Physics, Nanoscience, and Complexity
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Vectors and Beyond: Geometric Algebra and Its Philosophical
dialectica Vol. 63, N° 4 (2010), pp. 381–395 DOI: 10.1111/j.1746-8361.2009.01214.x Vectors and Beyond: Geometric Algebra and its Philosophical Significancedltc_1214 381..396 Peter Simons† In mathematics, like in everything else, it is the Darwinian struggle for life of ideas that leads to the survival of the concepts which we actually use, often believed to have come to us fully armed with goodness from some mysterious Platonic repository of truths. Simon Altmann 1. Introduction The purpose of this paper is to draw the attention of philosophers and others interested in the applicability of mathematics to a quiet revolution that is taking place around the theory of vectors and their application. It is not that new math- ematics is being invented – on the contrary, the basic concepts are over a century old – but rather that this old theory, having languished for many decades as a quaint backwater, is being rediscovered and properly applied for the first time. The philosophical importance of this quiet revolution is not that new applications for old mathematics are being found. That presumably happens much of the time. Rather it is that this new range of applications affords us a novel insight into the reasons why vectors and their mathematical kin find application at all in the real world. Indirectly, it tells us something general but interesting about the nature of the spatiotemporal world we all inhabit, and that is of philosophical significance. Quite what this significance amounts to is not yet clear. I should stress that nothing here is original: the history is quite accessible from several sources, and the mathematics is commonplace to those who know it. -
Acm Names Fellows for Innovations in Computing
CONTACT: Virginia Gold 212-626-0505 [email protected] ACM NAMES FELLOWS FOR INNOVATIONS IN COMPUTING 2014 Fellows Made Computing Contributions to Enterprise, Entertainment, and Education NEW YORK, January 8, 2015—ACM has recognized 47 of its members for their contributions to computing that are driving innovations across multiple domains and disciplines. The 2014 ACM Fellows, who hail from some of the world’s leading universities, corporations, and research labs, have achieved advances in computing research and development that are driving innovation and sustaining economic development around the world. ACM President Alexander L. Wolf acknowledged the advances made by this year’s ACM Fellows. “Our world has been immeasurably improved by the impact of their innovations. We recognize their contributions to the dynamic computing technologies that are making a difference to the study of computer science, the community of computing professionals, and the countless consumers and citizens who are benefiting from their creativity and commitment.” The 2014 ACM Fellows have been cited for contributions to key computing fields including database mining and design; artificial intelligence and machine learning; cryptography and verification; Internet security and privacy; computer vision and medical imaging; electronic design automation; and human-computer interaction. ACM will formally recognize the 2014 Fellows at its annual Awards Banquet in June in San Francisco. Additional information about the ACM 2014 Fellows, the awards event, as well as previous -
Journal of Applied Logic
JOURNAL OF APPLIED LOGIC AUTHOR INFORMATION PACK TABLE OF CONTENTS XXX . • Description p.1 • Impact Factor p.1 • Abstracting and Indexing p.1 • Editorial Board p.1 • Guide for Authors p.5 ISSN: 1570-8683 DESCRIPTION . This journal welcomes papers in the areas of logic which can be applied in other disciplines as well as application papers in those disciplines, the unifying theme being logics arising from modelling the human agent. For a list of areas covered see the Editorial Board. The editors keep close contact with the various application areas, with The International Federation of Compuational Logic and with the book series Studies in Logic and Practical Reasoning. Benefits to authors We also provide many author benefits, such as free PDFs, a liberal copyright policy, special discounts on Elsevier publications and much more. Please click here for more information on our author services. Please see our Guide for Authors for information on article submission. This journal has an Open Archive. All published items, including research articles, have unrestricted access and will remain permanently free to read and download 48 months after publication. All papers in the Archive are subject to Elsevier's user license. If you require any further information or help, please visit our Support Center IMPACT FACTOR . 2016: 0.838 © Clarivate Analytics Journal Citation Reports 2017 ABSTRACTING AND INDEXING . Zentralblatt MATH Scopus EDITORIAL BOARD . Executive Editors Dov M. Gabbay, King's College London, London, UK Sarit Kraus, Bar-llan University, -
Knowledge Representation in Bicategories of Relations
Knowledge Representation in Bicategories of Relations Evan Patterson Department of Statistics, Stanford University Abstract We introduce the relational ontology log, or relational olog, a knowledge representation system based on the category of sets and relations. It is inspired by Spivak and Kent’s olog, a recent categorical framework for knowledge representation. Relational ologs interpolate between ologs and description logic, the dominant formalism for knowledge representation today. In this paper, we investigate relational ologs both for their own sake and to gain insight into the relationship between the algebraic and logical approaches to knowledge representation. On a practical level, we show by example that relational ologs have a friendly and intuitive—yet fully precise—graphical syntax, derived from the string diagrams of monoidal categories. We explain several other useful features of relational ologs not possessed by most description logics, such as a type system and a rich, flexible notion of instance data. In a more theoretical vein, we draw on categorical logic to show how relational ologs can be translated to and from logical theories in a fragment of first-order logic. Although we make extensive use of categorical language, this paper is designed to be self-contained and has considerable expository content. The only prerequisites are knowledge of first-order logic and the rudiments of category theory. 1. Introduction arXiv:1706.00526v2 [cs.AI] 1 Nov 2017 The representation of human knowledge in computable form is among the oldest and most fundamental problems of artificial intelligence. Several recent trends are stimulating continued research in the field of knowledge representation (KR). -
The Construction of Spinors in Geometric Algebra
The Construction of Spinors in Geometric Algebra Matthew R. Francis∗ and Arthur Kosowsky† Dept. of Physics and Astronomy, Rutgers University 136 Frelinghuysen Road, Piscataway, NJ 08854 (Dated: February 4, 2008) The relationship between spinors and Clifford (or geometric) algebra has long been studied, but little consistency may be found between the various approaches. However, when spinors are defined to be elements of the even subalgebra of some real geometric algebra, the gap between algebraic, geometric, and physical methods is closed. Spinors are developed in any number of dimensions from a discussion of spin groups, followed by the specific cases of U(1), SU(2), and SL(2, C) spinors. The physical observables in Schr¨odinger-Pauli theory and Dirac theory are found, and the relationship between Dirac, Lorentz, Weyl, and Majorana spinors is made explicit. The use of a real geometric algebra, as opposed to one defined over the complex numbers, provides a simpler construction and advantages of conceptual and theoretical clarity not available in other approaches. I. INTRODUCTION Spinors are used in a wide range of fields, from the quantum physics of fermions and general relativity, to fairly abstract areas of algebra and geometry. Independent of the particular application, the defining characteristic of spinors is their behavior under rotations: for a given angle θ that a vector or tensorial object rotates, a spinor rotates by θ/2, and hence takes two full rotations to return to its original configuration. The spin groups, which are universal coverings of the rotation groups, govern this behavior, and are frequently defined in the language of geometric (Clifford) algebras [1, 2]. -
Contextuality, Cohomology and Paradox
The Sheaf Team Rui Soares Barbosa, Kohei Kishida, Ray Lal and Shane Mansfield Samson Abramsky Joint work with Rui Soares Barbosa, KoheiContextuality, Kishida, Ray LalCohomology and Shane and Mansfield Paradox (Department of Computer Science, University of Oxford)2 / 37 Contextuality. Key to the \magic" of quantum computation. Experimentally verified, highly non-classical feature of physical reality. And pervasive in logic, computation, and beyond. In a nutshell: data which is locally consistent, but globally inconsistent. We find a direct connection between the structure of quantum contextuality and classic semantic paradoxes such as \Liar cycles". Conversely, contextuality offers a novel perspective on these paradoxes. Cohomology. Sheaf theory provides the natural mathematical setting for our analysis, since it is directly concerned with the passage from local to global. In this setting, it is furthermore natural to use sheaf cohomology to characterise contextuality. Cohomology is one of the major tools of modern mathematics, which has until now largely been conspicuous by its absence, in logic, theoretical computer science, and quantum information. Our results show that cohomological obstructions to the extension of local sections to global ones witness a large class of contextuality arguments. Contextual Semantics Samson Abramsky Joint work with Rui Soares Barbosa, KoheiContextuality, Kishida, Ray LalCohomology and Shane and Mansfield Paradox (Department of Computer Science, University of Oxford)3 / 37 In a nutshell: data which is locally consistent, but globally inconsistent. We find a direct connection between the structure of quantum contextuality and classic semantic paradoxes such as \Liar cycles". Conversely, contextuality offers a novel perspective on these paradoxes. Cohomology. Sheaf theory provides the natural mathematical setting for our analysis, since it is directly concerned with the passage from local to global. -
The Genesis of Geometric Algebra: a Personal Retrospective
Adv. Appl. Clifford Algebras 27 (2017), 351–379 c 2016 The Author(s). This article is published with open access at Springerlink.com 0188-7009/010351-29 published online April 11, 2016 Advances in DOI 10.1007/s00006-016-0664-z Applied Clifford Algebras The Genesis of Geometric Algebra: A Personal Retrospective David Hestenes* Abstract. Even today mathematicians typically typecast Clifford Al- gebra as the “algebra of a quadratic form,” with no awareness of its grander role in unifying geometry and algebra as envisaged by Clifford himself when he named it Geometric Algebra. It has been my privilege to pick up where Clifford left off—to serve, so to speak, as principal architect of Geometric Algebra and Calculus as a comprehensive math- ematical language for physics, engineering and computer science. This is an account of my personal journey in discovering, revitalizing and ex- tending Geometric Algebra, with emphasis on the origin and influence of my book Space-Time Algebra. I discuss guiding ideas, significant re- sults and where they came from—with recollection of important events and people along the way. Lastly, I offer some lessons learned about life and science. 1. Salutations I am delighted and honored to join so many old friends and new faces in celebrating the 50th anniversary of my book Space-Time Algebra (STA). That book launched my career as a theoretical physicist and the journey that brought us here today. Let me use this opportunity to recall some highlights of my personal journey and offer my take on lessons to be learned. The first lesson follows: 2. -
The Relativistic Transactional Interpretation and Spacetime Emergence
The Relativistic Transactional Interpretation and Spacetime Emergence R. E. Kastner* 3/10/2021 “The flow of time is a real becoming in which potentiality is transformed into actuality.” -Hans Reichenbach1 This paper is an excerpt from Chapter 8 of the forthcoming second edition of my book, The Transactional Interpretation of Quantum Mechanics: A Relativistic Treatment, Cambridge University Press. Abstract. We consider the manner in which the spacetime manifold emerges from a quantum substratum through the transactional process, in which spacetime events and their connections are established. In this account, there is no background spacetime as is generally assumed in physical theorizing. Instead, the usual notion of a background spacetime is replaced by the quantum substratum, comprising quantum systems with nonvanishing rest mass. Rest mass corresponds to internal periodicities that function as internal clocks defining proper times, and in turn, inertial frames that are not themselves aspects of the spacetime manifold, but are pre- spacetime reference structures. Specific processes in the quantum substratum serve to distinguish absolute from relative motion. 1. Introduction and Background The Transactional Interpretation of quantum mechanics has been extended into the relativistic domain by the present author. That model is now known as the Relativistic Transactional Interpretation (RTI) (e.g., Kastner 2015, 2018, 2019, 2021; Kastner and Cramer, 2018). RTI is based on the direct-action theory of fields in its relativistic quantum form, -
Current Issue of FACS FACTS
Issue 2021-2 July 2021 FACS A C T S The Newsletter of the Formal Aspects of Computing Science (FACS) Specialist Group ISSN 0950-1231 FACS FACTS Issue 2021-2 July 2021 About FACS FACTS FACS FACTS (ISSN: 0950-1231) is the newsletter of the BCS Specialist Group on Formal Aspects of Computing Science (FACS). FACS FACTS is distributed in electronic form to all FACS members. Submissions to FACS FACTS are always welcome. Please visit the newsletter area of the BCS FACS website for further details at: https://www.bcs.org/membership/member-communities/facs-formal-aspects- of-computing-science-group/newsletters/ Back issues of FACS FACTS are available for download from: https://www.bcs.org/membership/member-communities/facs-formal-aspects- of-computing-science-group/newsletters/back-issues-of-facs-facts/ The FACS FACTS Team Newsletter Editors Tim Denvir [email protected] Brian Monahan [email protected] Editorial Team: Jonathan Bowen, John Cooke, Tim Denvir, Brian Monahan, Margaret West. Contributors to this issue: Jonathan Bowen, Andrew Johnstone, Keith Lines, Brian Monahan, John Tucker, Glynn Winskel BCS-FACS websites BCS: http://www.bcs-facs.org LinkedIn: https://www.linkedin.com/groups/2427579/ Facebook: http://www.facebook.com/pages/BCS-FACS/120243984688255 Wikipedia: http://en.wikipedia.org/wiki/BCS-FACS If you have any questions about BCS-FACS, please send these to Jonathan Bowen at [email protected]. 2 FACS FACTS Issue 2021-2 July 2021 Editorial Dear readers, Welcome to the 2021-2 issue of the FACS FACTS Newsletter. A theme for this issue is suggested by the thought that it is just over 50 years since the birth of Domain Theory1. -
Physics from Computer Science
Physics from Computer Science Samson Abramsky and Bob Coecke Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK. samson abramsky · [email protected] Where sciences interact. We are, respectively, a computer scientist interested in the logic and seman- tics of computation, and a physicist interested in the foundations of quantum mechanics. Currently we are pursuing what we consider to be a very fruitful collaboration as members of the same Computer Science department. How has this come about? It flows naturally from the fact that we are working in a field of computer science where physical theory starts to play a key role, that is, natural computation, with, of course, quantum computation as a special case. At this workshop there will be many advocates of this program present, and we are honoured to be part of that community. But there is more. Our joint research is both research on semantics for distributed computing with non-von Neumann architectures, and on the axiomatic foundations of physical theories. This dual character of our work comes without any compromise, and proves to be very fruitful. Computational architectures as toy models for physics. Computer science has something more to offer to the other sciences than the computer. Indeed, on the topic of mathematical and logical un- derstanding of fundamental transdisciplinary scientific concepts such as interaction, concurrency and causality, synchrony and asynchrony, compositional modelling and reasoning, open systems, qualitative versus quantitative reasoning, operational methodologies, continuous versus discrete, hybrid systems etc. computer science is far ahead of many other sciences, due to the challenges arising from the amaz- ing rapidity of the technology change and development it is constantly being confronted with. -
Reforming the Mathematical Language of Physics
Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics David Hestenes Department of Physics and Astronomy Arizona State University, Tempe, Arizona 85287-1504 The connection between physics teaching and research at its deepest level can be illuminated by Physics Education Research (PER). For students and scientists alike, what they know and learn about physics is profoundly shaped by the conceptual tools at their command. Physicists employ a miscellaneous assortment of mathematical tools in ways that contribute to a fragmenta- tion of knowledge. We can do better! Research on the design and use of mathematical systems provides a guide for designing a unified mathemati- cal language for the whole of physics that facilitates learning and enhances physical insight. This has produced a comprehensive language called Geo- metric Algebra, which I introduce with emphasis on how it simplifies and integrates classical and quantum physics. Introducing research-based re- form into a conservative physics curriculum is a challenge for the emerging PER community. Join the fun! I. Introduction The relation between teaching and research has been a perennial theme in academia as well as the Oersted Lectures, with no apparent progress on re- solving the issues. Physics Education Research (PER) puts the whole matter into new light, for PER makes teaching itself a subject of research. This shifts attention to the relation of education research to scientific research as the central issue. To many, the research domain of PER is exclusively pedagogical. Course content is taken as given, so the research problem is how to teach it most effec- tively. This approach to PER has produced valuable insights and useful results. -
Electron Paths, Tunnelling and Diffraction in the Spacetime Algebra
Electron Paths, Tunnelling and Diffraction in the Spacetime Algebra AUTHORS Stephen Gull Anthony Lasenby Chris Doran Found. Phys. 23(10), 1329-1356 (1993) 1 Abstract This paper employs the ideas of geometric algebra to investigate the physical content of Dirac’s electron theory. The basis is Hestenes’ discovery of the geometric significance of the Dirac spinor, which now represents a Lorentz transformation in spacetime. This transformation specifies a definite velocity, which might be interpreted as that of a real electron. Taken literally, this velocity yields predictions of tunnelling times through potential barriers, and defines streamlines in spacetime that would correspond to electron paths. We also present a general, first-order diffraction theory for electromagnetic and Dirac waves. We conclude with a critical appraisal of the Dirac theory. 2 1 Introduction In this, the last of a 4-paper series [1, 2, 3], we are concerned with one of the main areas of David Hestenes’ work — the Dirac equation. His thesis has always been that the theory of the electron is central to quantum mechanics and that the electron wavefunction, or Dirac spinor, contains important geometric information [4, 5, 6, 7]. This information is usually hidden by the conventional matrix notation, but it can be revealed by systematic use of a better mathematical language — the geometric (Clifford) algebra of spacetime, or spacetime algebra (STA) [8]. This algebra provides a powerful coordinate-free language for dealing with all aspects of relativistic physics — not just relativistic quantum mechanics [9]. Indeed, it is a formalism that makes the conventional 4-vector/tensor approach look decidedly primitive.