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