Rootsystems of simple Lie algebras

Rootsystems of nonclassical simple Lie algebras L = [summation operator]a[epsilon]R La such that a([e,f]) [not equal to] 0 for some e[epsilon]L'a, f[epsilon]L'-a for each a[epsilon]R - {0} either contain T2-sections or are irreducible Witt rootsystems. The irreducible Witt rootsystems of prime ranks 1, 2, 3 are W, W2, S2, W3, W[plus sign in circle](W [logical or] W), W[plus sign in circle]S2, S3, S3[plus sign in circle](W [logical or] W), S3(S2). Witt rootsystems having no sections S2, W[plus sign in circle](W [logical or] W) are classified as those rootsystems whose irreducible components are finite vector space subgroups. Since the latter are rootsystems of generalized Albert-Zassenhaus Lie algebras, it follows that the rootsystems of nonclassical simple Lie algebras L = [summation operator]a[epsilon]R La such that ([e,f]) [not equal to] 0 for some e[epsilon]L'a, f[epsilon]L'-a for each a[epsilon]R - {0} which contain no section of type T2, S2, or W[plus sign in circle](W [logical or] W) are classified up to isomorphism by finite vector space subgroups.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25510/1/0000051.pd

rotl: an R package to interact with the Open Tree of Life data

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135519/1/mee312593_am.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/135519/2/mee312593.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/135519/3/mee312593-sup-0001-AppendixS1.pd

Development and testing of a risk indexing framework to determine field-scale critical source areas of faecal bacteria on grassland.

This paper draws on lessons from a UK case study in the management of diffuse microbial pollution from grassland farm systems in the Taw catchment, south west England. We report on the development and preliminary testing of a field-scale faecal indicator organism risk indexing tool (FIORIT). This tool aims to prioritise those fields most vulnerable in terms of their risk of contributing FIOs to water. FIORIT risk indices were related to recorded microbial water quality parameters (faecal coliforms [FC] and intestinal enterococci [IE]) to provide a concurrent on-farm evaluation of the tool. There was a significant upward trend in Log[FC] and Log[IE] values with FIORIT risk score classification (r2 =0.87 and 0.70, respectively and P<0.01 for both FIOs). The FIORIT was then applied to 162 representative grassland fields through different seasons for ten farms in the case study catchment to determine the distribution of on-farm spatial and temporal risk. The high risk fields made up only a small proportion (1%, 2%, 2% and 3% for winter, spring, summer and autumn, respectively) of the total number of fields assessed (and less than 10% of the total area), but the likelihood of the hydrological connection of high FIO source areas to receiving watercourses makes them a priority for mitigation efforts. The FIORIT provides a preliminary and evolving mechanism through which we can combine risk assessment with risk communication to end-users and provides a framework for prioritising future empirical research. Continued testing of FIORIT across different geographical areas under both low and high flow conditions is now needed to initiate its long term development into a robust indexing tool

Generalized classical Albert-Zassenhaus Lie algebras

Two very large classes GCAZ and CAZK of Lie algebras are introduced, which contain all sums of classical, Albert-Zassenhaus, generalized Witt algebras of Kaplansky and associated holomorphs. Their rootsystems R are classified up to isomorphism. The group Aut L of automorphisms of L is shown to contain extensions of the Weyl group of R and the inner automorphism groups of classical Lie algebra complements of the Witt subalgebra of L. The Weyl group extension in Aut L acts transitively by conjugation on the classical complements, under general conditions.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25511/1/0000052.pd

Cartan decompositions and Engel subalgebra triangulability

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/23330/1/0000270.pd

Symmetric Lie algebras

Lie Rootsystems R are introduced, with axioms which reflect properties of the rootset of a Lie algebra L as structured by representations of compatible simple restricted rank 1 subquotients of L. The rank 1 Lie rootsystems and the rank 2 Lie rootsystems defined over p are classified up to isomorphism. Base, closure and core are discussed. The rootsystems of collapse on passage from R to Core R are shown to be of type Sm.Given any Lie rootsystem R, its independent root pairs are shown to fall into eleven classes. Where the eleventh (anomoly) pair T2 never occurs, it is shown that R is contained in R0 + S (not always equal), where R0 is a Witt rootsystem and S is a classical rootsystem. This result is of major importance to two papers (D. J. Winter, Generalized classical-Albert-Zassenhaus Lie algebras, to appear; Rootsystems of simple Lie algebras, to appear), since it implies that the rootsystems of the simple nonclassical Lie algebras considered there are Witt rootsystems.Toral Lie algebras and symmetric Lie algebras are introduced and studied as generalizations of classical-Albert-Zassenhaus Lie algebras. It is shown that their rootsystems are Lie rootsystems. The cores of toral Lie algebras are shown to be classical-Albert-Zassenhaus Lie algebras.These results form the basis for the abovementioned papers on rootsystems of simple Lie algebras and the classification of the rootsystems of two larger classes of Lie algebras, the generalized classical-Albert-Zassenhaus Lie algebras and the classical-Albert-Zassenhaus-Kaplansky Lie algebras.Symmetric Lie algebras are introduced as generalizations of classical-Albert-Zassenhaus Lie algebras. It is shown that their rootsets R are Lie rootsystems. Consequently, symmetric Lie algebras can be studied locally using the classification of rank 2 Lie rootsystems. This is done in detail for toral Lie algebras.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25509/1/0000050.pd

Achieving precise mechanical control in intrinsically noisy systems

How can precise control be realized in intrinsically noisy systems? Here, we develop a general theoretical framework that provides a way of achieving precise control in signal-dependent noisy environments. When the control signal has Poisson or supra-Poisson noise, precise control is not possible. If, however, the control signal has sub-Poisson noise, then precise control is possible. For this case, the precise control solution is not a function, but a rapidly varying random process that must be averaged with respect to a governing probability density functional. Our theoretical approach is applied to the control of straight-trajectory arm movement. Sub-Poisson noise in the control signal is shown to be capable of leading to precise control. Intriguingly, the control signal for this system has a natural counterpart, namely the bursting pulses of neurons-trains of Dirac-delta functions-in biological systems to achieve precise control performance

Symmetrysets

Combinatorial structures R are introduced which, in the presence of structure preserving symmetries at some or all points, determine a system or roots S(R) in the sense of Bourbaki with 0 added.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24208/1/0000467.pd

No evidence that elevated CO\u3csub\u3e2\u3c/sub\u3e gives tropical lianas an advantage over tropical trees

Recent studies indicate that lianas are increasing in size and abundance relative to trees in neotropical forests. As a result, forest dynamics and carbon balance may be altered through liana‐induced suppression of tree growth and increases in tree mortality. Increasing atmospheric CO2 is hypothesized to be responsible for the increase in neotropical lianas, yet no study has directly compared the relative response of tropical lianas and trees to elevated CO2. We explicitly tested whether tropical lianas had a larger response to elevated CO2 than co‐occurring tropical trees and whether seasonal drought alters the response of either growth form. In two experiments conducted in central Panama, one spanning both wet and dry seasons and one restricted to the dry season, we grew liana (n = 12) and tree (n = 10) species in open‐top growth chambers maintained at ambient or twice‐ambient CO2 levels. Seedlings of eight individuals (four lianas, four trees) were grown in the ground in each chamber for at least 3 months during each season. We found that both liana and tree seedlings had a significant and positive response to elevated CO2 (in biomass, leaf area, leaf mass per area, and photosynthesis), but that the relative response to elevated CO2 for all variables was not significantly greater for lianas than trees regardless of the season. The lack of differences in the relative response between growth forms does not support the hypothesis that elevated CO2 is responsible for increasing liana size and abundance across the neotropics

How Politics Become Personal: Sociohistorical Events and their Meanings in People's Lives

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/111993/1/josi12111.pd