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This is a contamination quite common in probability when properties of distributions are instead attributed to the associated random objects. Strictly speaking one should talk about isotropic (i.e., rotation invariant) measures or distributions rather than vectors. Yet another more recent example of this contamination is provided by so called "invariant ...


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A random vector $\mathbf{x}$ is called isotropic with respect to a norm $\mu$ (more generally, a quasinorm) if the equiprobability curves are given by $\mu(\mathbf{x})=\text{constant}$. If $\mu$ is the Euclidean norm, the equiprobability curves are rotationally invariant, hence the name "isotropic". This is a special case, more generally the ...


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Straw Man Proposal A straw-man (or straw-dog) proposal is a brainstormed simple draft proposal intended to generate discussion of its disadvantages and to provoke the generation of new and better proposals. https://en.wikipedia.org/wiki/Straw_man_proposal


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This notion of 'the sphere at infinity' is commonly encountered in hyperbolic geometries. Gromov, in particular, has used it in studying the behavior of discrete transformation groups on hyperbolic manifolds and you might also look at the works of Biquard on prescribing the geometry of the boundary at infinity of an Einstein manifold with negative Ricci ...


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For what it worth, in the terms of divergent integrals, your transform can be rewritten as $$\operatorname{reg} \int_1^\infty f(t)\frac{e^{-t s \omega _-}}{s t}dt$$ Looks like some kind of analog of Fourier transform, if you ask me...


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