A topos for algebraic quantum theory

The aim of this paper is to relate algebraic quantum mechanics to topos theory, so as to construct new foundations for quantum logic and quantum spaces. Motivated by Bohr's idea that the empirical content of quantum physics is accessible only through classical physics, we show how a C*-algebra of observables A induces a topos T(A) in which the amalgamation of all of its commutative subalgebras comprises a single commutative C*-algebra. According to the constructive Gelfand duality theorem of Banaschewski and Mulvey, the latter has an internal spectrum S(A) in T(A), which in our approach plays the role of a quantum phase space of the system. Thus we associate a locale (which is the topos-theoretical notion of a space and which intrinsically carries the intuitionistic logical structure of a Heyting algebra) to a C*-algebra (which is the noncommutative notion of a space). In this setting, states on A become probability measures (more precisely, valuations) on S(A), and self-adjoint elements of A define continuous functions (more precisely, locale maps) from S(A) to Scott's interval domain. Noting that open subsets of S(A) correspond to propositions about the system, the pairing map that assigns a (generalized) truth value to a state and a proposition assumes an extremely simple categorical form. Formulated in this way, the quantum theory defined by A is essentially turned into a classical theory, internal to the topos T(A).Comment: 52 pages, final version, to appear in Communications in Mathematical Physic

Global ocean wave fields show consistent regional trends between 1980 and 2014 in a multi-product ensemble

Historical trends in the direction and magnitude of ocean surface wave height, period, or direction are debated due to diverse data, time-periods, or methodologies. Using a consistent community-driven ensemble of global wave products, we quantify and establish regions with robust trends in global multivariate wave fields between 1980 and 2014. We find that about 30-40% of the global ocean experienced robust seasonal trends in mean and extreme wave height, period, and direction. Most of the Southern Hemisphere exhibited strong upward-trending wave heights (1-2 cm per year) and periods during winter and summer. Ocean basins with robust positive trends are far larger than those with negative trends. Historical trends calculated over shorter periods generally agree with satellite records but vary from product to product, with some showing a consistently negative bias. Variability in trends across products and time-periods highlights the importance of considering multiple sources when seeking robust change analyses

Global ocean wave fields show consistent regional trends between 1980 and 2014 in a multi-product ensemble

Historical trends in the direction and magnitude of ocean surface wave height, period, or direction are debated due to diverse data, time-periods, or methodologies. Using a consistent community-driven ensemble of global wave products, we quantify and establish regions with robust trends in global multivariate wave fields between 1980 and 2014. We find that about 30–40% of the global ocean experienced robust seasonal trends in mean and extreme wave height, period, and direction. Most of the Southern Hemisphere exhibited strong upward-trending wave heights (1–2 cm per year) and periods during winter and summer. Ocean basins with robust positive trends are far larger than those with negative trends. Historical trends calculated over shorter periods generally agree with satellite records but vary from product to product, with some showing a consistently negative bias. Variability in trends across products and time-periods highlights the importance of considering multiple sources when seeking robust change analyses.publishedVersio

Structures Related to the Emplacement of Shallow-Level Intrusions

A systematic view of the vast nomenclature used to describe the structures of shallow-level intrusions is presented here. Structures are organised in four main groups, according to logical breaks in the timing of magma emplacement, independent of the scales of features: (1) Intrusion-related structures, formed as the magma is making space and then develops into its intrusion shape; (2) Magmatic flow-related structures, developed as magma moves with suspended crystals that are free to rotate; (3) Solid-state, flow-related structures that formed in portions of the intrusions affected by continuing flow of nearby magma, therefore considered to have a syn-magmatic, non-tectonic origin; (4) Thermal and fragmental structures, related to creation of space and impact on host materials. This scheme appears as a rational organisation, helpful in describing and interpreting the large variety of structures observed in shallow-level intrusions