Academic literature on the topic 'Equivalence principles'

A persistent challenge for behaviorally-based accounts of learning has been providing an account of learning that occurs in the absence of systematically programmed contingencies of reinforcement. Symmetry, one type of emergent behavior, has been repeatedly demonstrated with humans, but has been considerably more difficult to demonstrate with non-humans. In this study, pigeons were exposed to a go/no-go procedure in which hue stimuli were presented full screen on a touchscreen monitor. Pigeons learned 12 baseline relations in less than 30 days. Traditional measures used to evaluate symmetry indicated that, during tests, three of the four birds responded more to the reverse of relations that were reinforced in training than to the reverse of relations that were not reinforced in training. However, additional analyses of these data suggests that these differences were driven by one of two trial types and that symmetry was only observed for one of the two predicted relations. These data systematically replicate and extend work by Urcuioli and colleagues and point to areas where further research is needed.

This thesis focusses on three main aspects of the foundations of any theory of gravity where the gravitational field admits a geometric interpretation: (a) the principles of equivalence; (b) their role as selection rules in the landscape of extended theories of gravity; and (c) the possible modifications of the spacetime structure at a "mesoscopic" scale, due to underlying, microscopic-level, quantum-gravitational effects. The first result of the work is the introduction of a formal definition of the Gravitational Weak Equivalence Principle, which expresses the universality of free fall of test objects with non-negligible self-gravity, in a matter-free environment. This principle extends the Galilean universality of free-fall world-lines for test bodies with negligible self-gravity (Weak Equivalence Principle). Second, we use the Gravitational Weak Equivalence Principle to build a sieve for some classes of extended theories of gravity, to rule out all models yielding non-universal free-fall motion for self-gravitating test bodies. When applied to metric theories of gravity in four spacetime dimensions, the method singles out General Relativity (both with and without the cosmological constant term), whereas in higher-dimensional scenarios the whole class of Lanczos--Lovelock gravity theories also passes the test. Finally, we focus on the traditional, manifold-based model of spacetime, and on how it could be modified, at a "mesoscopic" (experimentally attainable) level, by the presence of an underlying, sub-Planckian quantum regime. The possible modifications are examined in terms of their consequences on the hypotheses at the basis of von Ignatowski's derivation of the Lorentz transformations. It results that either such modifications affect sectors already tightly constrained (e.g. violations of the principle of relativity and/or of spatial isotropy), or they demand a radical breakdown of the operative interpretation of the coordinates as readings of clocks and rods.

First, we show that the Rothberger and 2-Rothberger games are equivalent. Then we adjust the former proof and introduce another game, the restricted Menger game, in order to obtain a broader result. This provides an answer in the context of Hausdorff spaces for an open question posed by Aurichi, Bella, and Dias.

Despite the steady growth of electronic commerce (e-commerce), Lesotho and SADC users are uncertain of how to securely sign e-communications practicably. This results in users' lack of confidence in the use of e-commerce. SADC and Lesotho regulatory bodies have developed legal instruments including model laws and bills in an attempt to regulate electronic signatures (e-signatures) in e-commerce to address this problem amongst others. However, it is unclear whether the approach adopted will ensure that the regulatory instruments effectively regulate e-signatures and consequently promote the growth of e-commerce and enhance the socio-economic development of the state. This study examines what the information and communications technology regulatory principles of functional equivalence and technology neutrality entail, their interpretation by the United Nations Commission on International Trade Law (UNCITRAL), and their appropriateness for effective regulation of e-signatures through conceptual analysis. In particular it examines the UNCITRAL Model Laws on e-commerce, UNCITRAL Model on e-signature and the United Nations Convention on the Use of Electronic Communications in International Contracts. The study describes the technical operation of different offline and online signatures in order to appreciate how e-signatures should be regulated. Through textual analysis, it examines whether regulatory instruments of Lesotho and SADC correctly apply the theories in a way that will render use of e-signatures practicable and their regulation effective. It also examines initiatives on regulation of e-signatures in South Africa, the United States of America and the European Union. The study reveals that the purpose of the signature formality is to promote certainty, prevent fraud and provide evidence of a contract despite the form of signature. Although not perfect, functional equivalence and technology neutrality principles render regulation of electronic signatures effective since rules that align with them promote equivalence of legal treatment between offline and online signatures. Consequently, the UNCITRAL's Convention reflects that ordinary e-signatures can meet purposes of the signature formality where appropriate if they observe its functional equivalence criteria. However, the reliability of such electronic signatures is a question of evidence as is the case in offline contracts. Thus, soft laws on electronic evidence must complement the e-signature rules to ensure equivalent legal treatment of signatures. The study reveals that the Lesotho instruments do not fully align with the regulatory principles whereas the SADC instrument closely aligns with them. To different extents, these instruments do not adequately address the problems of users and may inhibit the growth of e-commerce. It further found that the instruments erroneously exclude certain matters such as wills from e-signature application while they correctly exclude others such as negotiable instruments from e-signature application. Lastly it found that the UNCITRAL convention and the US instruments provide better models for effective regulation of e-signatures. By implementing amendments suggested by this study, Lesotho and SADC will address the challenges faced by e-commerce users and make the use of e-signatures feasible for all. Consequently, the instruments will effectively increase the growth of e-commerce and in turn enhance the development of socio economic growth of the SADC region.

The h-principle is a general homotopic way to solve partial differential equations and, more generally, partial differential relations. The theory was started by Y. Eliashberg, M. Gromov and A. V. Philips and it allows one to reduce a differential topological problem to an algebraic topological problem. A way to prove the h-principle is by Convex Integration Theory. Developed originally by Gromov, it is applied to solve relations in jet spaces, including certain classes of undetermined non-linear systems of partial differential equations. The h-principle occurs for instance in immersion problems, isometric immersion problems and other areas. A counter-intuitive result which can be proved by applying the h-principle is the sphere eversion without creasing or tearing. This thesis consists of three parts. In the first chapter we introduce the concept of fiber bundle, which is a space that is locally a product space but globally may have a different topological structure, and the concept of jet bundle, a construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. In the second chapter we develop the Gromov Convex Integration Theory that provides the main general topological method for solving the h-principle for a wide variety of problems in differential geometry and topology, with applications also to PDEs theory and to optimal control theory. Finally, in the last chapter, we examine some relations for which the h-principle holds.

A thorough treatment of the Strong Equivalence Principle is presented, demonstrating its failure in dealing with non-uniform gravitational fields. In particular, a calculation utilising the equivalence principle is shown to produce an incorrect rate of deflection of light. This calculation is used as a tool to investigate the nature of this deflection, and the meaning of the Strong Equivalence Principle. Using a generalised metric for outside a static, spherically symmetric gravitational source, it is shown that the failure of the equivalence principle is geometric and not due to any particular choice of metric. When transformed into a displaced rectangular coordinate system, the generalised metric consists of both diagonal and off-diagonal elements. Only the diagonal elements are equivalent to a flat, uniformly accelerating frame. The off-diagonal elements produce non-zero elements in the Riemann Curvature Tensor and are thus attributed to curvature. Therefore, the Strong Equivalence principle is only valid in the weak field limit, where the components of the Riemann curvature tensor vanish. In this case the metric becomes flat, which is the equivalent of a uniform gravitational field. Using the Schwarzschild metric in displaced rectangular coordinates, the effect of curvature on the rate of deflection of light are determined by tracing the effect of the off-diagonal elements. This calculation shows that only one-third of the deflection rate is due to acceleration in the local inertial frame, with the remaining two-thirds being the result of curvature. Because the rate of deflection is is an infinitesimal quantity defined locally, this shows the effects of curvature are important even for local measurements.