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# ðŸ“™ Finitely Supported Mathematics: An Introduction by Andrei Alexandru, Gabriel Ciobanu (auth.) â€” epub download

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In this book the authors present an alternative set theory dealing with a more relaxed notion of infiniteness, called finitely supported mathematics (FSM). It has strong connections to the Fraenkel-Mostowski (FM) permutative model of Zermelo-Fraenkel (ZF) set theory with atoms and to the theory of (generalized) nominal sets. More exactly, FSM is ZF mathematics rephrased in terms of finitely supported structures, where the set of atoms is infinite (not necessarily countable as for nominal sets). In FSM, 'sets' are replaced either by `invariant sets' (sets endowed with some group actions satisfying a finite support requirement) or by `finitely supported sets' (finitely supported elements in the powerset of an invariant set). It is a theory of `invariant algebraic structures' in which infinite algebraic structures are characterized by using their finite supports.

After explaining the motivation for using invariant sets in the experimental sciences as well as the connections with the nominal approach, admissible sets and Gandy machines (Chapter 1), the authors present in Chapter 2 the basics of invariant sets and show that the principles of constructing FSM have historical roots both in the definition of Tarski `logical notions' and in the Erlangen Program of Klein for the classification of various geometries according to invariants under suitable groups of transformations. Furthermore, the consistency of various choice principles is analyzed in FSM. Chapter 3 examines whether it is possible to obtain valid results by replacing the notion of infinite sets with the notion of invariant sets in the classical ZF results. The authors present techniques for reformulating ZF properties of algebraic structures in FSM. In Chapter 4 they generalize FM set theory by providing a new set of axioms inspired by the theory of amorphous sets, and so defining the extended Fraenkel-Mostowski (EFM) set theory. In Chapter 5 they define FSM semantics for certain process calculi (e.g., fusion calculus), and emphasize the links to the nominal techniques used in computer science. They demonstrate a complete equivalence between the new FSM semantics (defined by using binding operators instead of side conditions for presenting the transition rules) and the known semantics of these process calculi.

The book is useful for researchers and graduate students in computer science and mathematics, particularly those engaged with logic and set theory.

- Author:
**Andrei Alexandru, Gabriel Ciobanu (auth.)** - Year:
**2016** - Publisher:
**Springer International Publishing** - Language:
**English** - ISBN:
**978-3-319-42281-7,978-3-319-42282-4**

- File size:
**1 770 321** - Format:
**pdf**

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Striking a careful balance between mathematical rigor and engineering-oriented applications, this textbook aims to maximize the readers' understanding of both the mathematical and engineering aspects of control theory. The bedrock elements of classical...

This book is an introduction to a fascinating topic at the interface of functional analysis, algebra and numerical analysis, written for a broad audience of students, researchers and practitioners. It is concerned with the study of infinite matrices and t...

This introduction to 3D computer graphics emphasizes fundamentals and the mathematics underlying computer graphics, while also covering programming techniques using OpenGL, a platform-independent graphics programming environment. The minimal prerequisites...

This monograph contains a functional analytic introduction to Dirac's formalism. The first part presents some new mathematical notions in the setting of triples of Hilbert spaces, mentioning the concept of Dirac basis. The second part introduces a concept...

Part I gives a detailed, self-contained and mathematically rigorous exposition of classical conformal symmetry in n dimensions and its quantization in two dimensions. The conformal groups are determined and the appearence of the Virasoro algebra in the co...

An excellent introduction to feedback control system design, this book offers a theoretical approach that captures the essential issues and can be applied to a wide range of practical problems. Its explorations of recent developments in the field emphasiz...

The goal of this text is to present some of the basic ideas of fluid mechanics in a mathematically attractive manner, to present the physical background and motivation for some constructions that have been used in recent mathematical and numerical work on...

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as weil as the clasÂ sical techniques of applied mathe...

Basic Notions of Condensed Matter Physics is a clear introduction to some of the most significant concepts in the physics of condensed matter. The general principles of many-body physics and perturbation theory are emphasised, providing supportive mathema...

The papers published in this volume were presented at the Second International Conference on Ultra-WidebandiShort-Pulse (UWB/SP) Electromagnetics, ApriIS-7, 1994. To place this second international conference in proper perspective with respect to the firs...

From the President of the Research Society on Alcoholism In recent years, increasingly convincing evidence in support of a biobehavioral conceptual model of the etiology of alcoholism has emerged. In this model, the disorder is perceived as arising from t...

Diversity within and among living organisms is both a biological imperaÂ tive and a biological conundrum. Phenotypic and genotypic diversity is the critical currency ofecological interactions and the evolution of life. Thus, it is not unexpected to find v...

Volume 3 relates the demography of health behavior to developmental and diversity issues. Unique discussions of the health behaviors of gay males, lesbians, persons with HIV, and caregivers themselves are included. Each volume features extensive supplemen...

Often, a new area of science grows at the confines between recognised subject divisions, drawing upon techniques and intellectual perspectives from a diversity of fields. Such growth can remain unnoticed at first, until a characteristic fami ly of effects...

During the past two decades Membrane Science and Technology has made tremendous progress and has changed from a simple laboratory tool to large scale processes with numerous applications in Medicine and Industry. In this volume are collected papers presen...

Fundamentals.- Requirements for Systems and Subsystems (1,2).- Subsystems(1â€“6).- Systems.- Applications(1â€“4)....

In 1988 the Mossbauer effect community completed 30 years of continual contribution to the fields of nuclear physics, solid state science, and a variety of related disciplines. To celebrate this anniversary, Professor Gonser of the Universitat des Saarlan...

OpenStack provides a rich API that enables users to architect networks, create virtual machines, and scale their application as they see fit. However, it suffered limited network implementations, providing only basic networking through Linux VLANs and ipt...