Generators of simple Lie algebras in arbitrary characteristics

In this paper we study the minimal number of generators for simple Lie algebras in characteristic 0 or p > 3. We show that any such algebra can be generated by 2 elements. We also examine the 'one and a half generation' property, i.e. when every non-zero element can be completed to a generating pair. We show that classical simple algebras have this property, and that the only simple Cartan type algebras of type W which have this property are the Zassenhaus algebras.Comment: 26 pages, final version, to appear in Math. Z. Main improvements and corrections in Section 4.

Grasshopper Community Response to Climatic Change: Variation Along an Elevational Gradient

The impacts of climate change on phenological responses of species and communities are well-documented; however, many such studies are correlational and so less effective at assessing the causal links between changes in climate and changes in phenology. Using grasshopper communities found along an elevational gradient, we present an ideal system along the Front Range of Colorado USA that provides a mechanistic link between climate and phenology.This study utilizes past (1959-1960) and present (2006-2008) surveys of grasshopper communities and daily temperature records to quantify the relationship between amount and timing of warming across years and elevations, and grasshopper timing to adulthood. Grasshopper communities were surveyed at four sites, Chautauqua Mesa (1752 m), A1 (2195 m), B1 (2591 m), and C1 (3048 m), located in prairie, lower montane, upper montane, and subalpine life zones, respectively. Changes to earlier first appearance of adults depended on the degree to which a site warmed. The lowest site showed little warming and little phenological advancement. The next highest site (A1) warmed a small, but significant, amount and grasshopper species there showed inconsistent phenological advancements. The two highest sites warmed the most, and at these sites grasshoppers showed significant phenological advancements. At these sites, late-developing species showed the greatest advancements, a pattern that correlated with an increase in rate of late-season warming. The number of growing degree days (GDDs) associated with the time to adulthood for a species was unchanged across the past and present surveys, suggesting that phenological advancement depended on when a set number of GDDs is reached during a season.Our analyses provide clear evidence that variation in amount and timing of warming over the growing season explains the vast majority of phenological variation in this system. Our results move past simple correlation and provide a stronger process-oriented and predictive framework for understanding community level phenological responses to climate change

Alternating groups and moduli space lifting Invariants

Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r\ge n). Improves Fried-Serre on deciding when sphere covers with odd-order branching lift to unramified Spin covers. We produce Hurwitz-Torelli automorphic functions on Hurwitz spaces, and draw Inverse Galois conclusions. Example: Absolute spaces of 3-cycle covers with +1 (resp. -1) lift invariant carry canonical even (resp. odd) theta functions when r is even (resp. odd). For inner spaces the result is independent of r. Another use appears in, http://www.math.uci.edu/~mfried/paplist-mt/twoorbit.html, "Connectedness of families of sphere covers of A_n-Type." This shows the M(odular) T(ower)s for the prime p=2 lying over Hurwitz spaces first studied by, http://www.math.uci.edu/~mfried/othlist-cov/hurwitzLiu-Oss.pdf, Liu and Osserman have 2-cusps. That is sufficient to establish the Main Conjecture: (*) High tower levels are general-type varieties and have no rational points.For infinitely many of those MTs, the tree of cusps contains a subtree -- a spire -- isomorphic to the tree of cusps on a modular curve tower. This makes plausible a version of Serre's O(pen) I(mage) T(heorem) on such MTs. Establishing these modular curve-like properties opens, to MTs, modular curve-like thinking where modular curves have never gone before. A fuller html description of this paper is at http://www.math.uci.edu/~mfried/paplist-cov/hf-can0611591.html .Comment: To appear in the Israel Journal as of 1/5/09; v4 is corrected from proof sheets, but does include some proof simplification in \S

OBIS-USA: A Data-Sharing Legacy of the Census of Marine Life

This is the publisher's version, also available electronically from http://www.tos.org/oceanography.The United States Geological Survey's Biological Informatics Program hosts OBIS-USA, the US node of the Ocean Biogeographic Information System (OBIS). OBIS-USA gathers, coordinates, applies standard formats to, and makes widely available data on biological collections in marine waters of the United States and other areas where US investigators have collected data and, in some instances, specimens. OBIS-USA delivers its data to OBIS international, which then delivers its data to the Global Biodiversity Information Facility (GBIF) and other Web portals for marine biodiversity data. OBIS-USA currently has 145 data sets from 36 participants, representing over 6.5 million occurrence records of over 83,000 taxa from more than 888,000 locations. OBIS-USA, a legacy of the decade-long (2001–2010) international collaborative Census of Marine Life enterprise, continues to add data, including those from ongoing Census projects. Among the many challenges in creating OBIS, including OBIS-USA, were developing a community of trust and shared value among data providers, and demonstrating to providers the value of making their data accessible to others. Challenges also posed by the diversity of data sets relevant to marine biodiversity stored on thousands of computers, in a variety of formats, not all widely accessible, have been met in OBIS-USA by implementing a uniform standard and publishing platform that is easily accessible to a broad range of users

Bounding the dimensions of rational cohomology groups

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple simply-connected algebraic group over $k$ that is defined and split over the prime field $\mathbb{F}_p$. In this paper we investigate situations where the dimension of a rational cohomology group for $G$ can be bounded by a constant times the dimension of the coefficient module. We then demonstrate how our results can be applied to obtain effective bounds on the first cohomology of the symmetric group. We also show how, for finite Chevalley groups, our methods permit significant improvements over previous estimates for the dimensions of second cohomology groups.Comment: 13 page

On intermediate subfactors of Goodman-de la Harpe-Jones subfactors

In this paper we present a conjecture on intermediate subfactors which is a generalization of Wall's conjecture from the theory of finite groups. Motivated by this conjecture, we determine all intermediate subfactors of Goodman-Harpe-Jones subfactors, and as a result we verify that Goodman-Harpe-Jones subfactors verify our conjecture. Our result also gives a negative answer to a question motivated by a conjecture of Aschbacher-Guralnick.Comment: To appear in Comm. Math. Phy

On the trace of the antipode and higher indicators

We introduce two kinds of gauge invariants for any finite-dimensional Hopf algebra H. When H is semisimple over C, these invariants are respectively, the trace of the map induced by the antipode on the endomorphism ring of a self-dual simple module, and the higher Frobenius-Schur indicators of the regular representation. We further study the values of these higher indicators in the context of complex semisimple quasi-Hopf algebras H. We prove that these indicators are non-negative provided the module category over H is modular, and that for a prime p, the p-th indicator is equal to 1 if, and only if, p is a factor of dim H. As an application, we show the existence of a non-trivial self-dual simple H-module with bounded dimension which is determined by the value of the second indicator.Comment: additional references, fixed some typos, minor additions including a questions and some remark

Origins of Mass

Newtonian mechanics posited mass as a primary quality of matter, incapable of further elucidation. We now see Newtonian mass as an emergent property. Most of the mass of standard matter, by far, arises dynamically, from back-reaction of the color gluon fields of quantum chromodynamics (QCD). The equations for massless particles support extra symmetries - specifically scale, chiral, and gauge symmetries. The consistency of the standard model relies on a high degree of underlying gauge and chiral symmetry, so the observed non-zero masses of many elementary particles ($W$ and $Z$ bosons, quarks, and leptons) requires spontaneous symmetry breaking. Superconductivity is a prototype for spontaneous symmetry breaking and for mass-generation, since photons acquire mass inside superconductors. A conceptually similar but more intricate form of all-pervasive (i.e. cosmic) superconductivity, in the context of the electroweak standard model, gives us a successful, economical account of $W$ and $Z$ boson masses. It also allows a phenomenologically successful, though profligate, accommodation of quark and lepton masses. The new cosmic superconductivity, when implemented in a straightforward, minimal way, suggests the existence of a remarkable new particle, the so-called Higgs particle. The mass of the Higgs particle itself is not explained in the theory, but appears as a free parameter. Earlier results suggested, and recent observations at the Large Hadron Collider (LHC) may indicate, the actual existence of the Higgs particle, with mass $m_H \approx 125$ GeV. In addition to consolidating our understanding of the origin of mass, a Higgs particle with $m_H \approx 125$ GeV could provide an important clue to the future, as it is consistent with expectations from supersymmetry.Comment: Invited review for the Central European Journal of Physics. This is the supplement to my 2011 Solvay Conference talk promised there. It is adapted from an invited talk given at the Atlanta APS meeting, April 2012. 33 pages, 6 figures. v2: Added update section bringing in the CERN discovery announcemen

CWRML: representing crop wild relative conservation and use data in XML

Background Crop wild relatives are wild species that are closely related to crops. They are valuable as potential gene donors for crop improvement and may help to ensure food security for the future. However, they are becoming increasingly threatened in the wild and are inadequately conserved, both in situ and ex situ. Information about the conservation status and utilisation potential of crop wild relatives is diverse and dispersed, and no single agreed standard exists for representing such information; yet, this information is vital to ensure these species are effectively conserved and utilised. The European Community-funded project, European Crop Wild Relative Diversity Assessment and Conservation Forum, determined the minimum information requirements for the conservation and utilisation of crop wild relatives and created the Crop Wild Relative Information System, incorporating an eXtensible Markup Language (XML) schema to aid data sharing and exchange. Results Crop Wild Relative Markup Language (CWRML) was developed to represent the data necessary for crop wild relative conservation and ensure that they can be effectively utilised for crop improvement. The schema partitions data into taxon-, site-, and population-specific elements, to allow for integration with other more general conservation biology schemata which may emerge as accepted standards in the future. These elements are composed of sub-elements, which are structured in order to facilitate the use of the schema in a variety of crop wild relative conservation and use contexts. Pre-existing standards for data representation in conservation biology were reviewed and incorporated into the schema as restrictions on element data contents, where appropriate. Conclusion CWRML provides a flexible data communication format for representing in situ and ex situ conservation status of individual taxa as well as their utilisation potential. The development of the schema highlights a number of instances where additional standards-development may be valuable, particularly with regard to the representation of population-specific data and utilisation potential. As crop wild relatives are intrinsically no different to other wild plant species there is potential for the inclusion of CWRML data elements in the emerging standards for representation of biodiversity data