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A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains

The potential for sea ice-albedo feedback to give rise to nonlinear climate change in the Arctic Ocean – defined as a nonlinear relationship between polar and global temperature change or, equivalently, a time-varying polar amplification – is explored in IPCC AR4 climate models. Five models supplying SRES A1B ensembles for the 21 st century are examined and very linear relationships are found between polar and global temperatures (indicating linear Arctic Ocean climate change), and between polar temperature and albedo (the potential source of nonlinearity). Two of the climate models have Arctic Ocean simulations that become annually sea ice-free under the stronger CO 2 increase to quadrupling forcing. Both of these runs show increases in polar amplification at polar temperatures above-5 o C and one exhibits heat budget changes that are consistent with the small ice cap instability of simple energy balance models. Both models show linear warming up to a polar temperature of-5 o C, well above the disappearance of their September ice covers at about-9 o C. Below-5 o C, surface albedo decreases smoothly as reductions move, progressively, to earlier parts of the sunlit period. Atmospheric heat transport exerts a strong cooling effect during the transition to annually ice-free conditions. Specialized experiments with atmosphere and coupled models show that the main damping mechanism for sea ice region surface temperature is reduced upward heat flux through the adjacent ice-free oceans resulting in reduced atmospheric heat transport into the region.

Geophysical fluid dynamics.

From the Publisher:
"In many areas of mathematics, science and engineering, from computer graphics to inverse methods to signal processing it is necessary to estimate parameters, usually multidimensional, by approximation and interpolation. Radial basis functions are a modern and powerful tool which work well in very general circumstances, and so are becoming of widespread use, as the limitations of other methods, such as least squares, polynomial interpolation or wavelet-based, become apparent." This is the first book devoted to the subject and the author's aim is to give a thorough treatment from both the theoretical and practical implementation viewpoints. For example, he emphasises the many positive features of radial basis functions such as the unique solvability of the interpolation problem, the computation of interpolants, their smoothness and convergence, and provides a careful classification of the radial basis functions into types that have different convergence. A comprehensive bibliography rounds off what will prove a very valuable work.

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Radial Basis Functions

In the spring of 1988 an interagency consortium of Federal Land Managers and the Environmental Protection Agency initiated a national visibility and aerosol monitoring network to track spatial and temporal trends of visibility and visibility-reducing particles. The monitoring network consists of 36 stations located mostly in the western United States. The major visibility-reducing aerosol species, sulfates, nitrates, organics, light-absorbing carbon, and wind-blown dust are monitored as well as light scattering and extinction. Sulfates and organics are responsible for most of the extinction at most locations throughout the United States, while at sites in southern California nitrates are dominant. In the eastern United States, sulfates contribute to about two thirds of the extinction. In almost all cases, extinction and the major aerosol types are highest in the summer and lowest during the winter months.

Spatial and seasonal trends in particle concentration and optical extinction in the United States

Distributing many points on a sphere

1. Notation 2. Polynomials 3. Scalar Spherical Harmonics 4. Green's Functions 5. Spherical Radial Basis Functions 6. Spherical Splines 7. Approximate Integration 8. Spherical Singular Integrals 9. Gabor and Toeplitz Transform 10. Continuous Wavelet Transform 11. Discrete Wavelet Transform 12. Vector Spherical Harmonics 13. Vectorial Approximation Methods 14. Tensor Spherical Harmonics 15. Tensorial Approximation Methods References Subject Index

Constructive Approximation on the Sphere: With Applications to Geomathematics

In what follows, we introduce the classical Gamma function in Sect. 2.1. It is essentially understood to be a generalized factorial. However, there are many further applications, e.g., as part of probability distributions (see, e.g., Evans et al. 2000). The main properties of the Gamma function are explained in this chapter (for a more detailed discussion the reader is referred to, e.g., Artin (1964), Lebedev (1973), Muller (1998), Nielsen (1906), and Whittaker and Watson (1948) and the references therein).

The Gamma Function

Combined Spherical Harmonic and Wavelet Expansion—A Future Concept in Earth's Gravitational Determination

A concept of generalized discrepancy, which involves pseudodifferential operators to give a criterion of equidistributed pointsets, is developed on the sphere. A simply structured formula in terms of elementary functions is established for the computation of the generalized discrepancy. With the help of this formula five kinds of point systems on the sphere, namely lattices in polar coordinates, transformed two-dimensional sequences, rotations on the sphere, triangulations, and "sum of three squares sequence," are investigated. Quantitative tests are done, and the results are compared with one another. Our calculations exhibit different orders of convergence of the generalized discrepancy for different types of point systems.

Equidistribution on the Sphere

Basic Settings and Spherical Nomenclature- Scalar Spherical Harmonics- Green's Functions and Integral Formulas- Vector Spherical Harmonics- Tensor Spherical Harmonics- Scalar Zonal Kernel Functions- Vector Zonal Kernel Functions- Tensorial Zonal Kernel Functions- Zonal Function Modeling of Earth's Mass Distribution

Willi Freeden

Papers

Constructive Approximation on the Sphere: With Applications to Geomathematics

The Gamma Function

Combined Spherical Harmonic and Wavelet Expansion—A Future Concept in Earth's Gravitational Determination

Equidistribution on the Sphere

Spherical functions of mathematical geosciences : a scalar, vectorial, and tensorial setup