819 research outputs found

    Invariance: A Tale of Intellectual Migration

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    The plotline of the standard story told about the development of intellectual history at the end of the 19th/turn of the 20th century follows the move from absolutism to perspectivalism. The narrative takes us, on the one hand, from the scientism of late Enlightenment writers like Voltaire, Mill, D’Alebert, and Comte and the historical determinism of Hegel, all of which were based upon a universal picture of rationality, to, on the other hand, the relativistic physics of Einstein, the perspectival art of Picasso, and the individualism of Nietzsche and Kierkegaard leading to the phenomenology of Husserl and Heidegger to and on through the deconstructivist work of Derrida in which universal proclamations were deemed meaningless. In their place, was relative dependent upon subjective, political, and social factors, influences, and interpretations. Like all sketches, of course, the story is more complicated than that. There is another trend in the intellectual air of the early 20th century that gets left out of this oversimplified picture, one that threads a middle path between absolutism and perspectivalism, a path that considers both frame-dependent or covariant truths and frame-independent or invariant truths and examines the relations between them. Indeed, the notions of covariance and invariance play important roles in the development of the fields of mathematics, physics, philosophy, and psychology in the decades after the turn of the 20th century. The migration of the concepts of invariance and covariance illustrates not only the interconnectedness of the working communities of intellectuals, but also displays ways in which the personal, social, and political overlaps between groups of disciplinary thinkers are essential conduits for the conceptual cross-fertilization that aids in the health of our modern fields of study. [excerpt

    Minimax representation of nonexpansive functions and application to zero-sum recursive games

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    We show that a real-valued function on a topological vector space is positively homogeneous of degree one and nonexpansive with respect to a weak Minkowski norm if and only if it can be written as a minimax of linear forms that are nonexpansive with respect to the same norm. We derive a representation of monotone, additively and positively homogeneous functions on LL^\infty spaces and on Rn\mathbb{R}^n, which extend results of Kolokoltsov, Rubinov, Singer, and others. We apply this representation to nonconvex risk measures and to zero-sum games. We derive in particular results of representation and polyhedral approximation for the class of Shapley operators arising from games without instantaneous payments (Everett's recursive games)

    Las imágenes y la lógica del cono de luz: rastreando el giro postulacional de Robb en la física geométrica

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    Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron research to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics. Las discusiones anteriores de la obra de Robb acerca del espacio y el tiempo han ofrecido un enfoque filosófico de las interpretaciones de la teoría de la relatividad o un enfoque histórico de su empleo de la geometría no-euclidiana, o han ignorado enteramente las discusiones de la relatividad en Cambridge. En este artículo centro mi atención en la forma cómo la obra de Robb tomó contacto con esos mismos desarrollos fundacionales en la matemática y con sus aplicaciones. El contacto con las aplicaciones de la nueva lógica matemática en Göttingen y en Cambridge explica la transición de las investigaciones de Robb sobre los electrones a su tratamiento de la relatividad en 1911 y finalmente a su presentación axiomática de 1914. En el corazón de la óptica física de Robb estaba el modelo del cono de luz. Este modelo pasó de ser un modelo mecánico operante en el sentido cantabrigense maxwelliano de herramienta didáctica y heurística a ser un modelo semántico en el sentido lógico de la teoría de modelos. Robb marcó esta transición de la concepción del siglo XIX a la del siglo XX con el uso más temprano del término “modelo” en el nuevo sentido. Sitúo sus modelos de conos en una genealogía de modelos similares y uso su evolución para seguir la pista de cómo las investigaciones físicas de Robb dependían de su interés en la geometría, la lógica y los fundamentos de las matemáticas.

    Doubly-Special Relativity: Facts, Myths and Some Key Open Issues

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    I report, emphasizing some key open issues and some aspects that are particularly relevant for phenomenology, on the status of the development of "doubly-special" relativistic ("DSR") theories with both an observer-independent high-velocity scale and an observer-independent small-length/large-momentum scale, possibly relevant for the Planck-scale/quantum-gravity realm. I also give a true/false characterization of the structure of these theories. In particular, I discuss a DSR scenario without modification of the energy-momentum dispersion relation and without the κ\kappa-Poincar\'e Hopf algebra, a scenario with deformed Poincar\'e symmetries which is not a DSR scenario, some scenarios with both an invariant length scale and an invariant velocity scale which are not DSR scenarios, and a DSR scenario in which it is easy to verify that some observable relativistic (but non-special-relativistic) features are insensitive to possible nonlinear redefinitions of symmetry generators.Comment: This is the preprint version of a paper prepared for a special issue "Feature Papers: Symmetry Concepts and Applications" of the journal Symmetr
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