In this review paper, we discuss how gravity and spin can be obtained as the realization of the local Conformal-Affine group of symmetry transformations. In particular, we show how gravitation is a gauge theory which can be obtained starting from some local invariance as the Poincaré local symmetry. We review previous results where the inhomogeneous connection coefficients, transforming under the Lorentz group, give rise to gravitational gauge potentials which can be used to define covariant derivatives accommodating minimal couplings (...) of matter, gauge fields (and then spin connections). After we show, in a self-contained approach, how the tetrads and the Lorentz group can be used to induce the spacetime metric and then the Invariance Induced Gravity can be directly obtained both in holonomic and anholonomic pictures. Besides, we show how tensor valued connection forms act as auxiliary dynamical fields associated with the dilation, special conformal and deformation (shear) degrees of freedom, inherent to the bundle manifold. As a result, this allows to determine the bundle curvature of the theory and then to construct boundary topological invariants which give rise to a prototype (source free) gravitational Lagrangian. Finally, the Bianchi identities, the covariant field equations and the gauge currents are obtained determining completely the dynamics. (shrink)

Scale invariance provides a principled reason for the physical importance of Hilbert space, the Virasoro algebra, the string mode expansion, canonical commutators and Schrödinger evolution of states, independent of the assumptions of string theory and quantum theory. The usual properties of dimensionful fields imply an infinite, projective tower of conformal weights associated with the tangent space to scale-invariant spacetimes. Convergence and measurability on this tangent tower are guaranteed using a scale-invariant norm, restricted to conformally self-dual vectors. Maps on (...) the resulting Hilbert space are correspondingly restricted to semi-definite conformal weight. We find the maximally- and minimally-commuting, complete Lie algebras of definite-weight operators. The projective symmetry of the tower gives these algebras central charges, giving the canonical commutator and quantum Virasoro algebras, respectively. Using a continuous, m-parameter representation for rank-m tower tensors, we show that the parallel transport equation for the momentum-vector of a particle is the Schrödinger equation, while the associated definite-weight operators obey canonical commutation relations. Generalizing to the set of integral curves of general timelike, self-dual vector-valued weight maps gives a lifting such that the action of the curves parallel transports arbitrary tower vectors. We prove that the full set of Schrödinger-lifted integral curves of a general self-dual map gives an immersion of its 2-dim parameter space into spacetime, inducing a Lorentzian metric on the parameter space. This immersion is shown to satisfy the full variational equations of open string. (shrink)

A solution of the sourceless Einstein's equation with an infinite value for the cosmological constant Λ is discussed by using Inönü–Wigner contractions of the de Sitter groups and spaces. When Λ→∞, spacetime becomes a four-dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four-cone vertex to the infinity of Minkowski space, and the four-cone infinity to the Minkowski light-cone. The non-relativistic limit c→∞ is further considered, the kinematical group in this case being a modified Galilei (...) group in which the space and time translations are replaced by the non-relativistic limits of the corresponding proper conformal transformations. This group presents the same abstract Lie algebra as the Galilei group and can be named the conformal Galilei group. The results may be of interest to the early Universe Cosmology. (shrink)

This book aims to provide an overview of several topics in advanced Differential Geometry and Lie Group Theory, all of them stemming from mathematical problems in supersymmetric physical theories. It presents a mathematical illustration of the main development in geometry and symmetry theory that occurred under the fertilizing influence of supersymmetry/supergravity. The contents are mainly of mathematical nature, but each topic is introduced by historical information and enriched with motivations from high energy physics, which help the reader in getting a (...) deeper comprehension of the subject. (shrink)

In this paper we study a certain class of Lie algebras over commutative von Neumann algebras satisfying a certain finiteness condition. By using Boolean valued methods developed by Takeuti [8]-[11], we will establish the basic structure and representation theorems.

The rigidity of the real semisimple Lie algebra $${\mathfrak {so}}$$ so was first proved in a brief paper published by Fantappiè in 1954. The purpose of this note is to provide some historical context for this work and discuss why no further developments of this result were pursued by Italian mathematicians at the time.

The elementary theory of a nontrivial free Lie algebra over a commutative integral domain is unstable and has the strict order property.

The work aims to demonstrate that at the heart of Eriugena’s approach to Christian theology there lies a profoundly philosophical interest in the necessity of a cardinal shift in the paradigms of thinking – namely, that from the metaphysical to the dialectical one, which wins him a reputation of the ‘Hegel of the ninth century,’ as scholars in Post-Hegelian Germany called him. The prime concern of Eriugena’s discourse is to prove that the actual adoption of the salvific truth of Christ’s (...) revelation about all humans’ Sonship to God directly depends on the way the truth of God’s Oneness is consistently thought of. It is exactly the dialectic of the universal and particular which allows Eriugena both to tackle the dichotomy between being and non-being and to proceed towards raising a question how the totality of God’s being can be approached so as to let him radically reconsider a predominantly metaphysical view of creation the theological reflection traditionally relies on. According to the dialectical understanding of unity that Eriugena does adhere to, the reality of creation cannot be thought of, and therefore known, otherwise than in the way of being inseparable from the universal Principle of all. This is the Principle abandoned by nothing, unless the mind corrupted by the senses thinks otherwise and, following the metaphysical pattern of dichotomy, improperly sets the creation and its Principle apart. Restoration of the mind to the proper rational motion of recta ratio freed, as Eriugena argues, from the dictates of senses therefore becomes the way of both the epistemological breakthrough to the infinite whole and practical return from the world of finite things to living in the divine reality of creation. The work’s argument is based on the assumption of close affinity between Eriugena’s discourse and that of his Islamic contemporaries, who developed their dialectical ideas within the Mu’tazilah tradition of a philosophically disciplined approach to the truth of God’s Oneness. In particular, al-Nazzam’s engagement with Parmenides’ Periphyseon and his resistance to the danger of a dualistic interpretation of its ontology seem to provoke Eriugena’s innovative approach to Christian theology with a view to suggesting a mode of overcoming dualism as the main obstacle on the way to the Truth revealed. This vision of the meaning of Eriugena’s undertaking allows us not only to better understand the novelty of his approach to Christian theology, but also reconsider some of the key points of his discourse that seem to have become a sort of commonplace in Eriugenian studies: 1. Unlike the prevalent opinion, not the forms of the division of Nature but the modes of interpreting being and non-being are to be understood to constitute the genuine subject-matter of each book of the Periphyseon and, hence, of the five parts of his system. 2. The fourfold division of Nature is to be interpreted not as a basic structure of the system offered by Eriugena, but as a means of introducing dialectic to the body of theology by refuting Augustine’s metaphysical vision of a hierarchical model of the universe and indicating the way of resolution of the cardinally theological contradiction – God does and does not create at the same time. 3. All this gives reason to disagree with a general tendency of associating Eriugena’s work with exploration of the division of God’s Nature and to reinterpret it as an immense anti-division project to be understood as an important turn in the history of Christian thought entirely focused on the truth of God’s Oneness and human life in conformity to it. *** I affirm that this thesis is entirely my own work and has not been submitted for examination in any form elsewhere. (shrink)

In this paper, we develop the beginning of Lie-differential algebra, in the sense of Kolchin by using tools introduced by Hubert in [E. Hubert, Differential algebra for derivations with nontrivial commutation rules, J. Pure Appl. Algebra 200 163–190]. In particular it allows us to adapt the results of Tressl 3933–3951]) by showing the existence of a theory of Lie-differential fields of characteristic zero. This theory will serve as a model companion for every theory of large and Lie-differential (...) fields extending a model complete theory of pure fields. As an application, we introduce the Lie counterpart of classical theories of differential fields in several commuting derivations. (shrink)

Phase models for affine linear logic were independently devised by Lafont [10] and Piazza [15], although foreshadowed by Ono [14]. However, the existing semantics either contain no explicit directions for the construction of models in the general case, or else are forced to resort to additional conditions extending Girard's semantics. We dispense with these extra postulates - at least for the subexponential fragment of this logic - considering structures where the set of antiphases is concretely constructed. Moreover, we show (...) the equivalence of our phase models and the algebraic models of affine linear logic by means of a representation theorem. (shrink)

People who speak up about the unpleasant realities of environmental degradation, capitalist exploitation and the growth economy are likely to be criticised for 'negative framing' - while corporations undermine truths by casting them as social constructs with no objective validity. Environmentalists increasingly conform to the idea of telling nice stories using abstract metaphors rather than seeking to identify, specify and name systemic problems and their causes. Psychological pressures faced by scientists and activists, and personal strategies for coming to terms with (...) and combating harsh realities, are themes that run through this issue of Environmental Values. This reveals the necessity of radical activism and the expression of a more general concern over ethics and justice as counters to the stiflingly narrow hegemonic economic discourse that prevails in official circles. (shrink)

A simple model for the description of atomic and ionic species with spectra exhibiting a quantum defect is solved using the Lie algebra su(1, 1). The quantum defect of bound states is related to the phase shift of scattering states. The resonances are discussed in terms of the nonunitary representations of this algebra.

In this paper, constructions of free algebras corresponding to multiplicative classical linear logic, its affine variant, and their extensions with -contraction () are given. As an application, the cardinality problem of some one-variable linear fragments with -contraction is solved.

This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional.

Einstein's general relativity theory describes very well the gravitational phenomena in themacroscopic world. In themicroscopic domain of elementary particles, however, it does not exhibit gauge invariance or approximate Bjorken type scaling, properties which are believed to be indispensible for arenormalizable field theory. We argue that thelocal extension of space-time symmetries, such as of Lorentz and scale invariance, provides the clue for improvement. Eventually, this leads to aGL(4, R)-gauge approach to gravity in which the metric and the affine connection acquire (...) the status ofindependent fields. The Yang-Mills type field equations, the Noether identities, and conformal models of gravity are discussed within this framework. After symmetry breaking, Einstein's GR surfaces as an effective “low-energy” theory. (shrink)

Mind has played the starring role in the West's arts, humanities, and sciences, while an embodied notion of oneself, others, and the physical world has been customarily pushed under the rug. In view of radical new theories, methods and techniques that have emerged during the past century and a half, the notion of complementary, sympathetic co-participation, and its accompanying re-enchantment, merits attention. C. S. Peirce is at the crossroads between modernism, enchantment, and misplaced concreteness, on the one hand, and postmodernism (...) and its attendant poststructuralism, on the other. His categories of feeling, sensation, and thought, and his triadic, all-encompassing concept of the sign, can help bring all facets of human creativity, concrete living, and understanding into a proper balance. Three coined terms — homogeny, hierogeny, and heterogeny — serve to elucidate how Peirce's triadic vision can offer a middle path, a way of mediating between antagonistic modes of sensing, living, and thinking. This middle path follows traditional tenets of logic and reason — identity, non-contradiction, and excluded-middle — while at its fringes, it allows deviation — by way of process, inconsistency, and incompleteness — from the rigidity that lies therein. Studies of feminist postures and various cultural processes in Latin America bear out the premise that a dose of conformity accompanying a tendency toward resistance can synethnically help keep broad cultural processes on an even keel, balancing mind and body, conflicting theories and methods and modes of living, and styles of reason and unreason in equilibrium. (shrink)

This book presents the foundations of a general theory of algebras. Often called “universal algebra”, this theory provides a common framework for all algebraic systems, including groups, rings, modules, fields, and lattices. Each chapter is replete with useful illustrations and exercises that solidify the reader's understanding. The book begins by developing the main concepts and working tools of algebras and lattices, and continues with examples of classical algebraic systems like groups, semigroups, monoids, and categories. The essence of the book (...) lies in Chapter 4, which provides not only basic concepts and results of general algebra, but also the perspectives and intuitions shared by practitioners of the field. The book finishes with a study of possible uniqueness of factorizations of an algebra into a direct product of directly indecomposable algebras. There is enough material in this text for a two semester course sequence, but a one semester course could also focus primarily on Chapter 4, with additional topics selected from throughout the text. (shrink)

We define a Lie differential field as a field of characteristic 0 with an action, as derivations on , of some given Lie algebra . We assume that is a finite-dimensional vector space over some sub-field given in advance. As an example take the field of rational functions on a smooth algebraic variety, with .For every simple extension of Lie differential fields we find a finite system of differential equations that characterizes it. We then define, using first-order conditions, a (...) collection of allowed systems of differential equations s.t. the above characteristic systems are allowed. We prove that for every allowed system there exists a generic solution in some extension, and this solution is unique .We construct the model completion of the theory of Lie differential fields by adding axioms stating that every allowed system has almost generic solutions. The construction is a generalization of Blum's axioms for DCF0. We also show that this model completion is ω-stable. (shrink)

The CPT theorem states that any causal, Lorentz-invariant, thermodynamically well-behaved quantum field theory must also be invariant under a reflection symmetry that reverses the direction of time, flips spatial parity, and conjugates charge. Although its physical basis remains obscure, CPT symmetry appears to be necessary in order to unify quantum mechanics with relativity. This paper attempts to decipher the physical reasoning behind proofs of the CPT theorem in algebraic quantum field theory. Ultimately, CPT symmetry is linked to a systematic reversal (...) of the C*-algebraic Lie product that encodes the generating relationship between observables and symmetries. In any physically reasonable relativistic quantum field theory it is always possible to systematically reverse this generating relationship while preserving the dynamics, spectra, and localization properties of physical systems. Rather than the product of three separate reflections, CPT symmetry is revealed to be a single global reflection of the theory’s state space. (shrink)

All causal Lie products of solutions of the Klein-Gordon equation and the wave equation in Minkowski space are determined. The results shed light on the origin of the algebraic structures underlying quantum field theory.

This book is intended for mathematicians. Its origins lie in a course of lectures given by an algebraist to a class which had just completed a sub stantial course on abstract algebra. Consequently, our treatment ofthe sub ject is algebraic. Although we assurne a reasonable level of sophistication in algebra, the text requires little more than the basic notions of group, ring, module, etc. A more detailed knowledge of algebra is required for some of . the exercises. (...) We also assurne a familiarity with the main ideas of set theory, including cardinal numbers and Zorn's Lemma. In this book, we carry out a mathematical study of the logic used in mathematics. We do this by constructing a mathematical model oflogic and applying mathematics to analyse the properties of the model. We therefore regard all our existing knowledge of mathematics as being applicable to the analysis of the model, and in particular we accept set theory as part of the meta-Ianguage. We are not attempting to construct a foundation on which all mathematics is to be based-rather, any conclusions to be drawn about the foundations of mathematics co me only by analogy with the model, and are to be regarded in much the same way as the conclusions drawn from any scientific theory. (shrink)

In this paper, we study the field of algebraic numbers with a set of elements of small height treated as a predicate. We prove that such structures are not simple and have the independence property. A real algebraic integer is called a Salem number if α and are Galois conjugate and all other Galois conjugates of α lie on the unit circle. It is not known whether 1 is a limit point of Salem numbers. We relate the simplicity of a (...) certain pair with Lehmer's conjecture and obtain a model‐theoretic characterization of Lehmer's conjecture for Salem numbers. (shrink)

Lying is the deliberate act of conveying false information to deceive others using language. Examining the linguistic and content features of this act has been a point of interest for forensic linguists due to its potential significance in examining various legal contexts such as criminal investigations, accusations, eyewitness testimonies, allegations, and police interrogations. This article aimed to examine the content features of lying in Jordanian Colloquial Arabic and their social constraints. The study draws on the Reality Monitoring (RM) approach, a (...) forensic tool that has proven its effectiveness in lying detection. To achieve this aim, a corpus of 288 narratives, 144 lying and 144 truthful, was collected through face-to-face interviews with 48 speakers stratified by gender, age, and educational attainment. The findings of this study did not conform to the RM criteria. The study revealed that truthful accounts contain more cognitive details while lying accounts contain more spatial, temporal, affective, and perceptual details. (shrink)

Highly reputable bodies have said that lying is to be avoided when speaking with people living with dementia, unless it cannot be. And yet, the evidence is that many professionals looking after people who live with dementia have been lying to them. I wish to consider an underlying philosophical justification for the moral position that allows lying under some circumstances whilst still condemning it generally. It can seem difficult to ignore the immorality of lying, but thinkers have developed arguments to (...) get around the absolute prohibition. I shall argue that in concrete circumstances the object and the intended end of an action are not as clearly distinct as has been presumed. Further, looking at how language functions allows us to appeal to speech acts and to see the illocutionary force of a statement as way to broaden its purview. We need not think that the only options are between lying and not lying; there is also the possibility, in exigent circumstances, of ‘conforming to the reality’, which would allow a more nuanced account of moral acts, where the intentional nature of the act is no longer to lie. There are, thus, extreme concrete circumstances where not to speak the truth may be excusable, even if regrettable. (shrink)

This paper investigates the behavior of conformal fluctuations of space-time geometry that are admissible under the quantized version of Einstein's general relativity. The approach to quantum gravity is via path integrals. It is shown that considerable simplification results when only the conformal degrees of freedom are considered under this scheme, so much so that it is possible to write down a formal kernel in the most general case where the space-time contains arbitrary distributions of particles with no other (...) interaction except gravity. The behavior of this kernel near the classical space-time singularity then shows that quantum fluctuations inevitably diverge near the singularity. It is shown further that the root cause of this divergence lies in the fact that the Green's function for the conformally invariant scalar wave equation diverges at the singularity. The limitations on the validity of classical general relativity near the space-time singularity are discussed and it is argued that the notion of singularity itself needs to be radically modified once the quantum effects are taken into account. (shrink)

Although rarely appreciated, the collaboration that brought Felix Klein and Sophus Lie together in 1869 had mainly to do with their common interests in the new field of line geometry. As mathematicians, Klein and Lie identified with the latest currents in geometry. Not long before, Klein’s mentor Julius Plücker launched the study of first- and second-degree line complexes, which provided much inspiration for Klein and Lie, though both were busy exploring a broad range of problems and theories. Klein used invariant (...) theory and other algebraic methods to study the properties of line complexes, whereas Lie set his eyes on those aspects related to analysis and differential equations. Much later, historians and mathematicians came to treat the collaboration between Klein and Lie as a famous early chapter in the history of transformation groups, a development often identified with Klein’s “Erlangen Program” from 1872. The present detailed account of their joint work and mutual interests provides a very different picture of their early research, which had relatively little to do with group theory. This essay shows how the geometrical interests of Klein and Lie reflected contemporary trends by focusing on the central importance of quartic surfaces in line geometry. (shrink)

As part of his program to unify linear algebra and geometry using the language of Clifford algebra, David Hestenes has constructed a (well-known) isomorphism between the conformal group and the orthogonal group of a space two dimensions higher, thus obtaining homogeneous coordinates for conformal geometry.(1) In this paper we show that this construction is the Clifford algebra analogue of a hyperbolic model of Euclidean geometry that has actually been known since Bolyai, Lobachevsky, and Gauss, and (...) we explore its wider invariant theoretic implications. In particular, we show that the Euclidean distance function has a very simple representation in this model, as demonstrated by J. J. Seidel.(18). (shrink)

In this paper, constructions of free ordered algebras on one generator are given that correspond to some one-variable fragments of affine propositional classical logic and their extensions with n-contraction (n ≥ 2). Moreover, embeddings of the already known infinite free structures into the algebras introduced below are furnished with; thus, solving along the respective cardinality problems.

Bundles of C*-algebras can be used to represent limits of physical theories whose algebraic structure depends on the value of a parameter. The primary example is the ℏ→0 limit of the C*-algebras of physical quantities in quantum theories, represented in the framework of strict deformation quantization. In this paper, we understand such limiting procedures in terms of the extension of a bundle of C*-algebras to some limiting value of a parameter. We prove existence and uniqueness results for such extensions. Moreover, (...) we show that such extensions are functorial for the C*-product, dynamical automorphisms, and the Lie bracket (in the ℏ→0 case) on the fiber C*-algebras. (shrink)

Just as Kaplansky [4] has introduced the notion of an AW*-module as a generalization of a complex Hilbert space, we introduce the notion of an AL*-algebra, which is a generalization of that of an L*-algebra invented by Schue [9, 10]. By using Boolean valued methods developed by Ozawa [6–8], Takeuti [11–13] and others, we establish its basic properties including a fundamental structure theorem. This paper should be regarded as a continuation or our previous paper [5], the familiarity with (...) which is presupposed. MSC: 03C90, 03E40, 17B65, 46L10. (shrink)

I give a new argument for the moral difference between lying and misleading. First, following David Lewis, I hold that conventions of truthfulness and trust fix the meanings of our language. These conventions generate fair play obligations. Thus, to fail to conform to the conventions of truthfulness and trust is unfair. Second, I argue that the liar, but not the misleader, fails to conform to truthfulness. So the liar, but not the misleader, does something unfair. This account entails that bald-faced (...) lies are wrong, that we can lie nonlinguistically, and that linguistic innovation is morally significant. (shrink)

We introduce new first-order languages for the elementary n-dimensional geometry and elementary n-dimensional affine geometry , based on extending equation image and equation image, respectively, with new function symbols. Here, β stands for the betweenness relation and ≡ for the congruence relation. We show that the associated theories admit effective quantifier elimination.

The formulas for the Lie covariant differentiation of spinors are deduced from an algebraic viewpoint. The Lie covariant derivative of the spinor connection is calculated, and is given a geometric meaning. A theorem about the Lie covariant derivative of an operator in spin space that was stated in Part I of this work is discussed.

In the paper we construct a new kind of fractional dynamical model, i.e. the fractional relativistic Yamaleev oscillator model, and explore its dynamical behaviors. We will find that the fractional relativistic Yamaleev oscillator model possesses Lie algebraic structure and satisfies generalized Poisson conservation law. We will also give the Poisson conserved quantities of the model. Further, the relation between conserved quantities and integral invariants of the model is studied and it is proved that, by using the Poisson conserved quantities, we (...) can construct integral invariants of the model. Finally, the stability of the manifold of equilibrium states of the fractional relativistic Yamaleev oscillator model is studied. The paper provides a general method, i.e. fractional generalized Hamiltonian method, for constructing a family of fractional dynamical models of an actual dynamical system. (shrink)

In the original Ulam-Rényi game with _m_ lies/errors, Player I chooses a secret number \({\bar{x}}\) in a finite search space _S_, and Player II must guess \({\bar{x}}\) by adaptively asking Player I a minimum number of binary questions. Up to _m_ answers may be mendacious/erroneous or may be distorted before reaching Player II. In his monograph “Fault-Tolerant Search Algorithms. Reliable Computation with Unreliable Information”, F. Cicalese provides a comprehensive account of many models of the game and their applications in error-correcting (...) coding with noiseless feedback. Since for \(m>0\) lies the game is not called off by contradictory answers, and repeated answers to the same question are more informative than single answers, the underlying logic of the game with _m_ lies is not boolean. Indeed, the logic of Rényi-Ulam games is Łukasiewicz infinite-valued logic Ł \(_\infty \). In this paper we consider Ulam-Rényi games with variable numbers of lies over infinite search spaces. We characterize the MV-algebras and the unital Specker \(\ell \) -groups given by the logic of these games. (shrink)

A. THE TEXT The main importance of these two treatises lies in the insight they provide into Spinoza's conception of the relation between mathematics and certain disciplines not touched upon elsewhere in his major writings. The mathematics they involve are not the as those of the Ethics however, and the precise connection same between the geometrical order of this work and these excursions into optics and probability is by no means obvious. Add to this difficulty the knotty problems presented by (...) their editorial his tory, dating and scientific background, and it is not perhaps surprising that in spite of the fact that they provide such an excellent illustration of Spinoza's reaction to certain important developments in the history of physics and mathematics, they should not, so far, have attracted much attention. They were first published in 1687 by Levyn van Dyck (d. 1695), official printer to the town council in The Hague. Printing anything by Spinoza was not without its risks, and it is probably significant that during the same year van Dyck should also have published a lengthy and elaborate refutation of Spinozism by 1 the pious and eccentric physician ]. F. Helvetius. Spinoza's name was omitted from the title-page, possibly because the editor or publisher thought that his reputation as an atheist might prejudice the sale of the booklet, and it was not until 1860 that the Amsterdam bookseller Frederik Muller (1817-1881) identified him as its author. (shrink)

In a previous article, the writer explored the geometric foundation of the generally covariant spinor calculus. This geometric reasoning can be extended quite naturally to include the Lie covariant differentiation of spinors. The formulas for the Lie covariant derivatives of spinors, adjoint spinors, and operators in spin space are deduced, and it is observed that the Lie covariant derivative of an operator in spin space must vanish when taken with respect to a Killing vector. The commutator of two Lie covariant (...) derivatives is calculated; it is noted that the result is consistent with the geometric interpretation of the Jacobi identity for vectors. Lie current conservation is seen to spring from the result that the operator of spinor affine covariant differentiation commutes with the operator of spinor Lie covariant differentiation with respect to a Killing vector. It is shown that differentiations of the spinor field defined geometrically are Lorentz-covariant. (shrink)

An important factor in judging whether two retinal images arise from the same object viewed from different positions may be the presence of certain properties or cues that are 'qualitative invariants' with respect to the natural transformations, particularly affine transformations, associated with changes in viewpoint. To test whether observers use certain affine qualitative cues such as concavity, convexity, collinearity, and parallelism of the image elements, a 'same-different' discrimination experiment was carried out with planar patterns that were defined by (...) four points either connected by straight line segments (line patterns) or marked by dots (dot patterns). The first three points of each pattern were generated randomly; the fourth point fell on their diagonal bisector. According to the position of that point, the patterns were concave, triangular (three points being collinear), convex, or parallel sided. In a 'same' trial, an affine transformation was applied to one of two identical patterns; in a 'different' trial, the affine transformation was applied after the point lying on the diagonal bisector was perturbed a short, fixed distance along the bisector, inwards for one pattern and outwards for the other. Observers' ability to discriminate 'same' from 'different' pairs of patterns depended strongly on the position of the fourth, displaced, point: performance varied rapidly when the position of the displaced point was such that the patterns were nearly triangular or nearly parallel sided, consistent with observers using the hypothesised qualitative cues. The experimental data were fitted with a simple probabilistic model of discrimination performance that used a combination of these qualitative cues and a single quantitative cue. (shrink)

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that (...) emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

We look at a hierarchical arrangement of ideals in an MV -algebra. The principal classes of ideals studied are the maximals, the primes, the local and perfect ideals and the semi-locals. Beyond these special classes of ideals are the general ideals. Herein we study some relationships among these classes and, more specifically, the products of ideals of these classes. Among the results obtained are the square of a prime ideal is a local ideal, the finite product of prime ideals (...) is a semilocal ideal. We also show that the set of local ideals shares many of the properties of the class of prime ideals, as the Lying Over Theorem and the Going Up Theorem. (shrink)