When we interpret modal ◊ as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α , where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret (...) ◊ as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α , where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that ${X \in {\bf H}_{1}}$ iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that ${X \in {\bf H}_{n}}$ iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}}$ , and for a limit ordinal α, we have ${{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}}$ . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. (shrink)

Like bundles of homogeneous light or particles that are scattered in physics experiments, says Goris, the homogeneous field of metaphysics was dispersed in the 14th century. He connects that dispersion to the contemporary rejection of metaphysics. His topics are the analysis of concepts, the contested primacy, scattering the homogeneous field of me.

This book deconstructs the whole lineage of political philosophy, showing the ways democracy abuts and regularly undermines the sovereignist tradition across a range of texts from the Iliad to contemporary philosophy. Politics is an object of perennial difficulty for philosophy—as recalcitrant to philosophical mastery as is philosophy’s traditional adversary, poetry. That difficulty makes it an attractive topic for any deconstructive approach to the tradition from which we inherit our language and our concepts. Scatter 2 pursues that deconstruction, often starting (...) with, and sometimes departing from, the work of Jacques Derrida by attending to the concepts of sovereignty on the one hand and democracy on the other. The book begins by following the fate of a line from Homer’s Iliad, where Odysseus asserts that “the rule of many is no good thing, let there be one ruler, one king.” The line, Bennington shows, is quoted, misquoted, and progressively Christianized by Aristotle, Philo Judaeus, Suetonius, the early Church Fathers, Aquinas, Dante, Ockham, Marsilius of Padua, Jean Bodin, Etienne de la Boétie, up to Carl Schmitt and Erik Peterson, and even one of the defendants at the Nuremberg trials, before being discussed by Derrida himself. In the book’s second half, Bennington begins again with Plato and Aristotle and tracks the concept of democracy as it regularly abuts and undermines that sovereignist tradition. In detailed readings of Hobbes and Rousseau, Bennington develops a notion of “proto-democracy” as a possible name for the scatter that underlies and drives the political as such and that will always prevent politics from achieving its aim of bringing itself to an end. (shrink)

Can there be relational universals? If so, how can they be exemplified? A monadic universal is by definition capable of having a scattered spatiotemporal localization of its different exemplifications, but the problem of relational universals is that one single exemplification seems to have to be scattered in the many places where the relata are. The paper argues that it is possible to bite this bullet, and to accept a hitherto un-discussed kind of exemplification relation called ‘scattered exemplification’. It has no (...) immediate symbolic counterpart in any Indo-European natural language or in any so far constructed logical language. In order to remedy this, a notion called ‘many-place copula’ is introduced, too. (shrink)

A class of toposes is introduced and studied, suitable for semantical analysis of an extension of the Heyting predicate calculus admitting Gödel's provability interpretation.

In discussing the cultural history of the 19th century, Walter Benjamin diagnosed the emergence of the modern novel and its form of narration as the sign of a fracturing experience. The split in experience is related to the scattering of a homogeneous idea of space and time, constituted especially during the Enlightenment and in the German historicism. Benjamin's claim reflected the fracturing temporality of modern communities as well as the transformations in the understanding of the meaning of tradition. Here, I (...) begin by discussing Benjamin's conceptions of experience and memory in detail. Secondly, I consider his ideas on history in the framework of challenging the new forms of narration. In the end, I consider the loss of a unified community, especially by indicating ways in which the after-modern community reflects the relationship between aesthetics and politics in Jean-François Lyotard's thought. Key Words: Benjamin • community • experience • history • Lyotard • memory • narrative • remembrance • time. (shrink)

The problem of two relativistically-moving pointlike particles of constant mass is undertaken in an arbitrary Lorentz frame using the classical Lagrangian mechanics of Stückelberg, Horwitz, and Piron. The particles are assumed to interact at events along their world lines at a common “world time,” an invariant dynamical parameter which is not in general synchronous with the particle proper time. The Lorentz-scalar interaction is assumed to be the Coulomb potential (i.e., the inverse square spacetime potential) of the spacetime event separation. The (...) classical orbit equations are found in 1 + 1 spacetime dimensions in the hyperbolic angle coordinates for the reduced problem. The solutions to the reduced motion in these coordinates are the spacetime generalizations of the nonrelativistic Kepler solutions. and they introduce an invariant eccentricity which is a function of other known constants of the motion for the reduced problem. Solutions compatible with physical scattering are obtained by the assumption that the eccentricity is a given function of the ratio of the particle masses. (shrink)

This book offers the first comprehensive introduction to wave scattering in nonstationary materials. G. F. Roach's aim is to provide an accessible, self-contained resource for newcomers to this important field of research that has applications across a broad range of areas, including radar, sonar, diagnostics in engineering and manufacturing, geophysical prospecting, and ultrasonic medicine such as sonograms. New methods in recent years have been developed to assess the structure and properties of materials and surfaces. When light, sound, or some other (...) wave energy is directed at the material in question, "imperfections" in the resulting echo can reveal a tremendous amount of valuable diagnostic information. The mathematics behind such analysis is sophisticated and complex. However, while problems involving stationary materials are quite well understood, there is still much to learn about those in which the material is moving or changes over time. These so-called non-autonomous problems are the subject of this fascinating book. Roach develops practical strategies, techniques, and solutions for mathematicians and applied scientists working in or seeking entry into the field of modern scattering theory and its applications. Wave Scattering by Time-Dependent Perturbations is destined to become a classic in this rapidly evolving area of inquiry. (shrink)

I show that the Eternalist faces a trilemma. Given their theory of time, three claims are each very plausible, yet together form an inconsistent triad. Denying any one of these claims will have significant consequences for how they can conceive of the material realm. I urge that the best strategy is to deny the first claim, and show that this would have a significant consequence: Perdurantism is false.

A version of scattering theory that was developed many years ago to treat nuclear scattering processes, has provided a powerful tool to study universality in scattering processes involving open quantum systems with underlying classically chaotic dynamics. Recently, it has been used to make random matrix theory predictions concerning the statistical properties of scattering resonances in mesoscopic electron waveguides and electromagnetic waveguides. We provide a simple derivation of this scattering theory and we compare its predictions to those obtained from an exactly (...) solvable scattering model; and we use it to study the scattering of a particle wave from a random potential. This method may prove useful in distinguishing the effects of chaos from the effects of disorder in real scattering processes. (shrink)

A nonlinear integrodifferential equation is solved by the methods of S-matrix theory.

Background Recent publications reveal shortcomings in evidence review and summarization methods for patient decision aids. In the large-scale "Share to Care (S2C)" Shared Decision Making (SDM) project at the University Hospital Kiel, Germany, one of 4 SDM interventions was to develop up to 80 decision aids for patients. Best available evidence on the treatments' impact on patient-relevant outcomes was systematically appraised to feed this information into the decision aids. Aims of this paper were to (1) describe how PtDAs are developed (...) and how S2C evidence reviews for each PtDA are conducted, (2) appraise the quality of the best available evidence identified and (3) identify challenges associated with identified evidence. Methods The quality of the identified evidence was assessed based on GRADE quality criteria and categorized into high-, moderate-, low-, very low-quality evidence. Evidence appraisal was conducted across all outcomes assessed in an evidence review and for specific groups of outcomes, namely mortality, morbidity, quality of life, and treatment harms. Challenges in evidence interpretation and summarization resulting from the characteristics of decision aids and the type and quality of evidence are identified and discussed. Conclusions Evidence reviews in this project were carefully conducted and summarized. However, the evidence identified for our decision aids was indeed a "scattered landscape" and often poor quality. Facing a high prevalence of low-quality, non-directly comparative evidence for treatment alternatives doesn't mean it is not necessary to choose an evidence-based approach to inform patients. While there is an urgent need for high quality comparative trials, best available evidence nevertheless has to be appraised and transparently communicated to patients. (shrink)

In the paper Randomizations of Scattered Sentences, Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every \-sentence with few separable randomizations has countably many countable models.

Climate change is a temporally fragmented phenomenon: the causes and effects at work are dispersed over a remarkably long time period. Climate change exceeds human ability to forecast and quantify its effects in time. This creates serious epistemic, moral, and psychological difficulties and poses challenges to generating adequate ethical responses. Augustine’s understanding of time as a measure of imagination emphasizes the way in which human beings actively shape their sense of time. He sees “looking forward” in time as a matter (...) of spiritual vocation that collects the self out of dispersion and connects to a transgenerational collective. A notable example of how this “looking forward” may be practiced is singing the Psalms. The Augustinian “temporal imagination” links the imaginative, affective, moral, and vocational dimensions of measuring time. This offers some preliminary avenues for reimagining a sense of time responsive to climate change’s temporal fragmentation. (shrink)

The notion of a weakly scattered theory T is defined. T need not be scattered. For each a model of T, let sr() be the Scott rank of . Assume sr() ≤ ω\sp A \sb 1 for all a model of T. Let σ\sp T \sb 2 be the least Σ₂ admissible ordinal relative to T. If T admits effective k-splitting as defined in this paper, then θσ\cal Aθ\cal A$ a model of T.

Volume 500, 2009 On the Infrared Problem for the Dressed Non-Relativistic Electron in a Magnetic Field Laurent Amour, ...

The import of Petrarch's description of Laura extends well beyond the confines of his own poetic age; in subsequent times, his portrayal of feminine beauty became authoritative. As a primary canonical text, the Rime sparse consolidated and disseminated a Renaissance mode. Petrarch absorbed a complex network of descriptive strategies and then presented a single, transformed model. In this sense his role in the history of the interpretation and the internalization of woman's "image" by both men and women can scarcely be (...) overemphasized. When late-Renaissance theorists, poets, and painters represented woman's body, Petrarch's verse justified their aesthetic choices. His authority, moreover, extended beyond scholarly consideration to courtly conversation, beyond the treatise on beauty to the after-dinner game in celebration of it. The descriptive codes of others, both ancients and contemporaries, were, of course, not ignored, but the "scattered rhymes" undeniably enjoyed a privileged status: they informed the Renaissance norm of a beautiful woman.1· 1. On this "thoroughly self-conscious fashion," see. Elizabeth Cropper, "On Beautiful Women, Parmigianino, Petrarchismo, and the Vernacular Style," Art Bulletin 58 1976): 374-94.Nancy Vickers is an assistant professor of French and Italian at Dartmouth College. She has published articles on Dante and Petrarch and has recently completed a book, The Anatomy of Beauty: Woman's Body and Renaissance Blazon. (shrink)