Representations of the Random Fields on a Sphere

With practical applications in mind, a study is made of the representations of the random fields on a sphere based on the theory of representations of the rotation group. The representation of the rotation group is defined in this paper by a class of rotational shift transformations of the random fields generated by a homogeneous random field on the sphere, ‘homogeneous’ with respect to the rotational motions. First, the spectral decomposition of a homogeneous scalar random field on a sphere is given a simple interpratation : it is a sum of the invariant vectors in the irreducible representation spaces of the rotational shift transformations. Some representations of the random fields are given in terms of the stochastic integrals with respect to a homogeneous random measure on the sphere. A homogeneous l-vector random field with 2 l+ 1 components in an irreducible space of weight l representation is defined as an invariant tensor under rotational shift transformations. A ‘stochastic’ spherical harmonics is defined as one of such random fields, which is a stochastic version of the l-vector spherical harmonic. Similarly, a ‘stochastic’ solid harmonic is defined in terms of stochasic spherical harmonics and generalized spherical Bessel functions. It is expressible as a l-vector Fourier integral over a sphere as well as a tensorial Fourier integral, and satisfies the l-vector Helmholtz equation. Such ‘stochastic’ harmonic functions can be used effectively in dealing with the scattering problem associated with a random sphere

Simultaneous Spectral Representations of Isotropically Correlated Random Fields

The spectral representations for isotropically correlated two l₁- and l₂-vecotr random fields are given generally in terms of l-vector harmonics and random measures, where l-vector denotes a (2l+l)- dimensional vector in the irreducible representation space of weight l of the rotation group, l＝0 being a scalar and l＝1 an ordinary vector. The representation is derived by making use of the multi-dimensional moving average and the previous work on l-vector functions. The three special cases of interest are discussed in detail ; namely, two scalar random fields, two vector random fields and a scalar and a vector random field

Probability Distribution of Maxima of Random Surface

The joint probability distributions are obtained of the extremum value, principal curvatures and the angle of their orientation of a random surface described by a homogeneous Gaussian random field. The probability density for the extrema is calculated as a marginal distribution, and is cast into a Gram-Charlier series in a form convenient for numerical calculation. Probability densities are numerically calculated for some isotropic spectral densities for an application in the microwave backscattering from ocean waves. Key Words : random surface, extremum, curvature, probability density, Gram-Charlier serie

Heterogeneous Impacts of Grazing Animals and Vegetational Change in Japanese Native Pastures

Defoliation, defecation and trampling are the major modes whereby grazing animals give impacts on vegetation. Due to the uneven distribution, such grazing behavior can have profound effects on vegetation. For extensive grazing systems in native pastures, understandings of the plant-animal interaction are vital for adequate control of vegetation and animal conditions and sustainable use of natural resources. This paper reviews recent studies of the grazing impacts on vegetation in Japanese native pastures. Most of the studies were carried out in the Kawatabi Field Science Center (Kawatabi FSC), Tohoku University. 1.Native pastures in the Kawatabi FSC are composed of 61-155 plant species, of which cattle graze upon 26-76 species. Among these species, Miscanthus sinensis (Japanese plume-grass) was the most frequently grazed by cattle. The spatial distribution of available forage is a major factor affecting diet selection and consumption of cattle. Such selective grazing results in significant reduction of M. sinensis in native pastures. 2.Seed dispersal of plants by defecation of grazing animals can also result in significant vegetational change. Recent studies have shown that Carex spp. is the major plant whose seeds are dispersed by defecation of animals rotationally grazed in a native and a sown pasture. The mechanisms of the seed dispersal and its possible effects on vegetational succession are discussed. 3.Heavy trampling is known to degrade vegetative ground cover. Our research has shown that trampling by cattle promotes the invasion of a shrub, Weigela hortensis into Miscanthus-dominant pastures. Because the seeds of W. hortensis are light sensitive germinators, trampling by removing ground cover promotes its seed germination. These findings provide new perspectives on plantanimal interactions in Japanese native pastures and help estimate the impact of animals on plant succession. They also contribute to efforts to ensure sustainable grazing use of pastures