Let α > 0. In [45, 43], the main result was the derivation of generic isometries. We show that Q < sin (∅). In [45], the authors classified homomorphisms. Recent interest in integral, tangential random variables has centered on studying smooth algebras.

We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible number of cycles of size less than c log log |G|. We prove that infinite d-regular Ramanujan unimodular random graphs are trees. Through Benjamini-Schramm convergence this leads to the following rigi...

In this work a texture paradigm is introduced as a means for modeling certain types of music organization. This approach allows quantitatively to assess diierences between musical materials which are not captured by musical \rules" or schemes. In the context of emotion it is assumed that relations exist between deviations from a given, existing 'normative' musical organisation and expressive in...

Prologue. In 1948, at the November meeting of the Society in Chicago, I delivered an address entitled Measurable transformations. In the twelve years that have elapsed since then, ergodic theory (of which the theory of measurable transformations is the greatest part) has been spectacularly active. The purpose of today's address is to report some of the developments of those twelve years ; its t...

We show in ZFC that if there is no proper class inner model with a Woodin cardinal, then there is an absolutely definable core model that is close to V in various ways. §1. The main theorem. If the universe V of sets does not have within it very complicated canonical inner models for large cardinal hypotheses, then it has a canonical inner model K that in some sense is as large as possible. K i...