Logarithmic terms in trace expansions of Atiyah-Patodi-Singer problems

For a Dirac-type operator D with a spectral boundary condition, the associated heat operator trace has an expansion in powers and log-powers of t. Some of the log-coefficients vanish in the Atiyah-Patodi-Singer product case. We here investigate the effect of perturbations of D, by use of a pseudodifferential parameter-dependent calculus for boundary problems. It is shown that the first k log-terms are stable under perturbations of D vanishing to order k at the boundary (and the nonlocal power coefficients behind them are only locally perturbed). For perturbations of D from the APS product case by tangential operators commuting with the tangential part A, all the log-coefficients vanish if the dimension is odd.Comment: Published. Abstract added, small typos correcte

Extension theory for elliptic partial differential operators with pseudodifferential methods

This is a short survey on the connection between general extension theories and the study of realizations of elliptic operators A on smooth domains in R^n, n > 1. The theory of pseudodifferential boundary problems has turned out to be very useful here, not only as a formulational framework, but also for the solution of specific questions. We recall some elements of that theory, and show its application in several cases (including recent results), namely to the lower boundedness question, and the question of spectral asymptotics for differences between resolvents.Comment: 26 pages, style changed to LaTeX, new material added at the end, to appear in the Lecture Notes Series of the London Math. Soc. published by Cambridge Univ. Pres

Integration by parts and Pohozaev identities for space-dependent fractional-order operators

Consider a classical elliptic pseudodifferential operator $P$ on ${\Bbb R}^n$ of order $2a$ ($0<a<1)$ with even symbol. For example, $P=A(x,D)^a$ where $A(x,D)$ is a second-order strongly elliptic differential operator; the fractional Laplacian $(-\Delta )^a$ is a particular case. For solutions $u$ of the Dirichlet problem on a bounded smooth subset $\Omega \subset{\Bbb R}^n$, we show an integration-by-parts formula with a boundary integral involving $(d^{-a}u)|_{\partial\Omega }$, where $d(x)=\operatorname{dist}(x,\partial\Omega )$. This extends recent results of Ros-Oton, Serra and Valdinoci, to operators that are $x$-dependent, nonsymmetric, and have lower-order parts. We also generalize their formula of Pohozaev-type, that can be used to prove unique continuation properties, and nonexistence of nontrivial solutions of semilinear problems. An illustration is given with $P=(-\Delta +m^2)^a$. The basic step in our analysis is a factorization of $P$, $P\sim P^-P^+$, where we set up a calculus for the generalized pseudodifferential operators $P^\pm$ that come out of the construction.Comment: Final version to appear in J. Differential Equations, 42 pages. References adde

The sectorial projection defined from logarithms

For a classical elliptic pseudodifferential operator P of order m>0 on a closed manifold X, such that the eigenvalues of the principal symbol p_m(x,\xi) have arguments in \,]\theta,\phi [\, and \,]\phi, \theta +2\pi [\, (\theta <\phi <\theta +2\pi), the sectorial projection \Pi_{\theta, \phi}(P) is defined essentially as the integral of the resolvent along {e^{i\phi}R_+}\cup {e^{i\theta}R_+}. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that \P_{\theta, \phi}(P) is a \psi do of order 0; namely that p_m(x,\xi) cannot in general be modified to allow integration of (p_m(x,\xi)-\lambda)^{-1} along {e^{i\phi}R_+}\cup {e^{i\theta}R_+} simultaneously for all \xi . We show that the structure of \Pi_{\theta, \phi}(P) as a \psi do of order 0 can be deduced from the formula \Pi_{\theta, \phi}(P)= (i/(2\pi))(\log_\theta (P) - \log_\phi (P)) proved in an earlier work (coauthored with Gaarde). In the analysis of \log_\theta (P) one need only modify p_m(x,\xi) in a neighborhood of e^{i\theta}R_+; this is known to be possible from Seeley's 1967 work on complex powers.Comment: Quotations elaborated, 6 pages, to appear in Mathematica Scandinavic

Niches, rather than neutrality, structure a grassland pioneer guild

Pioneer species are fast-growing, short-lived gap exploiters. They are prime candidates for neutral dynamics because they contain ecologically similar species whose low adult density is likely to cause widespread recruitment limitation, which slows competitive dynamics. However, many pioneer guilds appear to be differentiated according to seed size. In this paper, we compare predictions from a neutral model of community structure with three niche-based models in which trade-offs involving seed size form the basis of niche differentiation. We test these predictions using sowing experiments with a guild of seven pioneer species from chalk grassland. We find strong evidence for niche structure based on seed size: specifically large-seeded species produce fewer seeds but have a greater chance of establishing on a per-seed basis. Their advantage in establishment arises because there are more microsites suitable for their germination and early establishment and not directly through competition with other seedlings. In fact, seedling densities of all species were equally suppressed by the addition of competitors' seeds. By the adult stage, despite using very high sowing densities, there were no detectable effects of interspecific competition on any species. The lack of interspecific effects indicates that niche differentiation, rather than neutrality, prevails

Remarks on nonlocal trace expansion coefficients

In a recent work, Paycha and Scott establish formulas for all the Laurent coefficients of Tr(AP^{-s}) at the possible poles. In particular, they show a formula for the zero'th coefficient at s=0, in terms of two functions generalizing, respectively, the Kontsevich-Vishik canonical trace density, and the Wodzicki-Guillemin noncommutative residue density of an associated operator. The purpose of this note is to provide a proof of that formula relying entirely on resolvent techniques (for the sake of possible generalizations to situations where powers are not an easy tool). - We also give some corrections to transition formulas used in our earlier works.Comment: Minor corrections. To appear in a proceedings volume in honor of K. Wojciechowski, "Analysis and Geometry of Boundary Value Problems", World Scientific, 19 page

"State Redemption of the Continental Dollar, 1779-1790"

Remittances of Continental Dollars to the national treasury from each state by year from 1779 through 1789 are used to determine state compliance with congressional resolutions regarding Continental-Dollar redemption. From 1781 through 1789, the states as a whole stayed well ahead of the remittance schedule set by Congress in 1779. Individual state compliance, however, varied considerably. By the time Congress changed redemption requirements with the Funding Act of 4 August 1790, a majority of the net new Continental Dollars ever emitted by Congress had already been redeemed by the states and remitted to the national treasury to be burned.American Revolution; US Constitution; credible commitment; debt retirement; state taxation