96 research outputs found
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving maps on a d-dimensional torus. The quantity we use to measure
the irreversibility of the dynamics is the dissipation time. We focus on the
asymptotic behaviour of this time in the limit of small noise. We derive
universal lower and upper bounds for the dissipation time in terms of various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
Fast linear algebra is stable
In an earlier paper, we showed that a large class of fast recursive matrix
multiplication algorithms is stable in a normwise sense, and that in fact if
multiplication of -by- matrices can be done by any algorithm in
operations for any , then it can be done
stably in operations for any . Here we extend
this result to show that essentially all standard linear algebra operations,
including LU decomposition, QR decomposition, linear equation solving, matrix
inversion, solving least squares problems, (generalized) eigenvalue problems
and the singular value decomposition can also be done stably (in a normwise
sense) in operations.Comment: 26 pages; final version; to appear in Numerische Mathemati
A weakly stable algorithm for general Toeplitz systems
We show that a fast algorithm for the QR factorization of a Toeplitz or
Hankel matrix A is weakly stable in the sense that R^T.R is close to A^T.A.
Thus, when the algorithm is used to solve the semi-normal equations R^T.Rx =
A^Tb, we obtain a weakly stable method for the solution of a nonsingular
Toeplitz or Hankel linear system Ax = b. The algorithm also applies to the
solution of the full-rank Toeplitz or Hankel least squares problem.Comment: 17 pages. An old Technical Report with postscript added. For further
details, see http://wwwmaths.anu.edu.au/~brent/pub/pub143.htm
An ecological future for weed science to sustain crop production and the environment. A review
Sustainable strategies for managing weeds are critical to meeting agriculture's potential to feed the world's population while conserving the ecosystems and biodiversity on which we depend. The dominant paradigm of weed management in developed countries is currently founded on the two principal tools of herbicides and tillage to remove weeds. However, evidence of negative environmental impacts from both tools is growing, and herbicide resistance is increasingly prevalent. These challenges emerge from a lack of attention to how weeds interact with and are regulated by the agroecosystem as a whole. Novel technological tools proposed for weed control, such as new herbicides, gene editing, and seed destructors, do not address these systemic challenges and thus are unlikely to provide truly sustainable solutions. Combining multiple tools and techniques in an Integrated Weed Management strategy is a step forward, but many integrated strategies still remain overly reliant on too few tools. In contrast, advances in weed ecology are revealing a wealth of options to manage weedsat the agroecosystem levelthat, rather than aiming to eradicate weeds, act to regulate populations to limit their negative impacts while conserving diversity. Here, we review the current state of knowledge in weed ecology and identify how this can be translated into practical weed management. The major points are the following: (1) the diversity and type of crops, management actions and limiting resources can be manipulated to limit weed competitiveness while promoting weed diversity; (2) in contrast to technological tools, ecological approaches to weed management tend to be synergistic with other agroecosystem functions; and (3) there are many existing practices compatible with this approach that could be integrated into current systems, alongside new options to explore. Overall, this review demonstrates that integrating systems-level ecological thinking into agronomic decision-making offers the best route to achieving sustainable weed management
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