Virtual Evidence: A Constructive Semantics for Classical Logics

This article presents a computational semantics for classical logic using constructive type theory. Such semantics seems impossible because classical logic allows the Law of Excluded Middle (LEM), not accepted in constructive logic since it does not have computational meaning. However, the apparently oracular powers expressed in the LEM, that for any proposition P either it or its negation, not P, is true can also be explained in terms of constructive evidence that does not refer to "oracles for truth." Types with virtual evidence and the constructive impossibility of negative evidence provide sufficient semantic grounds for classical truth and have a simple computational meaning. This idea is formalized using refinement types, a concept of constructive type theory used since 1984 and explained here. A new axiom creating virtual evidence fully retains the constructive meaning of the logical operators in classical contexts. Key Words: classical logic, constructive logic, intuitionistic logic, propositions-as-types, constructive type theory, refinement types, double negation translation, computational content, virtual evidenc

Statistics of the geomagnetic secular variation for the past 5Ma

A new statistical model is proposed for the geomagnetic secular variation over the past 5Ma. Unlike previous models, the model makes use of statistical characteristics of the present day geomagnetic field. The spatial power spectrum of the non-dipole field is consistent with a white source near the core-mantle boundary with Gaussian distribution. After a suitable scaling, the spherical harmonic coefficients may be regarded as statistical samples from a single giant Gaussian process; this is the model of the non-dipole field. The model can be combined with an arbitrary statistical description of the dipole and probability density functions and cumulative distribution functions can be computed for declination and inclination that would be observed at any site on Earth's surface. Global paleomagnetic data spanning the past 5Ma are used to constrain the statistics of the dipole part of the field. A simple model is found to be consistent with the available data. An advantage of specifying the model in terms of the spherical harmonic coefficients is that it is a complete statistical description of the geomagnetic field, enabling us to test specific properties for a general description. Both intensity and directional data distributions may be tested to see if they satisfy the expected model distributions

Geomagnetic field models incorporating physical constraints on the secular variation

This proposal has been concerned with methods for constructing geomagnetic field models that incorporate physical constraints on the secular variation. The principle goal that has been accomplished is the development of flexible algorithms designed to test whether the frozen flux approximation is adequate to describe the available geomagnetic data and their secular variation throughout this century. These have been applied to geomagnetic data from both the early and middle part of this century and convincingly demonstrate that there is no need to invoke violations of the frozen flux hypothesis in order to satisfy the available geomagnetic data

Water-rich bending faults at the Middle America Trench

Author Posting. © American Geophysical Union, 2015. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Geochemistry, Geophysics, Geosystems 16 (2015): 2582–2597, doi:10.1002/2015GC005927.The portion of the Central American margin that encompasses Nicaragua is considered to represent an end-member system where multiple lines of evidence point to a substantial flux of subducted fluids. The seafloor spreading fabric of the incoming Cocos plate is oriented parallel to the trench such that flexural bending at the outer rise optimally reactivates a dense network of normal faults that extend several kilometers into the upper mantle. Bending faults are thought to provide fluid pathways that lead to serpentinization of the upper mantle. While geophysical anomalies detected beneath the outer rise have been interpreted as broad crustal and upper mantle hydration, no observational evidence exists to confirm that bending faults behave as fluid pathways. Here we use seafloor electromagnetic data collected across the Middle America Trench (MAT) offshore of Nicaragua to create a comprehensive electrical resistivity image that illuminates the infiltration of seawater along bending faults. We quantify porosity from the resistivity with Archie's law and find that our estimates for the abyssal plain oceanic crust are in good agreement with independent observations. As the Cocos crust traverses the outer rise, the porosity of the dikes and gabbros progressively increase from 2.7% and 0.7% to 4.8% and 1.7%, peaking within 20 km of the trench axis. We conclude that the intrusive crust subducts twice as much pore water as previously thought, significantly raising the flux of fluid to the seismogenic zone and the mantle wedge.This work was supported by National Science Foundation grants OCE-0841114 and OCE-0840894, and the Seafloor Electromagnetic Methods Consortium at Scripps Institution of Oceanography.2016-02-1

Porosity and fluid budget of a water-rich megathrust revealed with electromagnetic data at the Middle America Trench

Author Posting. © American Geophysical Union, 2016. This article is posted here by permission of American Geophysical Union for personal use, not for redistribution. The definitive version was published in Geochemistry, Geophysics, Geosystems 17 (2016): 4495–4516, doi:10.1002/2016GC006556.At convergent margins, the distribution of fluids released from the downgoing slab modulates the state of stress and seismic coupling at the megathrust plate interface. However, existing geophysical data are unable to quantify the porosity along this interface. Here we use controlled-source electromagnetic data collected across the Middle America Trench offshore Nicaragua to image the electrical conductivity structure of the outer fore arc. Our results detect a highly conductive channel, inferred to be the region around the décollement, showing the entire section of water-rich seafloor sediments underthrust with the subducting lithosphere. We use an empirical model of the electrical conductivity of porous media to quantify the channel porosity. Our estimates are consistent with sediment compaction studies, showing a rapid decay of 65%–10% porosity from the trench to 25 km landward. We constrain the channel thickness and use the porosity estimates to determine the water budget, which represents the fraction taken up by fluid. The porosity and water budget estimates show significant lateral variations that we attribute to changes in subducted sediment thickness caused by outer rise bending faults. Between 18 and 23 km from the trench, the conductive channel broadens greatly to 1.5–2 km thick, possibly due to concentrated blind faults or sediment underplating, which suggests a sudden change in hydrogeologic structure at the plate interface. The impact of the anomalous conductor on the seismic coupling and mechanical properties of the megathrust is potentially related to the discrepancy in estimated fault slip between seismic and tsunami source inversions for the 1992 Nicaragua tsunami earthquake.National Science Foundation Grant Numbers: OCE-0841114 , OCE-0840894; Scripps Institution of Oceanography2017-05-1

PP-Wave / CFT_2 Duality

We investigate the pp-wave limit of the AdS_3\times S^3\times K3 compactification of Type IIB string theory from the point of view of the dual Sym_N(K3) CFT. It is proposed that a fundamental string in this pp-wave geometry is dual to the c=6 effective string of the Sym_N(K3) CFT, with the string bits of the latter being composed of twist operators. The massive fundamental string oscillators correspond to certain twisted Virasoro generators in the effective string. It is shown that both the ground states and the genus expansion parameter (at least in the orbifold limit of the CFT) coincide. Surprisingly the latter scales like J^2/N rather than the J^4/N^2 which might have been expected. We demonstrate a leading-order agreement between the pp-wave and CFT particle spectra. For a degenerate special case (one NS 5-brane) an intriguing complete agreement is found.Comment: JHEP3 LaTeX, 20 pages; discussion of WZW levels clarified, reference adde

Open Bar - a Brouwerian Intuitionistic Logic with a Pinch of Excluded Middle

One of the differences between Brouwerian intuitionistic logic and classical logic is their treatment of time. In classical logic truth is atemporal, whereas in intuitionistic logic it is time-relative. Thus, in intuitionistic logic it is possible to acquire new knowledge as time progresses, whereas the classical Law of Excluded Middle (LEM) is essentially flattening the notion of time stating that it is possible to decide whether or not some knowledge will ever be acquired. This paper demonstrates that, nonetheless, the two approaches are not necessarily incompatible by introducing an intuitionistic type theory along with a Beth-like model for it that provide some middle ground. On one hand they incorporate a notion of progressing time and include evolving mathematical entities in the form of choice sequences, and on the other hand they are consistent with a variant of the classical LEM. Accordingly, this new type theory provides the basis for a more classically inclined Brouwerian intuitionistic type theory

Models of Earth's main magnetic field incorporating flux and radial vorticity constraints

We describe a new technique for implementing the constraints on magnetic fields arising from two hypotheses about the fluid core of the Earth, namely the frozen-flux hypothesis and the hypothesis that the core is in magnetostrophic force balance with negligible leakage of current into the mantle. These hypotheses lead to time-independence of the integrated flux through certain ‘null-flux patches' on the core surface, and to time-independence of their radial vorticity. Although the frozen-flux hypothesis has received attention before, constraining the radial vorticity has not previously been attempted. We describe a parametrization and an algorithm for preserving topology of radial magnetic fields at the core surface while allowing morphological changes. The parametrization is a spherical triangle tesselation of the core surface. Topology with respect to a reference model (based on data from the Oersted satellite) is preserved as models at different epochs are perturbed to optimize the fit to the data; the topology preservation is achieved by the imposition of inequality constraints on the model, and the optimization at each iteration is cast as a bounded value least-squares problem. For epochs 2000, 1980, 1945, 1915 and 1882 we are able to produce models of the core field which are consistent with flux and radial vorticity conservation, thus providing no observational evidence for the failure of the underlying assumptions. These models are a step towards the production of models which are optimal for the retrieval of frozen-flux velocity fields at the core surfac