Relativistic geometrical action for a quantum particle in the superspace is analyzed from theoretical group point of view. To this end an alternative technique of quantization outlined by the authors in a previous work and that is based in the correct interpretation of the square root Hamiltonian, is used. The obtained spectrum of physical states and the Fock construction consist of Squeezed States which correspond to the representations with the lowest weights $\lambda=\frac{1}{4}$ and $\lambda=\frac{3}{4}$ with four possible (non-trivial) fractional representations (...) for the group decomposition of the spin structure. From the theory of semigroups the analytical representation of the radical operator in the superspace is constructed, the conserved currents are computed and a new relativistic wave equation is proposed and explicitly solved for the time-dependent case. The relation with the Relativistic Schrödinger equation and the Time-dependent Harmonic Oscillator is analyzed and discussed. (shrink)

It is proved that a system of linear equations over an arbitrary field has a solution if every finite subsystem has a solution provided that the set of variables can be well ordered.

The physical meaning of the particularly simple non-degenerate supermetric, introduced in the previous part by the authors, is elucidated and the possible connection with processes of topological origin in high energy physics is analyzed and discussed. New possible mechanism of the localization of the fields in a particular sector of the supermanifold is proposed and the similarity and differences with a 5-dimensional warped model are shown. The relation with gauge theories of supergravity based in the OSP(1/4) group is explicitly given (...) and the possible original action is presented. We also show that in this non-degenerate super-model the physic states, in contrast with the basic states, are observables and can be interpreted as tomographic projections or generalized representations of operators belonging to the metaplectic group Mp(2). The advantage of geometrical formulations based on non-degenerate super-manifolds over degenerate ones is pointed out and the description and the analysis of some interesting aspects of the simplest Riemannian superspaces are presented from the point of view of the possible vacuum solutions. (shrink)

Compact set valued iterations generalize classical point iterations quite naturally by replacing the function f with a tube f in the discrete iterations equation. In Section 3, some bifurcation results about logistic tube iterations are given. In Section 4, an analogous dynamical behaviour for the phase response tube involved in the entrainment of the respiratory rhythm is studied.

We prove that the category of Nachbin’s compact ordered spaces and order-preserving continuous maps between them is dually equivalent to a variety of algebras, with operations of at most countable arity. Furthermore, we observe that the countable bound on the arity is the best possible: the category of compact ordered spaces is not dually equivalent to any variety of finitary algebras. Indeed, the following stronger results hold: the category of compact ordered spaces is not dually equivalent to any finitely accessible (...) category, any first-order definable class of structures, and any class of finitary algebras closed under products and subalgebras. An explicit equational axiomatisation of the dual of the category of compact ordered spaces is obtained; in fact, we provide a finite one, meaning that our description uses only finitely many function symbols and finitely many equational axioms. In preparation for the latter result, we establish a generalisation of a celebrated theorem by Mundici: our result—whose proof is independent of Mundici’s theorem—asserts that the category of unital commutative distributive lattice-ordered monoids is equivalent to the category of what we call MV-monoidal algebras.Abstract taken directly from the thesis.E-mail: [email protected]: https://air.unimi.it/retrieve/handle/2434/812809/1698986/phd_unimi_R11882.pdf. (shrink)

We give a rough statement of the main result. Let D be a compact subset of ℝ3× ℝ. The propagation u of a wave can be noncomputable in any neighborhood of any point of D even though the initial conditions which determine the wave propagation uniquely are computable. A precise statement of the result appears below.

The Dirac operator arises naturally on $\mathbb{S}^1 \times \mathbb{S}^3 $ from the connection on the Lie group U(1)×SU(2) and maps spacetime rays into rays in the Lie algebra. We construct both simple harmonic and pulse solutions to the neutrino equations on $\mathbb{S}^1 \times \mathbb{S}^3 $ , classified by helicity and holonomy, using this map. Helicity is interpreted as the internal part of the Noether charge that arises from translation invariance; it is topologically quantized in integral multiples of a constant g (...) that converts a Lie-algebra phase shift into an action. The fundamental unit of helicity is associated with a full twist in u(1)×su(2) phase per global lightlike cycle. If we pass to the projective space ℝP1xℝP3, we get half-integral helicity. (shrink)

This article has not been written for specialists of exact solutions of Einstein's field equations but for physicists who are interested in nontrivial information on this topic. We recall the history and some basic properties of exact solutions of Einstein's vacuum equations. We show that the field equations for stationary axisymmetric vacuum gravitational fields can be expressed by only one nonlinear differential equation for a complex function. This compact form of the field equations allows the generation of almost all stationary (...) axisymmetric vacuum gravitational fields. We present a new stationary two-body solution of Einstein's equations as an application of this generation technique. This new solution proves the existence of a macroscopic, repulsive spin-spin interaction in general relativity. Some estimates that are related to this new two-body solution are given. (shrink)

We establish several first- or second-order properties of models of first-order theories by considering their elements as atoms of a new universe of set theory and by extending naturally any structure of Boolean model on the atoms to the whole universe. For example, complete f-rings are "boundedly algebraically compact" in the language $(+,-,\cdot,\wedge,\vee,\leq)$ , and the positive cone of a complete l-group with infinity adjoined is algebraically compact in the language (+, ∨, ≤). We also give an example with any (...) first-order language. The proofs can be translated into "naive set theory" in a uniform way. (shrink)

We study invariant measures on compact Hausdorff spaces using finitary integral logic. For each compact Hausdorff space X and any family equation image of its continuous transformations, we find equivalent conditions for the existence of an equation image-invariant measure on X. We give two proofs of the existence of Haar measure on compact groups.

This work presents a thorough derivation of the full-wave transmission-line equations on the basis of Maxwell’s theory. The multiconductor system is assumed to be composed of nonuniform thin wires. It is shown that the mixed potential integral equations are equivalent to generalized telegrapher equations. Novel, exact, and compact expressions for the multiconductor transmission-line parameters are derived, and their connection to radiation effects is shown. Iteration and perturbation procedures are proposed for the solution of the generalized transmission-line equations.

Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies (...) that the variety has equationally definable principal congruences. This result is then used to provide necessary and sufficient conditions for the existence of a model completion for theories of Hamiltonian varieties of pointed residuated lattices, a broad family of varieties that includes lattice-ordered abelian groups and MV-algebras. Notably, if the theory of a Hamiltonian variety of pointed residuated lattices has a model completion, it must have equationally definable principal congruences. In particular, the theories of lattice-ordered abelian groups and MV-algebras do not have a model completion, as first proved by Glass and Pierce, and Lacava, respectively. Finally, it is shown that certain varieties of pointed residuated lattices generated by their linearly ordered members, including lattice-ordered abelian groups and MV-algebras, can be extended with a binary operation to obtain theories that do have a model completion. (shrink)

Concentrating upon applications that are most relevant to modern physics, this valuable book surveys variational principles and examines their relationship to dynamics and quantum theory. Stressing the history and theory of these mathematical concepts rather than the mechanics, the authors provide many insights into the development of quantum mechanics and present much hard-to-find material in a remarkably lucid, compact form. After summarizing the historical background from Pythagoras to Francis Bacon, Professors Yourgrau and Mandelstram cover Fermat's principle of least time, the (...) principle of least action of Maupertuis, development of this principle by Euler and Lagrange, and the equations of Lagrange and Hamilton. Equipped by this thorough preparation to treat variational principles in general, they proceed to derive Hamilton's principle, the Hamilton-Jacobi equation, and Hamilton's canonical equations. An investigation of electrodynamics in Hamiltonian form covers next, followed by a resume of variational principles in classical dynamics. The authors then launch into an analysis of their most significant topics: the relation between variational principles and wave mechanics, and the principles of Feynman and Schwinger in quantum mechanics. Two concluding chapters extend the discussion to hydrodynamics and natural philosophy. Professional physicists, mathematicians, and advanced students with a strong mathematical background will find this stimulating volume invaluable reading. Extremely popular in its hardcover edition, this volume will find even wider appreciation in its first fine inexpensive paperbound edition. (shrink)

The Cartan equations defining simple spinors (renamed “pure” by C. Chevalley) are interpreted as equations of motion in compact momentum spaces, in a constructive approach in which at each step the dimensions of spinor space are doubled while those of momentum space increased by two. The construction is possible only in the frame of the geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and then momentum spaces result compact, isomorphic to invariant-mass-spheres (...) imbedded in each other, since the signatures result steadily Lorentzian; starting from dimension four (Minkowski) up to dimension ten with Clifford algebra ℂℓ(1, 9), where the construction naturally ends. The equations of motion met in the construction are most of those traditionally postulated ad hoc: from Weyl equations for neutrinos (and Maxwell's) to Majorana ones, to those for the electroweak model and for the nucleons interacting with the pseudoscalar pion, up to those for the 3 baryon-lepton families, steadily progressing from the description of lower energy phenomena to that of higher ones. The 3 division algebras: complex numbers, quaternions and octonions appear to be strictly correlated with Clifford algebras and then with this spinor-geometrical approach, from which they appear to gradually emerge in the construction, where they play a basic role for the physical interpretation: at the third step complex numbers generate U(1), possible origin of the electric charge and of the existence of charged—neutral fermion pairs, explaining also easily the opposite charges of proton-electron. Another U(1) appears to generate the strong charge at the fourth step. Quaternions generate the signature of space-time at the first step, the SU(2) internal symmetry of isospin and, in the gauge term, the SU(2) L one, of the electroweak model at the third step; they are also at the origin of 3 families; in number equal to that of quaternion imaginary units. At the fifth and last step octonions generate the SU(3) internal symmetry of flavour, with SU(2) isospin subgroup and, in the gauge term, the one of color, correlated with SU(2) L of the electroweak model. These 3 division algebras seem then to be at the origin of charges, families and of the groups of the Standard model. In this approach there seems to be no need of higher dimensional (>4) space-time, here generated by the four Poincaré translations, and dimensional reduction from ℂℓ(1,9) to ℂℓ(1,3) is equivalent to decoupling of the equations of motion. This spinor-geometrical approach is compatible with that based on strings, since these may be expressed bilinearly (as integrals) in terms of Majorana–Weyl simple or pure spinors which are admitted by ℂℓ(1, 9) = R(32). (shrink)

Suppose that fn is a sequence of nonnegative functions with compact support on a locally compact metric space, that T is a nonnegative linear functional, and that equation imageT fn < T f0. A result of Bishop, foundational to a constructive theory of functional analysis, asserts the existence of a point x such that equation imagefn < f0. This paper extends this result to arbitrary Hausdorff spaces, and gives short proofs using nonstandard analysis. While such arguments used are not themselves (...) constructive, they can illuminate where the difficulty lies in finding the point x. An algorithm for constructing x is then given, with a nonstandard proof that the algorithm converges to a correct value. (shrink)

Abstract. Gettier’s famous examples intended to show that knowledge cannot always be equated with justified true belief. The Gettier problem can also be considered as a problem for topological epistemic logic: If knowledge and justified belief are conceived as topological operators K and B on topological spaces (to be considered as universes of possible worlds), one may ask whether it happens that there is a proposition A such that KA ≠ A & BA or not. If this is the case, (...) the epistemological logic defined by the topological operators K and B may be said to be non-traditional since there is for them a “Gettier proposition” where knowledge does not coincide with justified true belief. As far as we know, for the first time, this topological Gettier problem is discussed in Steinvold’s PhD dissertation (Steinsvold 2012). In Baltag, Bezhanishvili, Özgün, and Smets (2013), Steinvold’s co-derived set account is criticized since it can be easily “Gettierized”. Baltag et alii claim (without proof) that their topological account of Stalnaker’s logic KB of knowledge and belief does not have this flaw. In the following we will give some topological conditions that determine whether a topological KB model does or does not cope with the Gettier challenge. First, it will be shown that every KB model defines a (topologically slightly simplified) model that is rather similar to it but a is a model of traditional JTB (knowledge = justified true belief) logic of knowledge and belief. The existence of such doppelgangers of all KB models may be read as a qualified and partial topological rehabilitation of traditional JTB epistemology. Second, we consider the Gettier problem for a special class of models of Stalnaker’s combined logic of knowledge and belief KB, namely, for T0 Alexandroff spaces. We prove that, that almost all Alexandroff spaces do not face Gettier counterexamples. Third, we prove that the Stone spaces of Boolean algebras of regular closed subsets of Hausdorff (or T2) spaces X do not face Gettier counterexamples if the X are not pseu–docompact, and these Stone spaces do face Gettier counterexamples for compact spaces X. Succinctly formulated, Stalnaker’s KB logic of knowledge and belief can avoid Gettier problems for some models, while it falls back to traditional pre-Gettier JTB epistemology for other models. In contrast, co-derived set semantics is doomed to fail always in the sense that its models always fall prey to Gettier counterexamples. (shrink)

The Stone-Weierstrass Theorem for compact Hausdorff spaces is a basic result of functional analysis with far-reaching consequences. We introduce an equational logic ⊨Δ associated with an infinitary variety Δ and show that the Stone-Weierstrass Theorem is a consequence of the Beth definability property of ⊨Δ, stating that every implicit definition can be made explicit. Further, we define an infinitary propositional logic ⊢Δ by means of a Hilbert-style calculus and prove a strong completeness result whereby the semantic notion of consequence (...) associated with ⊢Δ coincides with ⊨Δ. (shrink)

A new calculus, based upon the multivector derivative, is developed for Lagrangian mechanics and field theory, providing streamlined and rigorous derivations of the Euler-Lagrange equations. A more general form of Noether's theorem is found which is appropriate to both discrete and continuous symmetries. This is used to find the conjugate currents of the Dirac theory, where it improves on techniques previously used for analyses of local observables. General formulas for the canonical stress-energy and angular-momentum tensors are derived, with spinors and (...) vectors treated in a unified way. It is demonstrated that the antisymmetric terms in the stress-energy tensor are crucial to the correct treatment of angular momentum. The multivector derivative is extended to provide a functional calculus for linear functions which is more compact and more powerful than previous formalisms. This is demonstrated in a reformulation of the functional derivative with respect to the metric, which is then used to recover the full canonical stress-energy tensor. Unlike conventional formalisms, which result in a symmetric stress-energy tensor, our reformulation retains the potentially important antisymmetric contribution. (shrink)

Business cycle theory is a broad and disparate field. Different schools of thought offer alternative explanations for cycles, often using different mathematical methods. This book provides academics and graduate students of economics with a compact and accessible exposition of business cycle theory since Keynes. The author places the main theories -- Keynesian economics, monetarism, new classical economics, the real business cycles theory, and new Keynesian economics -- in an historical context by presenting them in the chronological order of their appearance (...) and highlighting their differences and commonalities. He minimizes the necessary mathematical prerequisites by using a unifying mathematical approach: stochastic second-order difference equations, which is explained in detail. Throughout the book, the international dimension of business cycles is acknowledged. The theoretical results obtained are set alongside empirical facts in separate boxes. Each chapter finishes with a set of problems designed to deepen the reader's understanding of the theories presented, and further reading sections which provide access to related material. (shrink)

In this paper, we present a non-geometrodynamic quantum Yang–Mills theory of gravity based on the homogeneous Lorentz group within the general framework of the Poincare gauge theories. The obstacles of this treatment are that first, on the one hand, the gauge group that is available for this purpose is non-compact. On the other hand, Yang–Mills theories with non-compact groups are rarely healthy, and only a few instances exist in the literature. Second, it is not clear how the direct observations of (...) space–time waves can be explained when space–time has no dynamics. We show that the theory is unitary and is renormalizable to the one-loop perturbation. Although in our proposal, gravity is not associated with any elementary particle analogous to the graviton, classical helicity-two space–time waves are explained. Five essential exact solutions to the field equations of our proposal are presented as well. We also discuss a few experimental tests that can falsify the presented Yang–Mills theory. (shrink)

The mathematical and physical aspects of the conformal symmetry of space-time and of physical laws are analyzed. In particular, the group classification of conformally flat space-times, the conformal compactifications of space-time, and the problem of imbedding of the flat space-time in global four-dimensional curved spaces with non-trivial topological and geometrical structure are discussed in detail. The wave equations on the compactified space-times are analyzed also, and the set of their elementary solutions constructed. Finally, the implications of global compactified space-times for (...) cosmology are discussed. It is argued that the recent discovery of periodic structure of matter distribution on large distances strongly suggests that the global cosmological space-time should be close. Next we analyze the inflation scalar field in the inflationary model of universe evolution considered on the spatially compact Robertson-Walker space-time. It is shown that the energy distribution in this model is periodic and the periods and density decrease with increasing distance, in striking agreement with experimental data. Our model of the universe also provides a definite predictions for the energy distribution, polar and azimuthal, considered as a function of angles θ and φ. These predictions should be tested with the new astronomical data. (shrink)

A model for U(1) gauge theories over a compact Lie group is described usingR×S 3 as background space. A comparison with other results is given. Electrodynamics equations are obtained. Finally, some considerations and observations about gravity onR×S 3 space are presented.

When a proposition is cumulatively entailed by a finite setA of premisses, there exists, trivially, a finite subsetB ofA such thatB B entails for all finite subsetsB that are entailed byA. This property is no longer valid whenA is taken to be an arbitrary infinite set, even when the considered inference operation is supposed to be compact. This leads to a refinement of the classical definition of compactness. We call supracompact the inference operations that satisfy the non-finitary analogue of (...) the above property. We show that for any arbitrary cumulative operationC, there exists a supracompact cumulative operationK(C) that is smaller thenC and agrees withC on finite sets. Moreover,K(C) inherits most of the properties thatC may enjoy, like monotonicity, distributivity or disjunctive rationality. The main part of the paper concerns distributive supracompact operations. These operations satisfy a simple functional equation, and there exists a representation theorem that provides a semantic characterization for this family of operations. We examine finally the case of rational operations and show that they can be represented by a specific kind of model particularly easy to handle. (shrink)

Compactified Minkowski spacetime is suggested by conformal covariance of Maxwell equations, while E. Cartan's definition of simple spinors leads to the idea of compactified momentum space. Assuming both diffeomorphic to (S 3 × S 1 )/Z 2 , one may obtain in the conformally flat stereographic projection field theories both infrared and ultraviolet regularized. On the compact manifold themselves instead, Fourier integrals of wave-field oscillations would have to be replaced by Fourier series summed over indices of spherical eigenfunctions: n, l, (...) m, m′. Tentatively identifying those wave structures with spacetime itself (in the frame of Big-Bang) and/or with matter and radiation distribution, some large-scale (hydrogenic) and small-scale (lattice) space structures are conjectured. (shrink)

This paper examines the Stark effect, as a first order perturbation of manifestly covariant hydrogen-like bound states. These bound states are solutions to a relativistic Schrödinger equation with invariant evolution parameter, and represent mass eigenstates whose eigenvalues correspond to the well-known energy spectrum of the nonrelativistic theory. In analogy to the nonrelativistic case, the off-diagonal perturbation leads to a lifting of the degeneracy in the mass spectrum. In the covariant case, not only do the spectral lines split, but they acquire (...) an imaginary part which is linear in the applied electric field, thus revealing induced bound state decay in first order perturbation theory. This imaginary part results from the coupling of the external field to the non-compact boost generator. In order to recover the conventional first order Stark splitting, we must include a scalar potential term. This term may be understood as a fifth gauge potential, which compensates for dependence of gauge transformations on the invariant evolution parameter. (shrink)

The Distributional Compositional Categorical model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that (...) could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, something which the other methods lack, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambekʼs pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambekʼs previous type-logic, the Lambek calculus. The latter and its extensions have been extensively used to formalise and reason about various linguistic phenomena. Hence, the apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings, and also reflects the structure of the parse trees of the Lambek calculus. (shrink)

Coalgebras of polynomial functors constructed from sets of observable elements have been found useful in modelling various kinds of data types and state-transition systems. This paper continues the study of equational logic and model theory for polynomial coalgebras begun in Goldblatt , where it was shown that Boolean combinations of equations between terms of observable type form a natural language of observable formulas for specifying properties of polynomial coalgebras, and for giving a Hennessy–Milner style logical characterisation of observational indistinguishability (...) of states.Here we give a structural characterisation of classes of coalgebras definable by observable formulas. This is an analogue for polynomial coalgebras of Birkhoff's celebrated characterisation of equationally definable classes of abstract algebras as being those closed under homomorphic images, subalgebras, and direct products. The coalgebraic characterisation involves new notions of observational ultraproduct and ultrapower of coalgebras, obtained from the classical construction of ultraproducts by deleting states that assign “nonstandard” values to terms of observable type. A class of polynomial coalgebras is shown to be the class of all models of a set of observable formulas if, and only if, it is closed under images of bisimulation relations, disjoint unions and observational ultrapowers.Observational ultraproducts are also used to discuss compactness—which holds only under limited conditions; to characterise finitely, axiomatisable classes of polynomial coalgebras; and to show that there are axiomatisable classes that are not finitely axiomatisable. (shrink)

Six characteristics of effective representational systems for conceptual learning in complex domains have been identified. Such representations should: (1) integrate levels of abstraction; (2) combine globally homogeneous with locally heterogeneous representation of concepts; (3) integrate alternative perspectives of the domain; (4) support malleable manipulation of expressions; (5) possess compact procedures; and (6) have uniform procedures. The characteristics were discovered by analysing and evaluating a novel diagrammatic representation that has been invented to support students' comprehension of electricity—AVOW diagrams (Amps, Volts, Ohms, (...) Watts). A task analysis is presented that demonstrates that problem solving using a conventional algebraic approach demands more effort than AVOW diagrams. In an experiment comparing two groups of learners using the alternative approaches, the group using AVOW diagrams learned more than the group using equations and were better able to solve complex transfer problems and questions involving multiple constraints. Analysis of verbal protocols and work scratchings showed that the AVOW diagram group, in contrast to the equations group, acquired a coherently organised network of concepts, learnt effective problem solving procedures, and experienced more positive learning events. The six principles of effective representations were proposed on the basis of these findings. AVOW diagrams are Law Encoding Diagrams, a general class of representations that have been shown to support learning in other scientific domains. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:BOOK REVIEWS 533 Le Cristologie contemporanee e le loro posizioni fondamentali al vaglio della dottrina di S. Tommaso. By DANIEL 0Ls, O.P., Studi Tomi· stici, 39. Citta Del Vaticano: Liberia Editrice Vaticana, 1991. Pp. 198 + 13. 25,000 Lire. The author's purpose in this compact but highly informative volume is to confront some of the more fundamental positions of current chris· tology with the christology of Aquinas, with the (...) further intention of deepening our appreciation of St. Thomas's thought (pp. 6 and 193). Ols is aware that, in speaking generically of " contemporary christology," he is entering into discussion with a large number of authors whose viewpoints often differ significantly from one another. He cites writers as diverse as Rahner, Schillebeeckx, Sobrino, and Duquoc, for example. Cognizant of the differences among them, he nonetheless sees points of convergence. There is, he says, a certain " negative unity " among them in that each holds himself at a certain distance from the doctrine of Chalcedon and from the " christology from above " which traditionally has been taken as proceeding from the Chalcedonian defi· nition of faith. (One of the reasons for using St. Thomas as interlocutor with the modern christologies is Aquinas's own development of such a "christology from above.") Along with this negative unity, there is another aspect that links the disparate modern christologies. Ols is aware that, in speaking generically of " contemporary christain anthropological focus which sees as normative " not the revealed message, but the ones to whom the message is directed" (p. 8). In developing his thesis, Ols in fact leaves to one side most of the major modern writers on christology and limits himself to a systematic analysis of the christologies of Rahner (especially as found in Foundations of Christian Faith and in A New Christology, which Rahner wrote with William Thiising) and Schillebeeckx. In five chapters (pp. 15-76) Ols examines the relevant points of the theology of each man and then separately contrasts the results with the theology of Aquinas. The examination of Rahner's thought is done not without a certain sympathy. Ols notes points of convergence between the transcendental Thomism of Rahner and the actual thought of Aquinas. There is a certain a priori methodology in both men, and there are similarities between Rahner's transcendental approach and Thomas's thought on the " natural desire" for the Beatific Vision. The harmonies, however, are often more apparent than real, says Ols; this is especially so in a point central to the Rahnerian anthropology: Rahner's a priori definition of man as the being capable of union with the deity. This capability Rahner equates with the Scholastic notion of potentia obedientalis. In fact, claims Ols, Rahncr is using the terminology in an equivocal sense 534 BOOK REVIEWS and means something quite different from what Aquinas meant by it. For Rahner, the obediential potency "is objectively identical with the essence of man" (p. 30, quoting from Rahner's Foundations); for Aquinas the obediential potency is nature's openness to being used by the Creator as He wills. As a result of the differences, the Incarnation, for Rahner, can be deduced as a possibility from the very nature of man; for Thomas, even the possibility can only be known retrospectively once the Incarnation has happened. The different starting points reflect different epistemologies: the one starts from man's potentialities and moves out toward the world and God; the other works from the world and God to understand man's possibilities. Thus, for the Thomist " a posteriori reflection on the convenience " of the Incarnation there is substituted the Rahnerian " a priori reflection on the deducibility of the Incarnation " (p. 53). Without going deeply into a philosophical critique of Rahner-Cornelio Fabro has already done that-Ols con· tends that this difference is due to the fact that Rahner's theology " is radically dependent on an idealistic thought-pattern which represents a mortal danger for theology" (p. 56). E. Schillebeeckx's thought likewise suffers, says Ols, from the influence of philosophical idealism. What is central to Schillebeeckx's christology is the " concrete experience of the primitive Christian community " (p. 71, quoting Schillebeeckx's Jesus). Consequently, what we find in... (shrink)

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT —a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual (...) DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case. (shrink)

As a sort of intellectual provocation and as a lateral thinking strategy for creativity, this chapter seeks to determine what the study of the dynamics of biodiversity can offer linguists. In recent years, the analogical equation "language = biological species" has become more widespread as a metaphorical source for conceptual renovation, and, at the same time, as a justification for the defense of language diversity. Language diversity would be protected in a way similar to the mobilization that has taken place (...) to protect endangered species. Nevertheless, one must be careful when uncritically transferring conceptualizations and theoretical frameworks from one field to another, since obviously, these two phenomena are quite different in the real world. The dialogue with bioecologists starts by asking about the formation of diversity, i.e., about specialization. Here, one can observe the similarity between the processes of linguistic and genetic fragmentation, in the sense that both phenomena have a (socio)geographical basis for dispersion and consequently, for the loss of their original compact nature and their intercommunication. The self-organizing and creative properties of human beings favor the development of specific varieties for each subset, varieties that continue to evolve constantly, through the unceasing "languaging" of humanity. Regarding the continuity of species or languages, one can also observe the decisive role of intragroup relations. The staying power of linguistic varieties will increase in direct proportion to the intensity of the relationship among the components of the subset. On the other hand, if exotic elements are introduced, especially if these elements are aggressive in nature, the alteration of the ecological niche may turn out to be fatal for the continuity of the previously existing forms. This suggests to us the need to make an in-depth study of what the minimal contextual conditions would be so that a set linguistic group could be assured a sustainable continuity within a framework of linguistic contact. What type of minimum (socio-)ecological niche would a language have to have if we wished to ensure its habitual reproduction? A proposal is made here to explore the ideas of "exclusive functions" and "non-hierarchical functional distribution" for codes in situations where there is high contact and a danger of disuse. As regards change, this phenomenon is seen as an inherent element in the tendency of life to create new developments, which may or not be accompanied by an adaptation to changing environmental conditions. It is pointed out that, similar to what happens in biology, much of linguistic innovation stems from a systemically reorganized mixture of solutions from different codes. An important research question would be, however, to determine why some of these innovations disappear and others survive and extend throughout the community. The big question mark, as Mufwene points out, is, then how to manage to understand how "the evolution of language proceeds by naturally selecting from among the competing alternatives available through the idiolects of individual speakers". Extinction, whether it be of languages or of species, is caused in most cases "by a combination of demographic processes and environmental changes", as Brown points out. Thus, the environment plays a fundamental role in the direction of evolution, since the "the survival of the fittest is, in actuality, the survival of those who fit into the context (Allen and Hoekstra)". This allows us to see the great degree of importance of political-economic contexts in the case of languages. In the same way, migratory movements are also one of the major variables determining the extinction of biodiversity and language diversity. Species and habitat form the basic unit of existence, and this is the major point of departure for understanding the problem of the preservation and recovery of species or languages. Given the increase in the degree of linguistic contact, the continuity of language diversity depends on determining, as exactly as possible, as Prigogine, the physicist, would say, what precise conditions of imbalance may prove to be stable. The great challenge is not so much avoiding contact but managing it. And "restorative ecology" can also be of help to us here. Being able to reach sustainable solutions for language diversity implies a profound knowledge of the dynamics for determining the ways in which language is used in contact situations. The general conclusion is that linguistics is still terminologically and conceptually ill prepared to deal with the dynamic character of human languages. The world and our objects must be conceived of as elements in a state of flux, as changing systems in an unstable equilibrium. With respect to language policy, it would be necessary to make an effort to manage to establish some general principles regarding the linguistic organization of the human species that would make it possible for local linguistic diversity and communication on a planetary scale, which must necessarily take place, to be compatible with each other. To succeed, it will be necessary to continue promoting an autonomous socio-ecological perspective devoted to the comprehension of language phenomena. Such a perspective would be based on a paradigm of complexity, and would, at the same time, place human beings at the center of its theoretical underpinnings. (shrink)

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra equation image such that equation image, equation image for every α < η and equation image. Especially, equation image and equation image can (...) be cardinal sequences of superatomic Boolean algebras. (shrink)

The vector product method developed in previous articles for space rotations and Lorentz transformations is extended to the cases of four-vectors, anti-symmetric tensors, and their transformations in Minkowski space. The electromagnetic fields are expressed in “six-vector” form using the notationH +iE, and this vector form is shown to be relativistically invariant. The wave equations of electromagnetism are derived using these vector products. The following three equations are deduced, which summarize electrodynamics in a compact form: (1) Maxwell's four equations expressed as (...) one, (2) the scalar and vector potential wave equations combined into one relation, and (3) the wave equations for the electric and magnetic fields and the continuity equation combined together. Space inversion, time reversal, and magnetic monopoles are also treated. (shrink)

A connection viewed from the perspective of integration has the Bianchi identities as constraints. It is shown that the removal of these constraints admits a natural solution on manifolds endowed with a metric and teleparallelism. In the process, the equations of structure and the Bianchi identities take standard forms of field equations and conservation laws.The Levi-Civita (part of the) connection ends up as the potential for the gravity sector, where the source is geometric and tensorial and contains an explicit gravitational (...) contribution.Nonlinear field equations for the torsion result. In a “low-energy” approximation (linearity andlow energy-momentumtransfer), the postulate that only charge and velocities contribute to the source transforms these equations into the Maxwell system. Moreover, the affine geodesics become the equations of motion of special relativity with Lorentz force in the same approximation [J. G. Vargas,Found. Phys. 21, 379 (1991)]. The field equations for the torsion must then be viewed as applying to an electromagnetic/strong interaction.A classical unified theory thus arises where the underlying geometry confers their contrasting characters to Maxwell-Lorentz electrodynamics and to an Einstein's-like theory of gravity. The highly compact field equations must, however, be developed in phase-spacetime, since the connection is velocity-dependent, i.e., Finsler-like.Further opportunities for similarities with present-day physics are discussed: (a) teleparallelism allows for the formulation of the torsion sector of the theory as a flat space theory with concomitant point-dependent transformations; (b) spinors should replace Lorentz frames in their role as the subjects to which the connection refers; (c) the Dirac equation consistent with the frame bundle for a velocity-dependent metric with Lorentz signature generates a weak-like interaction in the torsion sector. (shrink)

The use of the axial vector representing a three-dimensional rotation makes the rotation representation much more compact by extending the trigonometric functions to vectorial arguments. Similarly, the pure Lorentz transformations are compactly treated by generalizing a scalar rapidity to a vector quantity in spatial three-dimensional cases and extending hyperbolic functions to vectorial arguments. A calculation of the Wigner rotation simplified by using the extended functions illustrates the fact that the rapidity vector space obeys hyperbolic geometry. New representations bring a Lorentz-invariant (...) fundamental equation of motion corresponding to the Galilei-invariant equation of Newtonian mechanics. (shrink)

In this paper, DSEK model with fractional derivatives of the Atangana-Baleanu Caputo is proposed. This paper gives a brief overview of the ABC fractional derivative and its attributes. Fixed point theory has been used to establish the uniqueness and existence of solutions for the fractional DSEK model. According to this theory, we will define two operators based on Lipschitzian and prove that they are contraction mapping and relatively compact. Ulam-Hyers stability theorem is implemented to prove the fractional DSEK model’s stability (...) in Banach space. Also, fractional Euler’s numerical method is derived for initial value problems with ABC fractional derivative and implemented on fractional DSEK model. The symmetric properties contribute to determining the appropriate method for finding the correct solution to fractional differential equations. The numerical solutions generated using fractional Euler’s method have been plotted for different values of α where α ∈ 0,1 and different step sizes h. Result discussion will be given, describing the changes that occur due to the step size h. (shrink)

In this article, we introduce the notion of weakly measurable cardinal, a new large cardinal concept obtained by weakening the familiar concept of a measurable cardinal. Specifically, a cardinal κ is weakly measurable if for any collection equation image containing at most κ+ many subsets of κ, there exists a nonprincipal κ-complete filter on κ measuring all sets in equation image. Every measurable cardinal is weakly measurable, but a weakly measurable cardinal need not be measurable. Moreover, while the GCH cannot (...) fail first at a measurable cardinal, I will show that it can fail first at a weakly measurable cardinal. More generally, if κ is measurable, then we can make its weak measurability indestructible by the forcing Add for any η while forcing the GCH to hold below κ. Nevertheless, I shall prove that weakly measurable cardinals and measurable cardinals are equiconsistent. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

This note presents some new results from [1] about the Suszko operator and truth‐equational logics, following the works of Czelakowski [11] and Raftery [17]. It is proved that the Suszko operator relative to a truth‐equational logic preserves suprema and commutes with endomorphisms. Together with injectivity, proved by Raftery in [17], the Suszko operator relative to a truth‐equational logic is a structural representation, as defined in [15]. Furthermore, if is a quasivariety, then the Suszko operator relative to a (...) truth‐equational logic is continuous. Finally, it is proved that truth is equationally definable in the class if and only if is a ‐algebraic semantics for and the Suszko operator preserves suprema and commutes with substitutions. (shrink)

We describe which subdirectly irreducible flat algebras arise in the variety generated by an arbitrary class of flat algebras with absorbing bottom element. This is used to give an elementary translation of the universal Horn logic of algebras, and more generally still, partial structures into the equational logic of conventional algebras. A number of examples and corollaries follow. For example, the problem of deciding which finite algebras of some fixed type have a finite basis for their quasi-identities is shown (...) to be equivalent to the finite identity basis problem for the finite members of a finitely based variety with definable principal congruences. (shrink)

The Church-Turing Thesis (CTT) is often paraphrased as ``every computable function is computable by means of a Turing machine.'' The author has constructed a family of equational theories that are not Turing-decidable, that is, given one of the theories, no Turing machine can recognize whether an arbitrary equation is in the theory or not. But the theory is called pseudorecursive because it has the additional property that when attention is limited to equations with a bounded number of variables, one (...) obtains, for each number of variables, a fragment of the theory that is indeed Turing-decidable. In a 1982 conversation, Alfred Tarski announced that he believed the theory to be decidable, despite this contradicting CTT. The article gives the background for this proclamation, considers alternate interpretations, and sets the stage for further research. (shrink)

This paper investigates relative constructions as in The gifted mathematician that you claim to be should be able to solve this equation, in which the head noun is semantically dependent on an intensional operator in the relative clause, even though it is not c-commanded by it. This is the kind of situation that has led, within models of linguistic description that assume a syntactic level of Logical Form, to analyses in which the head noun is interpreted within the CP-internal gap (...) by reconstruction or interpretation of a lower element of a chain. We offer a solution that views surface representation as the input to semantics. The apparent inverted scope effects are traced back to the interpretation of the head nominal gifted mathematician as applying to individual concepts, and of the relative clause that you claim to be as including an equational statement. According to this view, the complex DP in question refers to the individual concept that exists just in the worlds that are compatible with what is generally supposed to be the case, is a gifted mathematician in those worlds, and is identical to you in those worlds. Our solution is related to the non-reconstructionist analysis of binding of pronouns that do not stand in a c-command relationship to their binder, as in The woman that every man hugged was his mother in Jacobson Proceedings of Semantics and Linguistic Theory IV:161-178, 1994) and Sharvit Proceedings of Semantics and Linguistic Theory VI:227-244, 1996), and allows us to capture both similarities with and differences from the latter type of construction. We point out and offer explanations for a number of properties of such relative clauses—in particular their need for an internal intensional operator, their incompatibility with any determiner other than the definite article, and the fact that some of their properties are shared by demonstrably distinct kinds of relative clauses. (shrink)

We give a systematic technical exposition of the foundations of the theory of computably compact metric spaces. We discover several new characterizations of computable compactness and apply these characterizations to prove new results in computable analysis and effective topology. We also apply the technique of computable compactness to give new and less combinatorially involved proofs of known results from the literature. Some of these results do not have computable compactness or compact spaces in their statements, and thus (...) these applications are not necessarily direct or expected. (shrink)