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Books > Science & Mathematics > Mathematics > Mathematical foundations > General
This monograph is devoted to a new class of non-commutative rings,
skew Poincare-Birkhoff-Witt (PBW) extensions. Beginning with the
basic definitions and ring-module theoretic/homological properties,
it goes on to investigate finitely generated projective modules
over skew PBW extensions from a matrix point of view. To make this
theory constructive, the theory of Groebner bases of left (right)
ideals and modules for bijective skew PBW extensions is developed.
For example, syzygies and the Ext and Tor modules over these rings
are computed. Finally, applications to some key topics in the
noncommutative algebraic geometry of quantum algebras are given,
including an investigation of semi-graded Koszul algebras and
semi-graded Artin-Schelter regular algebras, and the noncommutative
Zariski cancellation problem. The book is addressed to researchers
in noncommutative algebra and algebraic geometry as well as to
graduate students and advanced undergraduate students.
This book is a collection of contributions honouring Arnon Avron's
seminal work on the semantics and proof theory of non-classical
logics. It includes presentations of advanced work by some of the
most esteemed scholars working on semantic and proof-theoretical
aspects of computer science logic. Topics in this book include
frameworks for paraconsistent reasoning, foundations of relevance
logics, analysis and characterizations of modal logics and fuzzy
logics, hypersequent calculi and their properties,
non-deterministic semantics, algebraic structures for many-valued
logics, and representations of the mechanization of mathematics.
Avron's foundational and pioneering contributions have been widely
acknowledged and adopted by the scientific community. His research
interests are very broad, spanning over proof theory, automated
reasoning, non-classical logics, foundations of mathematics, and
applications of logic in computer science and artificial
intelligence. This is clearly reflected by the diversity of topics
discussed in the chapters included in this book, all of which
directly relate to Avron's past and present works. This book is of
interest to computer scientists and scholars of formal logic.
This exploration of a selection of fundamental topics and general
purpose tools provides a roadmap to undergraduate students who
yearn for a deeper dive into many of the concepts and ideas they
have been encountering in their classes whether their motivation is
pure curiosity or preparation for graduate studies. The topics
intersect a wide range of areas encompassing both pure and applied
mathematics. The emphasis and style of the book are motivated by
the goal of developing self-reliance and independent mathematical
thought. Mathematics requires both intuition and common sense as
well as rigorous, formal argumentation. This book attempts to
showcase both, simultaneously encouraging readers to develop their
own insights and understanding and the adoption of proof writing
skills. The most satisfying proofs/arguments are fully rigorous and
completely intuitive at the same time.
This book introduces new models based on R-calculus and theories of
belief revision for dealing with large and changing data. It
extends R-calculus from first-order logic to propositional logic,
description logics, modal logic and logic programming, and from
minimal change semantics to subset minimal change,
pseudo-subformula minimal change and deduction-based minimal change
(the last two minimal changes are newly defined). And it proves
soundness and completeness theorems with respect to the minimal
changes in these logics. To make R-calculus computable, an
approximate R-calculus is given which uses finite injury priority
method in recursion theory. Moreover, two applications of
R-calculus are given to default theory and semantic inheritance
networks. This book offers a rich blend of theory and practice. It
is suitable for students, researchers and practitioners in the
field of logic. Also it is very useful for all those who are
interested in data, digitization and correctness and consistency of
information, in modal logics, non monotonic logics,
decidable/undecidable logics, logic programming, description
logics, default logics and semantic inheritance networks.
This book describes some basic principles that allow developers of
computer programs (computer scientists, software engineers,
programmers) to clearly think about the artifacts they deal with in
their daily work: data types, programming languages, programs
written in these languages that compute from given inputs wanted
outputs, and programs that describe continuously executing systems.
The core message is that clear thinking about programs can be
expressed in a single universal language, the formal language of
logic. Apart from its universal elegance and expressiveness, this
"logical" approach to the formal modeling of and reasoning about
computer programs has another advantage: due to advances in
computational logic (automated theorem proving, satisfiability
solving, model checking), nowadays much of this process can be
supported by software. This book therefore accompanies its
theoretical elaborations by practical demonstrations of various
systems and tools that are based on respectively make use of the
presented logical underpinnings.
This book delves into finite mathematics and its application in
physics, particularly quantum theory. It is shown that quantum
theory based on finite mathematics is more general than standard
quantum theory, whilst finite mathematics is itself more general
than standard mathematics.As a consequence, the mathematics
describing nature at the most fundamental level involves only a
finite number of numbers while the notions of limit,
infinite/infinitesimal and continuity are needed only in
calculations that describe nature approximately. It is also shown
that the concepts of particle and antiparticle are likewise
approximate notions, valid only in special situations, and that the
electric charge and baryon- and lepton quantum numbers can be only
approximately conserved.
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New Perspectives in Algebra, Topology and Categories
- Summer School, Louvain-la-Neuve, Belgium, September 12-15, 2018 and September 11-14, 2019
(Hardcover, 1st ed. 2021)
Maria Manuel Clementino, Alberto Facchini, Marino Gran
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R2,219
Discovery Miles 22 190
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Ships in 10 - 15 working days
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This book provides an introduction to some key subjects in algebra
and topology. It consists of comprehensive texts of some hours
courses on the preliminaries for several advanced theories in
(categorical) algebra and topology. Often, this kind of
presentations is not so easy to find in the literature, where one
begins articles by assuming a lot of knowledge in the field. This
volume can both help young researchers to quickly get into the
subject by offering a kind of " roadmap " and also help master
students to be aware of the basics of other research directions in
these fields before deciding to specialize in one of them.
Furthermore, it can be used by established researchers who need a
particular result for their own research and do not want to go
through several research papers in order to understand a single
proof. Although the chapters can be read as " self-contained "
chapters, the authors have tried to coordinate the texts in order
to make them complementary. The seven chapters of this volume
correspond to the seven courses taught in two Summer Schools that
took place in Louvain-la-Neuve in the frame of the project Fonds
d'Appui a l'Internationalisation of the Universite catholique de
Louvain to strengthen the collaborations with the universities of
Coimbra, Padova and Poitiers, within the Coimbra Group.
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