2,112 research outputs found
Volcanological constraints of Archaean tectonics
Volcanological and trace element geochemical data can be integrated to place some constraints upon the size, character and evolutionary history of Archean volcanic plumbing, and hence indirectly, Archean tectonics. The earliest volcanism in any greenhouse belt is almost universally tholeitic basalt. Archean mafic magma chambers were usually the site of low pressure fractionation of olivine, plagioclase and later Cpx + or - an oxide phase during evolution of tholeitic liquids. Several models suggest basalt becoming more contaminated by sial with time. Data in the Uchi Subprovince shows early felsic volcanics to have fractionated REE patterns followed by flat REE pattern rhyolites. This is interpreted as initial felsic liquids produced by melting of a garnetiferous mafic source followed by large scale melting of LIL-rich sial. Rare andesites in the Uchi Subprovince are produced by basalt fractionation, direct mantle melts and mixing of basaltic and tonalitic liquids. Composite dikes in the Abitibi Subprovince have a basaltic edge with a chill margin, a rhyolitic interior with no basalt-rhyolite chill margin and partially melted sialic inclusions. Ignimbrites in the Uchi and Abitibi Subprovinces have mafic pumice toward the top. Integration of these data suggest initial mantle-derived basaltic liquids pond in a sialic crust, fractionate and melt sial. The inirial melts low in heavy REE are melts of mafic material, subsequently melting of adjacent sial produces a chamber with a felsic upper part underlain by mafic magma
Area Inequalities for Embedded Disks Spanning Unknotted Curves
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric
inequality that bounds the area of an embedded disk spanning the curve in terms
of two parameters: the length L of the curve and the thickness r (maximal
radius of an embedded tubular neighborhood) of the curve. For fixed length, the
expression giving the upper bound on the area grows exponentially in 1/r^2. In
the direction of lower bounds, we give a sequence of length one curves with r
approaching 0 for which the area of any spanning disk is bounded from below by
a function that grows exponentially with 1/r. In particular, given any constant
A, there is a smooth, unknotted length one curve for which the area of a
smallest embedded spanning disk is greater than A.Comment: 31 pages, 5 figure
Spherical structures on torus knots and links
The present paper considers two infinite families of cone-manifolds endowed
with spherical metric. The singular strata is either the torus knot or the torus link . Domains of existence for a
spherical metric are found in terms of cone angles and volume formul{\ae} are
presented.Comment: 17 pages, 5 figures; typo
Reconstructing the global topology of the universe from the cosmic microwave background
If the universe is multiply-connected and sufficiently small, then the last
scattering surface wraps around the universe and intersects itself. Each circle
of intersection appears as two distinct circles on the microwave sky. The
present article shows how to use the matched circles to explicitly reconstruct
the global topology of space.Comment: 6 pages, 2 figures, IOP format. To be published in the proceedings of
the Cleveland Cosmology and Topology Workshop 17-19 Oct 1997. Submitted to
Class. Quant. Gra
Geophysical characteristics and crustal structure of greenstone terranes: Canadian Shield
Geophysical studies in the Canadian Shield have provided some insights into the tectonic setting of greenstone belts. Greenstone belts are not rooted in deep crustal structures. Geophysical techniques consistently indicate that greenstones are restricted to the uppermost 10 km or so of crust and are underlain by geophysically normal crust. Gravity models suggest that granitic elements are similarly restricted, although magnetic modelling suggests possible downward extension to the intermediate discontinuity around approx. 18 km. Seismic evidence demonstrates that steeply-dipping structure, which can be associated with the belts in the upper crust, is not present in the lower crust. Horizontal intermediate discontinuities mapped under adjacent greenstone and granitic components are not noticeably disrupted in the boundary zone. Geophysical evidence points to the presence of discontinuities between greenhouse-granite and adjacent metasedimentary erranes. Measured stratigraphic thicknesses of greenstone belts are often twice or more the vertical thicknesses determined from gravity modelling. Explantations advanced for the discrepancy include stratigraphy repeated by thrust faulting and/or listric normal faulting, mechanisms which are consistent with certain aspects of conceptual models of greenstone development. Where repetition is not a factor the gravity evidence points to removal of the root zones of greenstone belts. For one region, this has been attributed to magmatic stopping during resurgent caldera activity
Cohomology of groups of diffeomorphims related to the modules of differential operators on a smooth manifold
Let be a manifold and be the cotangent bundle. We introduce a
1-cocycle on the group of diffeomorphisms of with values in the space of
linear differential operators acting on When is the
-dimensional sphere, , we use this 1-cocycle to compute the
first-cohomology group of the group of diffeomorphisms of , with
coefficients in the space of linear differential operators acting on
contravariant tensor fields.Comment: arxiv version is already officia
Cosmology, cohomology, and compactification
Ashtekar and Samuel have shown that Bianchi cosmological models with compact
spatial sections must be of Bianchi class A. Motivated by general results on
the symmetry reduction of variational principles, we show how to extend the
Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as
defined, e.g., by Singer and Thurston. In particular, it is shown that any
m-dimensional homogeneous space G/K admitting a G-invariant volume form will
allow a compact discrete quotient only if the Lie algebra cohomology of G
relative to K is non-vanishing at degree m.Comment: 6 pages, LaTe
Quadri-tilings of the plane
We introduce {\em quadri-tilings} and show that they are in bijection with
dimer models on a {\em family} of graphs arising from rhombus
tilings. Using two height functions, we interpret a sub-family of all
quadri-tilings, called {\em triangular quadri-tilings}, as an interface model
in dimension 2+2. Assigning "critical" weights to edges of , we prove an
explicit expression, only depending on the local geometry of the graph ,
for the minimal free energy per fundamental domain Gibbs measure; this solves a
conjecture of \cite{Kenyon1}. We also show that when edges of are
asymptotically far apart, the probability of their occurrence only depends on
this set of edges. Finally, we give an expression for a Gibbs measure on the
set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs
measures, and conjecture it to be that of minimal free energy per fundamental
domain.Comment: Revised version, minor changes. 30 pages, 13 figure
Twin paradox and space topology
If space is compact, then a traveller twin can leave Earth, travel back home
without changing direction and find her sedentary twin older than herself. We
show that the asymmetry between their spacetime trajectories lies in a
topological invariant of their spatial geodesics, namely the homotopy class.
This illustrates how the spacetime symmetry invariance group, although valid
{\it locally}, is broken down {\it globally} as soon as some points of space
are identified. As a consequence, any non--trivial space topology defines
preferred inertial frames along which the proper time is longer than along any
other one.Comment: 6 pages, latex, 3 figure
Diagnosing students' difficulties in learning mathematics
This study considers the results of a diagnostic test of student difficulty and contrasts the difference in performance between the lower attaining quartile and the higher quartile. It illustrates a difference in qualitative thinking between those who succeed and those who fail in mathematics, illustrating a theory that those who fail are performing a more difficult type of mathematics (coordinating procedures) than those who succeed (manipulating concepts). Students who have to coordinate or reverse processes in time will encounter far greater difficulty than those who can manipulate symbols in a flexible way. The consequences of such a dichotomy and implications for remediation are then considered
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