We give a new geometric construction of the big projective module in the principal block of the BGG category O, or rather the corresponding D-module on the flag variety. Namely, given a one-parameter family of nondegenerate additive characters of the unipotent radical of a Borel subgroup which degenerate to the trivial character, there is a corresponding one-parameter family of Whittaker sheave...

Despite much impressive work (e.g. [1], [3], [4], [5], [7]) since the introduction of the Shannon capacity in [8], many natural questions regarding this parameter remain widely open (see [2], [6] for surveys). Let G1 + G2 denote the disjoint union of the graphs G1 and G2. It is easy to see that c(G1 + G2) ≥ c(G1) + c(G2). Shannon [8] conjectured that c(G1 + G2) = c(G1) + c(G2), but this was dis...

11 The Zagier–Broadhurst formula Theorem 41. For any n ≥ 1, ζ ï¿¿ {3, 1} n ï¿¿ = 1 2 2n ζ ï¿¿ {4} n ï¿¿ .

We consider an infinite graph G whose vertex set is the set of natural numbers and adjacency depends solely on the difference between vertices. We study the largest cardinality of a set of permutations of [n] any pair of which differ somewhere in a pair of adjacent vertices of G and determine it completely in an interesting special case. We give estimates for other cases and compare the results...

Prioritizing areas for management of non-native invasive plants is critical, as invasive plants can negatively impact plant community structure. Extensive and multi-jurisdictional inventories are essential to prioritize actions aimed at mitigating the impact of invasions and changes in disturbance regimes. However, previous work devoted little effort to devising sampling methods sufficient to a...

The Sinharaja rainforest in southwestern Sri Lanka is a protected forest largely agriculture-dominated landscape. In keeping with global UNESCO biosphere reserves planning, the surrounded by buffer zone of regenerating and villages small tea plots multi-strata tree gardens (homegardens). Globally, however, conservation planning lacks standards on management. We ask what relationships exist betw...

Let A be the adeles of the function eld Fq (T). If E(g; s) is the Eisenstein series associated with the spherical vector in the principal series representation of the double cover of GL2(A), then the Fourier coeecients (Whittaker functions) of E(g; s) have an Euler product and satisfy a `Riemann Hypothesis' condition. Namely, the zeros occur only on the line <(s) = 0. These Whittaker functions ...

Generalizing work of Gross–Zagier and Schofer on singular moduli, we study the CM values of regularized theta lifts of harmonic Whittaker forms. We compute the archimedian part of the height pairing of arithmetic special divisors and CM cycles on Shimura varieties associated to quadratic spaces over an arbitrary totally real base field. As a special case, we obtain an explicit formula for the n...

Fire is an indissoluble component of ecosystems, however quantifying the effects of fire on vegetation is challenging task as fire lies outside the typical experimental design attributes. A recent simulation study showed that under increased fire regimes positive tree-tree interactions were recorded . Data from experimental burning plots in an African savanna, the Kruger National Park, were col...