Normal Domains Arising from Graph Theory

Determining whether an arbitrary subring R of k[x1±1,...,xn±1] is a normal domain is, in general, a nontrivial problem, even in the special case of a monomial generated domain. First, we determine normality in the case where R is a monomial generated domain where the generators have the form (xixj)±1. Using results for this special case we generalize to the case when R is a monomial generated domain where the generators have the form xi±1xj±1. In both cases, for the ring R, we consider the combinatorial structure that assigns an edge in a mixed directed signed graph to each monomial of the ring. We then use this relationship to provide a combinatorial characterization of the normality of R, and, when R is not normal, we use the combinatorial characterization to compute the normalization of R. Using this construction, we also determine when the ring R satisfies Serre\u27s R1 condition. We also discuss generalizations of this to directed graphs with a homogenizing variable and a special class of hypergraphs

Young children's research: children aged 4-8 years finding solutions at home and at school

Children's research capacities have become increasingly recognised by adults, yet children remain excluded from the academy, with reports of their research participation generally located in adults' agenda. Such practice restricts children's freedom to make choices in matters affecting them, underestimates children’s capabilities and denies children particular rights. The present paper reports on one aspect of a small-scale critical ethnographic study adopting a constructivist grounded approach to conceptualise ways in which children's naturalistic behaviours may be perceived as research. The study builds on multi-disciplinary theoretical perspectives, embracing 'new' sociology, psychology, economics, philosophy and early childhood education and care (ECEC). Research questions include: 'What is the nature of ECEC research?' and 'Do children’s enquiries count as research?' Initially, data were collected from the academy: professional researchers (n=14) confirmed 'finding solutions' as a research behaviour and indicated children aged 4-8 years, their practitioners and primary carers as 'theoretical sampling'. Consequently, multi-modal case studies were constructed with children (n=138) and their practitioners (n=17) in three ‘good’ schools, with selected children and their primary carers also participating at home. This paper reports on data emerging from children aged 4-8 years at school (n=17) and at home (n=5). Outcomes indicate that participating children found diverse solutions to diverse problems, some of which they set themselves. Some solutions engaged children in high order thinking, whilst others did not; selecting resources and trialing activities engaged children in 'finding solutions'. Conversely, when children's time, provocations and activities were directed by adults, the quality of their solutions was limited, they focused on pleasing adults and their motivation to propose solutions decreased. In this study, professional researchers recognised 'finding solutions' as research behaviour and children aged 4-8 years naturalistically presented with capacities for finding solutions; however, the children's encounters with adults affected the solutions they found

The link between volcanism and plutonism in epizonal magma systems; high-precision U–Pb zircon geochronology from the Organ Mountains caldera and batholith, New Mexico

The Organ Mountains caldera and batholith expose the volcanic and epizonal plutonic record of an Eocene caldera complex. The caldera and batholith are well exposed, and extensive previous mapping and geochemical analyses have suggested a clear link between the volcanic and plutonic sections, making this an ideal location to study magmatic processes associated with caldera volcanism. Here we present high-precision thermal ionization mass spectrometry U–Pb zircon dates from throughout the caldera and batholith, and use these dates to test and improve existing petrogenetic models. The new dates indicate that Eocene volcanic and plutonic rocks in the Organ Mountains formed from ~44 to 34 Ma. The three largest caldera-related tuff units yielded weighted mean [superscript 206]Pb/[superscript 238]U dates of 36.441 ± 0.020 Ma (Cueva Tuff), 36.259 ± 0.016 Ma (Achenback Park tuff), and 36.215 ± 0.016 Ma (Squaw Mountain tuff). An alkali feldspar granite, which is chemically similar to the erupted tuffs, yielded a synchronous weighted mean [superscript 206]Pb/[superscript 238]U date of 36.259 ± 0.021 Ma. Weighted mean [superscript 206]Pb/[superscript 238]U dates from the larger volume syenitic phase of the underlying Organ Needle pluton range from 36.130 ± 0.031 to 36.071 ± 0.012 Ma, and the youngest sample is 144 ± 20 to 188 ± 20 ka younger than the Squaw Mountain and Achenback Park tuffs, respectively. Younger plutonism in the batholith continued through at least 34.051 ± 0.029 Ma. We propose that the Achenback Park tuff, Squaw Mountain tuff, alkali feldspar granite and Organ Needle pluton formed from a single, long-lived magma chamber/mush zone. Early silicic magmas generated by partial melting of the lower crust rose to form an epizonal magma chamber. Underplating of the resulting mush zone led to partial melting and generation of a high-silica alkali feldspar granite cap, which erupted to form the tuffs. The deeper parts of the chamber underwent continued recharge and crystallization for 144 ± 20 ka after the final eruption. Calculated magmatic fluxes for the Organ Needle pluton range from 0.0006 to 0.0030 km3/year, in agreement with estimates from other well-studied plutons. The petrogenetic evolution proposed here may be common to many small-volume silicic volcanic systems

Evidence for the generation of juvenile granitic crust during continental extension, Mineral Mountains Batholith, Utah

This is the published version. Copyright 1976 American Geophysical Union. All Rights Reserved.Field, chemical and isotopic data from the Miocene Mineral Mountains batholith in southwest Utah are consistent with the batholith being derived through differentiation of material recently separated from the lithospheric mantle, with little involvement of pre-Oligocene crust. The batholith ranges in composition and texture from diabase and gabbro to high-silica rhyolite and granite and is distinctly calcalkaline in nature. Field evidence for anatexis of intermediate-composition Oligocene crust and magma mixing suggest that fractional melting and mixing were important processes during the evolution of the batholith. Major oxide and rare earth element data for the batholith are consistent with chemical evolution of the magma system being controlled by fractionation of hornblende, plagioclase and sphene (all of which occur in restitic portions of Miocene migmatites exposed in the field area) during partial melting, and mixing between gabbro and granite. Isotopic data indicate a lithospheric mantle source for mafic rocks in the study area and, on the basis of field data and their similarity in isotopic composition, granitic rocks are interpreted to be derived indirectly from the same source during Basin and Range extension. Evolution of the granites is hypothesized to involve a series of partial melting steps, one of which is exposed in the batholith, which refine mantle-derived gabbros into high-silica rocks. Thus the Mineral Mountains batholith represents juvenile granitic material added to the crust during extension. This raises the possibility that extension may be an important granitic crust-forming event. Furthermore, this suggests that pure-shear igneous inflation of the crust by the mantle can be an important mechanism during extensional deformation. Data presented here indicate that fractional melting of young mafic crust may be an important process in the evolution of isotopically homogeneous intrusive suites which span a broad compositional range. Furthermore, the data support the idea that lithospheric mantle in the Great Basin region may be Proterozoic in age

Provenance of the upper Eocene Castle Rock Conglomerate, south Denver Basin, Colorado, U.S.A.

The Castle Rock Conglomerate contains distinctive clasts from the Colorado Front Range, and when combined with detrital zircon ages, the unit can be subdivided into two lithofacies. Precambrian quartzites and stretched-pebble conglomerates from Coal Creek Canyon (to the northwest of the Castle Rock Conglomerate outcrop belt) and detrital zircons from Precambrian and Tertiary igneous rocks identify a northern provenance with detritus derived from tens of kilometers northwest of Denver, Colorado. A second source, composed of mainly granite from the Pikes Peak batholith, lies in the southern Front Range west of the Castle Rock Conglomerate outcrop belt. Both the north and west lithofacies can be mapped in the Castle Rock Conglomerate outcrop belt by using the presence (north) and absence (west) of Coal Creek Canyon quartzite clasts. This distinction is confirmed by detrital zircon ages. The north lithofacies dominates the present-day, northernmost outcrops, but dilution and interbedding with west lithofacies increase as the southeast-flowing basin axial paleodrainage meets piedmont tributaries that carried Pikes Peak batholith detritus from the west and southwest. The basin axial drainage transported coarse conglomerate southward about 120 km during Castle Rock Conglomerate deposition (36.7-34.0 Ma). The Precambrian quartzite exposed in Coal Creek Canyon is interpreted to be an important point source that can be useful in provenance studies of sediments shed from the Colorado Front Range. Additionally, detrital zircons from Laramide-age igneous rocks show potential for improved stratigraphic resolution in Paleogene strata of the Denver Basin