Projection Theorems Using Effective Dimension

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand\u27s projection theorem, which shows that for every analytic set E, for almost every line L, the Hausdorff dimension of the orthogonal projection of E onto L is maximal. We use Kolmogorov complexity to give two new results on the Hausdorff and packing dimensions of orthogonal projections onto lines. The first shows that the conclusion of Marstrand\u27s theorem holds whenever the Hausdorff and packing dimensions agree on the set E, even if E is not analytic. Our second result gives a lower bound on the packing dimension of projections of arbitrary sets. Finally, we give a new proof of Marstrand\u27s theorem using the theory of computing

Renormalization group equations as 'decoupling' theorems

We propose a simple derivation of renormalization group equations and Callan-Symanzik equations as decoupling theorems of the structures underlying effective field theories.Comment: 7 pages, no figure, revtex4, additional typos removed, published in Phys. Lett.

Simple normal crossing Fano varieties and log Fano manifolds

A projective log variety (X, D) is called "a log Fano manifold" if X is smooth and if D is a reduced simple normal crossing divisor on X with -(K_X+D) ample. The n-dimensional log Fano manifolds (X, D) with nonzero D are classified in this article when the log Fano index r of (X, D) satisfies either r\geq n/2 with \rho(X)\geq 2 or r\geq n-2. This result is a partial generalization of the classification of logarithmic Fano threefolds by Maeda.Comment: 38 pages, minor revision; correct Theorem 2.7 and add reference