Koppelman formulas on Grassmannians

We construct Koppelman formulas on Grassmannians for forms with values in any holomorphic line bundle as well as in the tautological vector bundle and its dual. As a consequence we obtain some vanishing theorems of the Bott-Borel-Weil type. We also relate the projection part of our formulas to the Bergman kernels associated to the line bundles.Comment: sections 5.3 and 6.2 are ne

Weighted integral formulas on manifolds

We present a method of finding weighted Koppelman formulas for $(p,q)$-forms on $n$-dimensional complex manifolds $X$ which admit a vector bundle of rank $n$ over $X \times X$, such that the diagonal of $X \times X$ has a defining section. We apply the method to \Pn and find weighted Koppelman formulas for $(p,q)$-forms with values in a line bundle over \Pn. As an application, we look at the cohomology groups of $(p,q)$-forms over \Pn with values in various line bundles, and find explicit solutions to the \dbar-equation in some of the trivial groups. We also look at cohomology groups of $(0,q)$-forms over \Pn \times \Pm with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.Comment: 25 page

Evaluating partial cutting in broadleaved temperate forest under strong experimental control: Short-term effects on herbaceous plants

Partial harvesting of forest for biofuel and other products may be less harmful to biodiversity than clear-cutting, and may even be beneficial for some species or groups of organisms such as herbs. There are, however, few well-controlled experiments evaluating positive and negative effects, such as species losses directly after harvest. In closed canopy mixed oak forest in Sweden, about 25% of the tree basal area and 50-90% of the understory was removed (mainly spruce, birch, aspen, lime, rowan and hazel). In each of six forests, we studied herbs in an experimental (cutting) plot and a control plot (undisturbed) before, and in the first summer, after the harvest (conducted in winter). Losses of species were similar in experimental and control plots (15-16%). The harvest increased species richness by 4-31% (mean 18%); also species diversity (H) increased. Several ruderals increased in experimental plots, but most changes occurred in grassland and forest species; partial cutting led to complex, partly unpredictable early changes in the herb community. A review of early effects of partial cutting (eight experiments) indicated that it increases herb species richness in stands of broadleaves, but apparently not in conifer stands; there was no evidence that partial cutting increases species losses. Thus, with respect to early changes after harvest, we found no negative effects of partial cutting on herbs. We suggest, however, that some proportion of closed-canopy mixed oak forest should not be harvested, to protect rare, potentially sensitive herbs, and to create stand diversity. (c) 2005 Elsevier B.V. All rights reserved

Koppelman formulas on Grassmannians

We construct Koppelman formulas on Grassmannians for forms with values in any holomorphic line bundle as well as in the tautological vector bundle and its dual. As an application we obtain new explicit proofs of some vanishing theorems of the Bott-Borel-Weil type by solving the corresponding -equation. We also relate the projection part of our formulas to the Bergman kernels associated to the line bundles

Non-divergence form parabolic equations associated with non-commuting vector fields : boundary behavior of nonnegative solutions

In a cylinder Omega(T) = Omega x (0, T) subset of R-+(n+1) we study the boundary behavior of nonnegative solutions of second order parabolic equations of the form H u = Sigma(m)(i,j=1) a(ij)(x, t)XiX (j)u - partial derivative(t)u = 0, (x, t) is an element of R-+(n+1), where X = {X-l, . . . , X-m} is a system of C-infinity vector fields inR(n) satisfying Hormander's rank condition (1.2), and Omega is a non-tangentially accessible domain with respect to the Carnot-Caratheodory distance d induced by X. Concerning the matrix-valued function A = {a(ij)}, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries a(ij) are Holder continuous with respect to the parabolic distance associated with d. Our main results are: I) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Holder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator H (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20, 39]. With one proviso: in those papers the authors assume that the coefficients a(ij) be only bounded and measurable, whereas we assume Holder continuity with respect to the intrinsic parabolic distance