mathematical tools for the analysis. Furthermore, the concise
overview of adaptive BEM, hp-version BEM, and coupling of FEM-BEM
provides efficient computational tools for solving practical
problems with applications in science and engineering.
This book includes up-to-date contributions in the broadly defined
area of probabilistic analysis of voting rules and decision
mechanisms. Featuring papers from all fields of social choice and
game theory, it presents probability arguments to allow readers to
gain a better understanding of the properties of decision rules and
of the functioning of modern democracies. In particular, it focuses
on the legacy of William Gehrlein and Dominique Lepelley, two
prominent scholars who have made important contributions to this
field over the last fifty years. It covers a range of topics,
including (but not limited to) computational and technical aspects
of probability approaches, evaluation of the likelihood of voting
paradoxes, power indices, empirical evaluations of voting rules,
models of voters' behavior, and strategic voting. The book gathers
articles written in honor of Gehrlein and Lepelley along with
original works written by the two scholars themselves.
Susanna Epp's DISCRETE MATHEMATICS WITH APPLICATIONS, 4e,
International Edition provides a clear introduction to discrete
mathematics. Renowned for her lucid, accessible prose, Epp explains
complex, abstract concepts with clarity and precision. This book
presents not only the major themes of discrete mathematics, but
also the reasoning that underlies mathematical thought. Students
develop the ability to think abstractly as they study the ideas of
logic and proof. While learning about such concepts as logic
circuits and computer addition, algorithm analysis, recursive
thinking, computability, automata, cryptography, and combinatorics,
students discover that the ideas of discrete mathematics underlie
and are essential to the science and technology of the computer
age. Overall, Epp's emphasis on reasoning provides students with a
strong foundation for computer science and upper-level mathematics
courses.
Functions and their properties have been part of the rigorous
precollege curriculum for decades. And functional equations have
been a favorite topic of the leading national and international
mathematical competitions. Yet the subject has not received equal
attention by authors at an introductory level. The majority of the
books on the topic remain unreachable to the curious and
intelligent precollege student. The present book is an attempt to
eliminate this disparity. The book opens with a review chapter on
functions, which collects the relevant foundational information on
functions, plus some material potentially new to the reader. The
next chapter presents a working definition of functional equations
and explains the difficulties in trying to systematize the theory.
With each new chapter, the author presents methods for the solution
of a particular group of equations. Each chapter is complemented
with many solved examples, the majority of which are taken from
mathematical competitions and professional journals. The book ends
with a chapter of unsolved problems and some other auxiliary
material. The book is an invaluable resource for precollege and
college students who want to deepen their knowledge of functions
and their properties, for teachers and instructors who wish to
enrich their curricula, and for any lover of mathematical
problem-solving techniques.
Financial market modeling is a prime example of a real-life
application of probability theory and stochastics. This
authoritative book discusses the discrete-time approximation and
other qualitative properties of models of financial markets, like
the Black-Scholes model and its generalizations, offering in this
way rigorous insights on one of the most interesting applications
of mathematics nowadays.
This monograph presents a general theory of weakly implicative
logics, a family covering a vast number of non-classical logics
studied in the literature, concentrating mainly on the abstract
study of the relationship between logics and their algebraic
semantics. It can also serve as an introduction to (abstract)
algebraic logic, both propositional and first-order, with special
attention paid to the role of implication, lattice and residuated
connectives, and generalized disjunctions. Based on their recent
work, the authors develop a powerful uniform framework for the
study of non-classical logics. In a self-contained and didactic
style, starting from very elementary notions, they build a general
theory with a substantial number of abstract results. The theory is
then applied to obtain numerous results for prominent families of
logics and their algebraic counterparts, in particular for
superintuitionistic, modal, substructural, fuzzy, and relevant
logics. The book may be of interest to a wide audience, especially
students and scholars in the fields of mathematics, philosophy,
computer science, or related areas, looking for an introduction to
a general theory of non-classical logics and their algebraic
semantics.
Flatland is a fascinating nineteenth century work - an utterly
unique combination of multi-plane geometry, social satire and
whimsy. Although its original publication went largely unnoticed,
the discoveries of later physicists brought it new recognition and
respect, and its popularity since has justly never waned. It
remains a charming and entertaining read, and a brilliant
introduction to the concept of dimensions beyond those we can
perceive. This is a reworking of the expanded 2nd edition of 1884,
with particularly large, clear text, and all the original author's
illustrations.
Hulchul: The Common Ingredient of MotionMotionMotionMotion and Time
Author, Sohan Jain, proposes the following in the book: Instants of
Motion, Instants of Time and Time Outage: Just as time has instants
of time, motion has instants of motion, too. Instants of time and
motion can be divided into three classes: pure instants of time,
pure instants of motion, and composite instants of time and motion.
The sequences of the three types of instants are interspersed into
a single sequence of their occurrences. A body does not experience
time during pure instants of motion, a phenomenon we will call time
outage -the cause of time dilation. Time outage is not continuous;
it is intermittent. Internal and external motion of a body and
their inheritance: Each body has, generally, two kinds of motions:
internal motion and external motion. A body goes, wherever its
outer bodies go. An inner body inherits external motion of its
outer bodies. An outer body inherits internal motion of its inner
bodies. Photons and light do not inherit motion; may be, this is
why their motions are independent of their sources. Prime ticks,
the building blocks of time and any motion: Motion of a common body
is not continuous; it is intermittent. Any kind of motion is
perceived to be made of discrete, indivisible tiny movements,
called prime ticks (p-ticks). P-ticks are to motion what elementary
particles are to matter or what photons are to light. There is time
only because there is motion. Prime ticks are events and imply
motion. Events have concurrency, which implies time. Total
concurrency hulchul, a universal constant: Concurrency events of
external and internal p-ticks of a body are precisely the instants
of motion and time. The sum of the two is called the total
concurrency hulchul (c-hulchul). Total c-hulchul is the same for
all bodies. The proposed theory possibly explains: Why a particle
accelerator works. Why atoms have compartmentalized internal
structure. Why lighter bodies, such as elementary particles and
photons, have wavy straight motion rather than straight motion. The
theory predicts: The sharing of an electron by two atoms is not
continuous; it alternates between the two atoms.
Mechanics is quite obviously geometric, yet the traditional
approach to the subject is based mainly on differential equations.
Setting out to make mechanics both accessible and interesting for
non-mathematicians, Richard Talman augments this approach with
geometric methods such as differential geometry, differential
forms, and tensor analysis to reveal qualitative aspects of the
theory.
For easy reference, the author treats Lagrangian, Hamiltonian,
and Newtonian mechanics separately - exploring their geometric
structure through vector fields, symplectic geometry, and gauge
invariance respectively.
This second, fully revised edition has been expanded to further
emphasize the importance of the geometric approach. Starting from
Hamilton's principle, the author shows, from a geometric
perspective, how "all" of classical physics can be subsumed within
classical mechanics. Having developed the formalism in the context
of classical mechanics, the subjects of electrodynamics,
relativistic strings and general relativity are treated as examples
of classical mechanics. This modest unification of classical
physics is intended to provide a background for the far more
ambitious "grand unification" program of quantum field theory.
The final chapters develop approximate methods for the analysis
of mechanical systems. Here the emphasis is more on practical
perturbative methods than on the canonical transformation
formalism. "Geometric Mechanics" features numerous illustrative
examples and assumes only basic knowledge of Lagrangian
mechanics.
This book features more than 20 papers that celebrate the work of
Hajnal Andreka and Istvan Nemeti. It illustrates an interaction
between developing and applying mathematical logic. The papers
offer new results as well as surveys in areas influenced by these
two outstanding researchers. They also provide details on the
after-life of some of their initiatives. Computer science connects
the papers in the first part of the book. The second part
concentrates on algebraic logic. It features a range of papers that
hint at the intricate many-way connections between logic, algebra,
and geometry. The third part explores novel applications of logic
in relativity theory, philosophy of logic, philosophy of physics
and spacetime, and methodology of science. They include such
exciting subjects as time travelling in emergent spacetime. The
short autobiographies of Hajnal Andreka and Istvan Nemeti at the
end of the book describe an adventurous journey from electric
engineering and Maxwell's equations to a complex system of computer
programs for designing Hungary's electric power system, to
exploring and contributing deep results to Tarskian algebraic logic
as the deepest core theory of such questions, then on to
applications of the results in such exciting new areas as
relativity theory in order to rejuvenate logic itself.
This book is a collection of papers devoted to the emergence and
development in Bulgarian Academy of Sciences of some of the areas
of informatics, including artificial intelligence. The papers are
prepared by specialists from the Academy, some of whom are among
the founders of these scientific and application areas in Bulgaria
and in some cases - in the world. The book is interesting for
specialists in informatics and computer science and researchers in
history of sciences.
Composites have been studied for more than 150 years, and interest
in their properties has been growing. This classic volume provides
the foundations for understanding a broad range of composite
properties, including electrical, magnetic, electromagnetic,
elastic and viscoelastic, piezoelectric, thermal, fluid flow
through porous materials, thermoelectric, pyroelectric,
magnetoelectric, and conduction in the presence of a magnetic field
(Hall effect). Exact solutions of the PDEs in model geometries
provide one avenue of understanding composites; other avenues
include microstructure-independent exact relations satisfied by
effective moduli, for which the general theory is reviewed;
approximation formulae for effective moduli; and series expansions
for the fields and effective moduli that are the basis of numerical
methods for computing these fields and moduli. The range of
properties that composites can exhibit can be explored either
through the model geometries or through microstructure-independent
bounds on the properties. These bounds are obtained through
variational principles, analytic methods, and Hilbert space
approaches. Most interesting is when the properties of the
composite are unlike those of the constituent materials, and there
has been an explosion of interest in such composites, now known as
metamaterials. The Theory of Composites surveys these aspects,
among others, and complements the new body of literature that has
emerged since the book was written. It remains relevant today by
providing historical background, a compendium of numerous results,
and through elucidating many of the tools still used today in the
analysis of composite properties. This book is intended for applied
mathematicians, physicists, and electrical and mechanical
engineers. It will also be of interest to graduate students.
This book reports on the latest knowledge concerning critical
phenomena arising in fluid-structure interaction due to movement
and/or deformation of bodies. The focus of the book is on reporting
progress in understanding turbulence and flow control to improve
aerodynamic / hydrodynamic performance by reducing drag, increasing
lift or thrust and reducing noise under critical conditions that
may result in massive separation, strong vortex dynamics,
amplification of harmful instabilities (flutter, buffet), and flow
-induced vibrations. Theory together with large-scale simulations
and experiments have revealed new features of turbulent flow in the
boundary layer over bodies and in thin shear layers immediately
downstream of separation. New insights into turbulent flow
interacting with actively deformable structures, leading to new
ways of adapting and controlling the body shape and vibrations to
respond to these critical conditions, are investigated. The book
covers new features of turbulent flows in boundary layers over
wings and in shear layers immediately downstream: studies of
natural and artificially generated fluctuations; reduction of noise
and drag; and electromechanical conversion topics. Smart actuators
as well as how smart designs lead to considerable benefits compared
with conventional methods are also extensively discussed. Based on
contributions presented at the IUTAM Symposium "Critical Flow
Dynamics involving Moving/Deformable Structures with Design
applications", held in June 18-22, 2018, in Santorini, Greece, the
book provides readers with extensive information about current
theories, methods and challenges in flow and turbulence control,
and practical knowledge about how to use this information together
with smart and bio-inspired design tools to improve aerodynamic and
hydrodynamic design and safety.
This volume explores the connections between mathematical modeling,
computational methods, and high performance computing, and how
recent developments in these areas can help to solve complex
problems in the natural sciences and engineering. The content of
the book is based on talks and papers presented at the conference
Modern Mathematical Methods and High Performance Computing in
Science & Technology (M3HPCST), held at Inderprastha
Engineering College in Ghaziabad, India in January 2020. A wide
range of both theoretical and applied topics are covered in detail,
including the conceptualization of infinity, efficient domain
decomposition, high capacity wireless communication, infectious
disease modeling, and more. These chapters are organized around the
following areas: Partial and ordinary differential equations
Optimization and optimal control High performance and scientific
computing Stochastic models and statistics Recent Trends in
Mathematical Modeling and High Performance Computing will be of
interest to researchers in both mathematics and engineering, as
well as to practitioners who face complex models and extensive
computations.
Precise approach with definitions, theorems, proofs, examples and
exercises. Topics include partial differentiation, vectors,
differential geometry, Stieltjes integral, infinite series, gamma
function, Fourier series, Laplace transform, much more. Numerous
graded exercises with selected answers.
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