We estimate the costs of climate change to US agriculture, and associated potential benefits of abating greenhouse gas emissions. Five major crops' yield responses to climatic variation are modeled ...empirically, and the results combined with climate projections for a no-policy, high-warming future, as well as moderate and stringent mitigation scenarios. Unabated warming reduces yields of wheat and soybeans by 2050, and cotton by 2100, but moderate warming increases yields of all crops except wheat. Yield changes are monetized using the results of economic simulations within an integrated climate-economy modeling framework. Uncontrolled warming's economic effects on major crops are slightly positive-annual benefits <$4 B. These are amplified by emission reductions, but subject to diminishing returns-by 2100 reaching $17 B under moderate mitigation, but only $7 B with stringent mitigation. Costs and benefits are sensitive to irreducible uncertainty about the fertilization effects of elevated atmospheric carbon dioxide, without which unabated warming incurs net costs of up to $18 B, generating benefits to moderate (stringent) mitigation as large as $26 B ($20 B).
We develop a new, coordinate-free formulation of Hamiltonian mechanics on the dual of a Lie algebroid. Our approach uses a connection, rather than coordinates in a local trivialization, to obtain ...global expressions for the horizontal and vertical dynamics. We show that these dynamics can be obtained in two equivalent ways: (1) using the canonical Lie–Poisson structure, expressed in terms of the connection; or (2) using a novel variational principle that generalizes Hamilton’s phase space principle.
We show that a cooperative game may be decomposed into a sum of component games, one for each player, using the combinatorial Hodge decomposition on a graph. This decomposition is shown to satisfy ...certain efficiency, null-player, symmetry, and linearity properties. Consequently, we obtain a new characterization of the classical Shapley value as the value of the grand coalition in each player's component game. We also relate this decomposition to a least-squares problem involving inessential games (in a similar spirit to previous work on least-squares and minimum-norm solution concepts) and to the graph Laplacian. Finally, we generalize this approach to games with weights and/or constraints on coalition formation.
We sharpen the classic
a priori
error estimate of Babuška for Petrov–Galerkin methods on a Banach space. In particular, we do so by (1) introducing a new constant, called the
Banach–Mazur constant
, ...to describe the geometry of a normed vector space; (2) showing that, for a nontrivial projection
P
, it is possible to use the Banach–Mazur constant to improve upon the naïve estimate
|
|
I
-
P
|
|
≤
1
+
|
|
P
|
|
; and (3) applying that improved estimate to the Petrov–Galerkin projection operator. This generalizes and extends a 2003 result of Xu and Zikatanov for the special case of Hilbert spaces.
...the franchise disclosure document has, in recent times, grown greatly in length and complexity for the benefit of franchisors. ...it is not unheard of for a franchise disclosure document to exceed ...500 pages, or 7.5 pounds, of material.12 See a typical FDD on the next page.
A recent paper of Arnold, Falk, and Winther (Bull. Am. Math. Soc. 47:281–354,
2010
) showed that a large class of mixed finite element methods can be formulated naturally on Hilbert complexes, where ...using a Galerkin-like approach, one solves a variational problem on a finite-dimensional subcomplex. In a seemingly unrelated research direction, Dziuk (Lecture Notes in Math., vol. 1357, pp. 142–155,
1988
) analyzed a class of nodal finite elements for the Laplace–Beltrami equation on smooth 2-surfaces approximated by a piecewise-linear triangulation; Demlow later extended this analysis (SIAM J. Numer. Anal. 47:805–827,
2009
) to 3-surfaces, as well as to higher-order surface approximation. In this article, we bring these lines of research together, first developing a framework for the analysis of variational crimes in abstract Hilbert complexes, and then applying this abstract framework to the setting of finite element exterior calculus on hypersurfaces. Our framework extends the work of Arnold, Falk, and Winther to problems that violate their subcomplex assumption, allowing for the extension of finite element exterior calculus to approximate domains, most notably the Hodge–de Rham complex on approximate manifolds. As an application of the latter, we recover Dziuk’s and Demlow’s a priori estimates for 2- and 3-surfaces, demonstrating that surface finite element methods can be analyzed completely within this abstract framework. Moreover, our results generalize these earlier estimates dramatically, extending them from nodal finite elements for Laplace–Beltrami to mixed finite elements for the Hodge Laplacian, and from 2- and 3-dimensional hypersurfaces to those of arbitrary dimension. By developing this analytical framework using a combination of general tools from differential geometry and functional analysis, we are led to a more geometric analysis of surface finite element methods, whereby the main results become more transparent.
Preservation of linear and quadratic invariants by numerical integrators has been well studied. However, many systems have linear or quadratic observables that are not invariant, but which satisfy ...evolution equations expressing important properties of the system. For example, a time-evolution PDE may have an observable that satisfies a local conservation law, such as the multisymplectic conservation law for Hamiltonian PDEs. We introduce the concept of
functional equivariance
, a natural sense in which a numerical integrator may preserve the dynamics satisfied by certain classes of observables, whether or not they are invariant. After developing the general framework, we use it to obtain results on methods preserving local conservation laws in PDEs. In particular, integrators preserving quadratic invariants also preserve local conservation laws for quadratic observables, and symplectic integrators are multisymplectic.
PDF
