The isotropy of products of Pfister forms is studied. In particular, an improved lower bound on the value of their first Witt index is obtained. This result and certain of its corollaries are applied to the study of the weak isotropy index (or equivalently, the sublevel) of arbitrary quadratic forms. The relationship between this invariant and the level of the form is investigated. The problem ...

Tissot indicatrices have provided visual measures of local area and isotropy distortions. Here we show how large scale distortions of flexion (bending) and skewness (lopsidedness) can be measured. Area and isotropy distortions depend on the map projection metric, flexion and skewness, which manifest themselves on continental scales, depend on the first derivatives of the metric. We introduce ne...

We examine two models for human perception of shape from texture, based on two assumptions about the surface texture: isotropy and homogeneity. Observers made orientation judgments on planar textured surfaces. Surface textures were either isotropic or anisotropically stretched or compressed. If subjects used an isotropy assumption, they would make biased orientation estimates for the anisotropi...

The breakdown of statistical homogeneity and isotropy of cosmic perturbations is a generic feature of ultra large scale structure of the cosmos, in particular, of non trivial cosmic topology. The statistical isotropy (SI) of the Cosmic Microwave Background temperature fluctuations (CMB anisotropy) is sensitive to this breakdown on the largest scales comparable to, and even beyond the cosmic hor...

K 〈x, θ〉2dx = LK ∀θ ∈ Sn−1 where LK is a constant independent of θ, which is called the isotropy constant of K. Here 〈·, ·〉 denotes the standard scalar product in R. It is well known that for every convex body K ∈ R there exists an affine map T such that TK is isotropic. Furthermore, K and TK are both isotropic if and only if T is an orthogonal transformation. In this case the isotropy constant...

Some useful information is known about the fundamental domain for certain Hilbert modular groups. The six nonequivalent points with nontrivial isotropy in the fundamental domains under the action of the modular group for Q( √ 5), Q( √ 2), and Q( √ 3) have been determined previously by Gundlach. In finding these points, use was made of the exact size of the isotropy groups. Here we show that the...

It is well known that Landau's phenomenological description' of a continuous phase transition is very useful in classifying syminetry changes which can take place during such a transition. A good illustrative example is given by Lyubarskii. The procedure can be summarized as follows: (1) Find the space group 6 of the high-symmetry phase of a physical system (crystal, alloy, etc.) and the irredu...