The scattering-phase theorem states that the values of scattering and reduced scattering coefficients of the bulk random media are proportional to the variance of the phase and the variance of the phase gradient, respectively, of the phase map of light passing through one thin slice of the medium. We report a new derivation of the scattering phase theorem and provide the correct form of the rel...

This paper presents a mechanisation of some basic computability theory. The mechanisation uses two models: the recursive functions and the λcalculus, and shows that they have equivalent computational power. Results proved include the Recursion Theorem, an instance of the s-m-n theorem, the existence of a universal machine, Rice’s Theorem, and closure facts about the recursive and recursively en...

Part 2. Computability Theory: Very Powerful Models 15 7. Turing Machines: 1/28/14 15 8. Recognizability, Decidability, and Reductions: 1/30/14 18 9. Reductions, Undecidability, and the Post Correspondence Problem: 2/4/14 21 10. Oracles, Rice’s Theorem, the Recursion Theorem, and the Fixed-Point Theorem: 2/6/14 23 11. Self-Reference and the Foundations of Mathematics: 2/11/14 26 12. A Universal ...

The goal of this paper is to describe all closed, aspherical Riemannian manifolds M whose universal covers M̃ have have a nontrivial amount of symmetry. By this we mean that Isom(M̃) is not discrete. By the well-known theorem of Myers-Steenrod [MS], this condition is equivalent to [Isom(M̃) : π1(M)] = ∞. Also note that if any cover of M has a nondiscrete isometry group, then so does its universal ...

The Ricci curvature at an isolated singularity of an immersed hypersurface exhibits a local behaviour echoing a global property of the Ricci curvature on a complete hypersurface in euclidean space (i.e., the Efimov theorem [12], that sup Ric ≥ 0). This local behaviour takes the form of a near universal bound on the decay of the Ricci curvature at a simple singularity (eg. a cone singularity) in...

We provide a new proof of a recent theorem of Ben-Yaacov, Melleray, and Tsankov. If G is a Polish group and X is a minimal, metrizable G-flow with all orbits meager, then the universal minimal flow M(G) is non-metrizable. In particular, we show that given X as above, the universal highly proximal extension of X is non-metrizable.

We continue our study of quadratic forms using Geometry of Numbers methods by considering universal quaternary positive definite integral forms of square discriminant. We give a small multiple theorem for such forms and use it to prove universality for all nine universal diagonal forms. The most interesting case is x2 + 2y2 + 5z2 + 10w2, which required computer calculations.

Hippocratic randomness is defined in a similar way to Martin-Löf randomness, however it does not assume computability of the probability and the existence of universal test is not assured. We introduce the notion of approximation of probability and show the existence of the universal test (Levin-Schnorr theorem) for Hippocratic randomness when the logarithm of the probability is approximated wi...

Figure 2: Near rectangular dissections in applications. Note that in the applications the areas of the rectangles of a dissection are relevant, actually in most cases these areas are prescribed. A rectangular dissection is area-universal if for any assignment of positive weights to the rectangles there is an equivalent dissection such that the areas of rectangles are equal to the given weights....