Punctured polygons and polyominoes on the square lattice

We use the finite lattice method to count the number of punctured staircase and self-avoiding polygons with up to three holes on the square lattice. New or radically extended series have been derived for both the perimeter and area generating functions. We show that the critical point is unchanged by a finite number of punctures, and that the critical exponent increases by a fixed amount for each puncture. The increase is 1.5 per puncture when enumerating by perimeter and 1.0 when enumerating by area. A refined estimate of the connective constant for polygons by area is given. A similar set of results is obtained for finitely punctured polyominoes. The exponent increase is proved to be 1.0 per puncture for polyominoes.Comment: 36 pages, 11 figure

Estimation and inversion across the spectrum of carbon cycle modeling

Understanding of the carbon cycle is particularly important because of the role of carbon dioxide as a greenhouse gas. Carbon cycle models play an essential role in the interpretation of observational data. The analysis of the carbon cycle involves statistical estimation in various contexts. These include various types of model calibration, including the estimation of feedbacks. A range of inverse calculations are involved in estimating the spatial and/or temporal dependence of carbon dioxide sources and sinks, given observations of concentrations. The uncertainties in these estimates propagate into uncertainties in projections of future carbon cycle behavior. These disparate analyses are discussed in terms of a modeling spectrum that runs from empirical statistical models through to reductionist mechanistic models. The use of the modeling spectrum allows a comparison of di erent modeling approaches. Comparing di erent levels of modeling can provide a basis for assessing the extent to which estimation is being applied consistently

Assessing the Information Content in Environmental Modelling: A Carbon Cycle Perspective

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Tangents, adjoints and computational complexity in terrestrial carbon modelling

Differentiation enters modelling through initialisation, calibration, sensitivity analysis and data assimilation. Automatic differentiation provides tools for augmenting models to calculate the derivatives. Adjoint transformations lead to computational gains in such analyses. The calculation of tangent models by operator overloading provides a reference case against which to assess such gains. This article uses a vector space representation to analyse how special localisation characteristics of the land surface within the earth system might change the computational complexity of calculating derivatives. References B. D. Craven. Control and Optimization. Chapman and Hall, London, 1995. I. G. Enting. Inverse Problems in Atmospheric Constituent Transport. CUP, Cambridge, UK, 2002. I. G. Enting, D. M. Etheridge, and M. J. Fielding. A perturbation analysis of the climate benefit of geosequestration. Int. J. Greenhouse Gas Control, 2:289--296, 2008. doi:10.1016/j.ijggc.2008.02.005 R. Giering. Tangent linear and adjoint biogeochemical models. In P. Kasibhatla et al., editor, Inverse Methods in Global Biogeochemical Cycles. (Geophysical Monograph no. 114), pages 33--48. AGU, Washington, DC, 2000. A. Griewank. Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation. SIAM, Philadelphia, 2000. C. S. Potter, J. T. Randerson, C. B. Field, P. A. Matson, P. M. Vitousek, and H. A. Mooney. Terrestrial ecosystem production: A process model based on global satellite and surface data. Global Biogeochemical Cycles, 7:811--841, 1993. doi:10.1029/93GB02725 P. J. Rayner, M. Scholze, W. Knorr, T. Kaminski, R. Giering, and H. Widmann. Two decades of terrestrial carbon fluxes from a carbon cycle data assimilation system {(ccdas)}. Global Biogeochemical Cycles, 19:GB2026, 2005. doi:10.1029/2004GB002254 C. Rodenbeck. Estimating {co}$_2$ sources and sinks from atmospheric mixing ratio measurements using a global inversion of atmospheric transport. Max-Planck-Institut fur Biogeochemie: Technical Paper 6, 2005. http://www.bgc-jena.mpg.de/mpg/websiteBiogeochemie/Publikationen/Technical_Reports/tech_report6.pdf, 2005 W. Steffen, editor. Blueprint for Australian Carbon Cycle Research. Australian Greenhouse Office, Canberra, 2005. http://www.globalcarbonproject.org/global/pdf/Australia.CarbonBluePrint_2005.pdf C. W. Straka. ADF95: Tool for automatic differentiation of a FORTRAN code designed for large numbers of independent variables. Comput. Phys. Commun., 168:123--139, 2005. doi:10.1016/j.cpc.2005.01.011 A. Tarantola. Inverse Problem Theory: Methods for Data Fitting and Model Parameter Estimation. SIAM, Philadelphia, 2005. Y.-P. Wang, C. M. Trudinger, and I. G. Enting. A review of applications of model-data fusion to studies of terrestrial carbon fluxes at different scales. Agricultural and Forest Meteorology, 149:1829--1842, 2009. doi:10.1016/j.agrformet.2009.07.00