This book introduces and develops new algebraic methods to work with relations, often conceived as Boolean matrices, and applies them to topology. Although these objects mirror the matrices that appear throughout mathematics, numerics, statistics, engineering, and elsewhere, the methods used to work with them are much less well known. In addition to their purely topological applications, the volume also details how the techniques may be successfully applied to spatial reasoning and to logics of computer science. Topologists will find several familiar concepts presented in a concise and algebraically manipulable form which is far more condensed than usual, but visualized via represented relations and thus readily graspable. This approach also offers the possibility of handling topological problems using proof assistants.
If there is a central conceptual framework that has reliably borne the weight of modern physics as it ascends into the twenty-first century, it is the framework of quantum mechanics. Because of its enduring stability in experimental application, physics has today reached heights that not only inspire wonder, but arguably exceed the limits of intuitive vision, if not intuitive comprehension. For many physicists and philosophers, however, the currently fashionable tendency toward exotic interpretation of the theoretical formalism is recognized not as a mark of ascent for the tower of physics, but rather an indicator of sway—one that must be dampened rather than encouraged if practical progress is to continue. In this unique two-part volume, designed to be comprehensible to both specialists and non-specialists, the authors chart out a pathway forward by identifying the central deficiency in most interpretations of quantum mechanics: That in its conventional, metrical depiction of extension, inherited from the Enlightenment, objects are characterized as fundamental to relations—i.e., such that relations presuppose objects but objects do not presuppose relations. The authors, by contrast, argue that quantum mechanics exemplifies the fact that physical extensiveness is fundamentally topological rather than metrical, with its proper logico-mathematical framework being category theoretic rather than set theoretic. By this thesis, extensiveness fundamentally entails not only relations of objects, but also relations of relations. Thus, the fundamental quanta of quantum physics are properly defined as units of logico-physical relation rather than merely units of physical relata as is the current convention. Objects are always understood as relata, and likewise relations are always understood objectively. In this way, objects and relations are coherently defined as mutually implicative. The conventional notion of a history as “a story about fundamental objects” is thereby reversed, such that the classical “objects” become the story by which we understand physical systems that are fundamentally histories of quantum events. These are just a few of the novel critical claims explored in this volume—claims whose exemplification in quantum mechanics will, the authors argue, serve more broadly as foundational principles for the philosophy of nature as it evolves through the twenty-first century and beyond.
General Topology and Its Relations to Modern Analysis and Algebra II is comprised of papers presented at the Second Symposium on General Topology and its Relations to Modern Analysis and Algebra, held in Prague in September 1966. The book contains expositions and lectures that discuss various subject matters in the field of General Topology. The topics considered include the algebraic structure for a topology; the projection spectrum and its limit space; some special methods of homeomorphism theory in infinite-dimensional topology; types of ultrafilters on countable sets; the compactness operator in general topology; and the algebraic generalization of the topological theorems of Bolzano and Weierstrass. This publication will be found useful by all specialists in the field of Topology and mathematicians interested in General Topology.
The book constitutes the joint refereed proceedings of the 9th International Conference on Relational Methods in Computer Science, RelMiCS 2006, and the 4th International Workshop on Applications of Kleene Algebras, AKA 2006, held in Manchester, UK in August/September 2006. The 25 revised full papers presented together with two invited papers and the abstract of an invited talk were carefully reviewed and selected from 44 submissions.
During the 1980s the Marshallian concept of industrial district (ID) became widely popular due to the resurgence of interest in the reasons that make the agglomeration of specialised industries a territorial phenomenon worth being analysed. The analysis of clusters and IDs has often been limited, considering only the local dimension of the created business networks. The external links of these systems have been systematically under-evaluated. This book offers a deep insight into the evolution of these systems and the internal-external mechanism of knowledge circulation and learning. This means that the access to external knowledge (information or R&D cooperative research) or to productive networks (global supply chains) is studied in order to describe how external knowledge is absorbed and how local clusters or districts become global systems. It provides a unified approach; showing that existing capabilities expand when locally embedded knowledge is combined with accessible external knowledge. In this view, external knowledge linkages reduce the danger of cognitive ‘lock-in’ and ‘over-embeddedness’, which may become important obstacles to local learning and innovation when technological trajectories and global economic conditions change. A selection of international experts
This book constitutes the thoroughly refereed post-conference proceedings of the 13th International Conference on Relational and Algebraic Methods in Computer Science, RAMiCS 13, held in Cambridge, UK, in September 2012. The 23 revised full papers presented were carefully selected from 39 submissions in the general area of relational and algebraic methods in computer science, adding special focus on formal methods for software engineering, logics of programs and links with neighboring disciplines. The papers are structured in specific fields on applications to software specification and correctness, mechanized reasoning in relational algebras, algebraic program derivation, theoretical foundations, relations and algorithms, and properties of specialized relations.
In the last decade we have seen the emergence of a new inter-disciplinary field concentrating on the understanding large networks which are dynamic, large, open, and have a structure that borders order and randomness. The field of Complex Networks has helped us better understand many complex phenomena such as spread of decease, protein interaction, social relationships, to name but a few. The field of Complex Networks has received a major boost caused by the widespread availability of huge network data resources in the last years. One of the most surprising findings is that real networks behave very distinct from traditional assumptions of network theory. Traditionally, real networks were supposed to have a majority of nodes of about the same number of connections around an average. This is typically modeled by random graphs. But modern network research could show that the majority of nodes of real networks is very low connected, and, by contrast, there exists some nodes of very extreme connectivity (hubs). The current theories coupled with the availability of data makes the field of Complex Networks (sometimes called Network Sciences) one of the most promising interdisciplinary disciplines of today. This sample of works in this book gives as a taste of what is in the horizon such controlling the dynamics of a network and in the network, using social interactions to improve urban planning, ranking in music, and the understanding knowledge transfer in influence networks.
