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Content (measure theory)

About: Content (measure theory) is a(n) research topic. Over the lifetime, 3986 publication(s) have been published within this topic receiving 111026 citation(s). The topic is also known as: finitely additive measure.

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Open accessJournal Article
Shaun Cole1, Will J. Percival, John A. Peacock, Peder Norberg2  +27 moreInstitutions (2)
Abstract: We present a power spectrum analysis of the final 2dF Galaxy Redshift Survey, employing a direct Fourier method. The sample used comprises 221,414 galaxies with measured redshifts. We investigate in detail the modelling of the sample selection. A new angular mask is derived, based on revisions to the photometric calibration. The redshift selection function is determined by dividing the survey according to rest-frame colour, and deducing a self-consistent treatment of k-corrections and evolution for each population. The covariance matrix for the power-spectrum estimates is determined using two different approaches to the construction of mock surveys which are used to demonstrate that the input cosmological model can be correctly recovered. We are confident that the 2dFGRS power spectrum can be used to infer the matter content of the universe. On large scales, our estimated power spectrum shows evidence for the `baryon oscillations' that are predicted in CDM models. Fitting to a CDM model, assuming a primordial $n_{s}=1$ spectrum, $h=0.72$ and negligible neutrino mass, the preferred parameters are $\Omega_{M} h = 0.168 \pm 0.016$ and a baryon fraction $\Omega_{b} /\Omega_{M} = 0.185\pm0.046$ (1$\sigma$ errors). The value of $\Omega_{M} h$ is $1\sigma$ lower than the $0.20 \pm 0.03$ in our 2001 analysis of the partially complete 2dFGRS. This shift is largely due to the signal from the newly-sampled regions of space, rather than the refinements in the treatment of observational selection. This analysis therefore implies a density significantly below the standard $\Omega_{M} =0.3$: in combination with CMB data from WMAP, we infer $\Omega_{M} =0.231\pm 0.021$. (Abridged.)

Topics: , , Spectral density (55%) ...read more

1,516 Citations

Journal Article
D. J. Ellis1, David H. Green1Institutions (1)
Abstract: A series of basaltic compositions and compositions within the simple system CaO-MgO-FeO-Al2O3-SiO2 have been crystallized to garnetclinopyroxene bearing mineral assemblages in the range 24–30 kb pressure, 750°–1,300° C temperature. Microprobe analyses of coexisting garnet and clinopyroxene show that K D(Fe2+/MgG+/Fe2+/MgCpx) for the Fe-Mg exchange reaction between coexisting garnet and clinopyroxene is obviously dependent upon the Ca-content and apparently independent of the Mg/(Mg+Fe) content of the clinopyroxene and garnet. The Ca-effect is believed to be due to a combination of non-ideal Ca-Mg substitutions in the garnet and clinopyroxene. Our data and interpretation reconciles previous inconsistencies in the temperature dependence of K D − values determined in experimental studies of simple systems, complex basalt, grospydite and garnet peridotite compositions. Previous differences between the effect of pressure upon K Das predicted from simple system theory (Banno, 1970), and that observed in experiments on multicomponent natural rock compositions (Raheim and Green, 1974a) can now be resolved. We have determined K Das a function of P, T, and X Gt Ca (grossular) and derived the empirical relation $$T\left( {^\circ {\text{K}}} \right) = \frac{{3104X_{{\text{Ca}}}^{{\text{Gt}}} + 3030 + 10.86P\left( {{\text{kb}}} \right)}}{{\ln K_{\text{D}} + 1.9034}}$$ . This empirical relationship has been applied to garnet-clinopyroxene bearing rocks from a wide range of geological environments. The geothermometer yields similar estimates for garnet-clinopyroxene equilibration for neighbouring rocks of different composition and different K Dvalues. In addition, temperature estimates using the above relationship are more consistent with independent temperature estimates based on other geothermometers than previous estimates which did not correct for the Ca-effect. An alternative approach to the above empirical geothermometer was attempted using regular solution models to derive Margules parameters for various solid solutions in garnets and clinopyroxenes. The derived Margules parameters are broadly consistent with those determined from binary solution studies, but caution must be exercised in interpreting them in terms of actual thermodynamic properties of the relevant crystalline solid solutions because of the assumptions which necessarily have to be made in this approach.

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1,449 Citations

Journal Article
E. B. Watson1, David A. Wark1, Jay B. Thomas1Institutions (1)
Abstract: Zircon and rutile are common accessory minerals whose essential structural constituents, Zr, Ti, and Si can replace one another to a limited extent. Here we present the combined results of high pressure–temperature experiments and analyses of natural zircons and rutile crystals that reveal systematic changes with temperature in the uptake of Ti in zircon and Zr in rutile. Detailed calibrations of the temperature dependencies are presented as two geothermometers—Ti content of zircon and Zr content of rutile—that may find wide application in crustal petrology. Synthetic zircons were crystallized in the presence of rutile at 1–2 GPa and 1,025–1,450°C from both silicate melts and hydrothermal solutions, and the resulting crystals were analyzed for Ti by electron microprobe (EMP). To augment and extend the experimental results, zircons hosted by five natural rocks of well-constrained but diverse origin (0.7–3 GPa; 580–1,070°C) were analyzed for Ti, in most cases by ion microprobe (IMP). The combined experimental and natural results define a log-linear dependence of equilibrium Ti content (expressed in ppm by weight) upon reciprocal temperature: $$\log ({\text{Ti}}_{{{\text{zircon}}}}) = (6.01 \pm 0.03) - \frac{{5080 \pm 30}}{{T\;(\hbox{K})}}.$$ In a strategy similar to that used for zircon, rutile crystals were grown in the presence of zircon and quartz (or hydrous silicic melt) at 1–1.4 GPa and 675–1,450°C and analyzed for Zr by EMP. The experimental results were complemented by EMP analyses of rutile grains from six natural rocks of diverse origin spanning 0.35–3 GPa and 470–1,070°C. The concentration of Zr (ppm by weight) in the synthetic and natural rutiles also varies in log-linear fashion with T −1: $$\log ({\text{Zr}}_{{{\text{rutile}}}}) = (7.36 \pm 0.10) - \frac{{4470 \pm 120}}{{T\;(\hbox{K})}}.$$ The zircon and rutile calibrations are consistent with one another across both the synthetic and natural samples, and are relatively insensitive to changes in pressure, particularly in the case of Ti in zircon. Applied to natural zircons and rutiles of unknown provenance and/or growth conditions, the thermometers have the potential to return temperatures with an estimated uncertainty of ±10 ° or better in the case of zircon and ±20° or better in the case of rutile over most of the temperature range of interest (∼400–1,000°C). Estimates of relative temperature or changes in temperature (e.g., from zoning profiles in a single mineral grain) made with these thermometers are subject to analytical uncertainty only, which can be better than ±5° depending on Ti or Zr concentration (i.e., temperature), and also upon the analytical instrument (e.g., IMP or EMP) and operating conditions.

Topics: , Zircon (52%), Rutile (51%)

1,280 Citations

Journal Article
John M. Ferry1, E. B. Watson2Institutions (2)
Abstract: The models recognize that ZrSiO4, ZrTiO4, and TiSiO4, but not ZrO2 or TiO2, are independently variable phase components in zircon. Accordingly, the equilibrium controlling the Zr content of rutile coexisting with zircon is ZrSiO4 = ZrO2 (in rutile) + SiO2. The equilibrium controlling the Ti content of zircon is either ZrSiO4 + TiO2 = ZrTiO4 + SiO2 or TiO2 + SiO2 = TiSiO4, depending whether Ti substitutes for Si or Zr. The Zr content of rutile thus depends on the activity of SiO2 $$(a_{\text{SiO}_{2}})$$ as well as T, and the Ti content of zircon depends on $$a_{\text{SiO}_{2}}$$ and $$a_{\text{TiO}_{2}}$$ as well as T. New and published experimental data confirm the predicted increase in the Zr content of rutile with decreasing $$a_{\text{SiO}_{2}},$$ and unequivocally demonstrate that the Ti content of zircon increases with decreasing $$a_{\text{SiO}_{2}}$$ . The substitution of Ti in zircon therefore is primarily for Si. Assuming a constant effect of P, unit $$a_{\text{ZrSiO}_{4}},$$ and that $$a_{\text{ZrO}_{2}}$$ and $$a_{\text{ZrTiO}_{4}}$$ are proportional to ppm Zr in rutile and ppm Ti in zircon, [log(ppm Zr-in-rutile) + log $$a_{\text{SiO}_{2}}$$ ] = A1 + B1/T(K) and [log(ppm Ti-in-zircon) + log $$a_{\text{SiO}_{2}}$$ − log $$a_{\text{TiO}_{2}}$$ ] = A2 + B2/T, where the A and B are constants. The constants were derived from published and new data from experiments with $$a_{\text{SiO}_{2}}$$ buffered by either quartz or zircon + zirconia, from experiments with $$a_{\text{SiO}_{2}}$$ defined by the Zr content of rutile, and from well-characterized natural samples. Results are A1 = 7.420 ± 0.105; B1 = −4,530 ± 111; A2 = 5.711 ± 0.072; B2 = −4,800 ± 86 with activity referenced to α-quartz and rutile at P and T of interest. The zircon thermometer may now be applied to rocks without quartz and/or rutile, and the rutile thermometer applied to rocks without quartz, provided that $$a_{\text{SiO}_{2}}$$ and $$a_{\text{TiO}_{2}}$$ are estimated. Maximum uncertainties introduced to zircon and rutile thermometry by unconstrained $$a_{\text{SiO}_{2}}$$ and $$a_{\text{TiO}_{2}}$$ can be quantitatively assessed and are ≈60 to 70°C at 750°C. A preliminary assessment of the dependence of the two thermometers on P predicts that an uncertainty of ±1 GPa introduces an additional uncertainty at 750°C of ≈50°C for the Ti-in-zircon thermometer and of ≈70 to 80°C for the Zr-in-rutile thermometer.

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1,258 Citations

Journal Article
Max W. Schmidt1Institutions (1)
Abstract: The Al-in-hornblende barometer, which correlates Altot content of magmatic hornblende linearly with crystallization pressure of intrusion (Hammarstrom and Zen 1986), has been calibrated experimentally under water-saturated conditions at pressures of 2.5–13 kbar and temperatures of 700–655°C. Equilibration of the assemblage hornlende-biotite-plagioclase-orthoclasequartz-sphene-Fe-Ti-oxide-melt-vapor from a natural tonalite 15–20° above its wet solidus results in hornblende compositions which can be fit by the equation: P(±0.6 kbar) = −3.01 + 4.76 Al hbl tot r 2=0.99, where Altot is the total Al content of hornblende in atoms per formula unit (apfu). Altot increase with pressure can be ascribed mainly to a tschermak-exchange ( $$t\vec k,{\text{ Mg}}_{{\text{ - 1 }}} {\text{Al}}^{{\text{VI}}} {\text{Si}}_{{\text{ - 1}}} {\text{ Al}}^{{\text{IV}}}$$ ) accompanied by minor plagioclase-substitution ( $$\vec pl,{\text{ Ca}}_{{\text{ - 1 }}} {\text{Na}}^{{\text{M(4)}}} {\text{ Al}}_{{\text{ - 1}}}^{{\text{IV}}} {\text{ Si}}$$ ). This experimental calibration agrees well with empirical field calibrations, wherein pressures are estimated by contact-aureole barometry, confirming that contact-aureole pressures and pressures calculated by the Al-in-hornblende barometer are essentially identical. This calibration is also consistent with the previous experimental calibration by Johnson and Rutherford (1989b) which was accomplished at higher temperatures, stabilizing the required buffer assemblage by use of mixed H2O-CO2 fluids. The latter calibration yields higher Altot content in hornblendes at corresponding pressures, this can be ascribed to increased edenite-exchange ( $$\vec ed,{\text{ }}\square _{{\text{ }} - {\text{ }}1}^{ A} {\text{ Na}}^{\text{A}} {\text{Si }}_{ - {\text{ }}1} {\text{Al}}^{{\text{IV}}}$$ ) at elevated temperatures. The comparison of both experimental calibrations shows the important influence of the fluid composition, which affects the solidus temperature, on equilibration of hornblende in the buffering phase assemblage.

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1,118 Citations

Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
2021398
2020290
2019261
2018207
2017176

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Topic's top 5 most impactful authors

Desika Narayanan

6 papers, 57 citations

S. Blusk

4 papers, 108 citations

Peyman Nasehpour

4 papers, 16 citations

Brian C. Sales

4 papers, 167 citations

Kamal A. Aly

4 papers, 4 citations