The theorem of Madsen and Weiss [MW] identifies the homology of mapping class groups of surfaces, in a stable dimension range, with the homology of a certain infinite loopspace. This result is not only intrinsically interesting, showing that two objects that appear to be quite different turn out to be homologically equivalent, but it also allows explicit calculations of the stable homology, rat...

as a, a + p1(d), a + p2(d), . . . , a + pk−1(d) where pi(d) = id. Why these functions? We ponder replacing pi with other functions. The following remarkable theorem was first proved by Bergelson and Leibman [1]. They proved it by first proving the polynomial version of the Hales-Jewitt Theorem [2] (see Section 4 for a statement and proof of the original Hales-Jewitt Theorem), from which Theorem...

In this paper, we address an issue concerning the application of an ontological layer over some federal jurisprudences repositories. After a brief state of art, we designed a domain ontology performing a substantial modification of an existing one, adding some federal features through experts interview. This approach seems to open promising ways towards a semantic exposition of Brazilian jurisp...

The 3SUM problem asks if there are three integers a, b, c summing to 0 in a given set of n integers of magnitude poly(n). This problem can be easily solved in time Õ(n). (Throughout this note, Õ and Ω̃ hide subpolynomial factors n.) It seems natural to believe that this problem also requires time Ω̃(n), and this has been confirmed in some restricted models.[Eri99, AC05] The importance of this bel...

Abstract A construction of Bourgain [Bou05] gave the first 2-source extractor to break the min-entropy rate 1/2 barrier. In this note, we write an exposition of his result, giving a high level way to view his extractor construction. We also include a proof of a generalization of Vazirani’s XOR lemma that seems interesting in its own right, and an argument (due to Boaz Barak) that shows that any...

The branch of philosophical logic which has become known as “belief change” has, in the course of its development, become alienated from its epistemological origins. However, as formal criteria do not suffice to defend a principled choice between competing systems for belief change, we do need to take their epistemological embedding into account. Here, on the basis of a detailed examination of ...

A NONTECENICAL EXPOSITION OVERVIEW In his insightful and groundbreaking essay entitled The Structure of Scientific Revolutions, Thomas S. Kuhn discussed the notion of scientific advancement. In particular, he introduces the idea of a paradigm, a body of knowledge and beliefs which provides direction for research. The development and acceptance of a paradigm is a key factor in transforming a gro...

The purpose of this paper is to give an elementary and relatively self-contained proof of the following basic theorem, apparently due to Fricke [6, 7]. LetG = SL(2,C) and letG act onH = SL(2,C)×SL(2,C) by g · (ξ, η) 7−→ (gξg, gηg). Let κ(x, y, z) = x + y + z − xyz − 2. Theorem. Let f : H −→ C be a regular function which is invariant under the above G-action. Then there exists a polynomial funct...

This essay is published for the first time here and, concurrently , as a chapter in New Directions in Austrian Economics