Measurable cardinals and good Σ1(κ)\Sigma_1(\kappa)-wellorderings

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $\kappa$ with the property that the collection of all initial segments of the wellordering is definable by a $\Sigma_1$-formula with parameter $\kappa$. A short argument shows that the existence of a measurable cardinal $\delta$ implies that such wellorderings do not exist at $\delta$-inaccessible cardinals of cofinality not equal to $\delta$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that interferes with the existence of such wellorderings at uncountable cardinals and we generalize the above result to the minimal model containing two measurable cardinals.Comment: 14 page

Improving Retrieval Results with discipline-specific Query Expansion

Choosing the right terms to describe an information need is becoming more difficult as the amount of available information increases. Search-Term-Recommendation (STR) systems can help to overcome these problems. This paper evaluates the benefits that may be gained from the use of STRs in Query Expansion (QE). We create 17 STRs, 16 based on specific disciplines and one giving general recommendations, and compare the retrieval performance of these STRs. The main findings are: (1) QE with specific STRs leads to significantly better results than QE with a general STR, (2) QE with specific STRs selected by a heuristic mechanism of topic classification leads to better results than the general STR, however (3) selecting the best matching specific STR in an automatic way is a major challenge of this process.Comment: 6 pages; to be published in Proceedings of Theory and Practice of Digital Libraries 2012 (TPDL 2012

The Hurewicz dichotomy for generalized Baire spaces

By classical results of Hurewicz, Kechris and Saint-Raymond, an analytic subset of a Polish space $X$ is covered by a $K_\sigma$ subset of $X$ if and only if it does not contain a closed-in-$X$ subset homeomorphic to the Baire space ${}^\omega \omega$. We consider the analogous statement (which we call Hurewicz dichotomy) for $\Sigma^1_1$ subsets of the generalized Baire space ${}^\kappa \kappa$ for a given uncountable cardinal $\kappa$ with $\kappa=\kappa^{<\kappa}$, and show how to force it to be true in a cardinal and cofinality preserving extension of the ground model. Moreover, we show that if the Generalized Continuum Hypothesis (GCH) holds, then there is a cardinal preserving class-forcing extension in which the Hurewicz dichotomy for $\Sigma^1_1$ subsets of ${}^\kappa \kappa$ holds at all uncountable regular cardinals $\kappa$, while strongly unfoldable and supercompact cardinals are preserved. On the other hand, in the constructible universe L the dichotomy for $\Sigma^1_1$ sets fails at all uncountable regular cardinals, and the same happens in any generic extension obtained by adding a Cohen real to a model of GCH. We also discuss connections with some regularity properties, like the $\kappa$-perfect set property, the $\kappa$-Miller measurability, and the $\kappa$-Sacks measurability.Comment: 33 pages, final versio

Expectation Propagation on the Maximum of Correlated Normal Variables

Many inference problems involving questions of optimality ask for the maximum or the minimum of a finite set of unknown quantities. This technical report derives the first two posterior moments of the maximum of two correlated Gaussian variables and the first two posterior moments of the two generating variables (corresponding to Gaussian approximations minimizing relative entropy). It is shown how this can be used to build a heuristic approximation to the maximum relationship over a finite set of Gaussian variables, allowing approximate inference by Expectation Propagation on such quantities.Comment: 11 pages, 7 figure

Discursive Killings: Intertextuality, Aestheticization, and Death in Nabokov's Lolita

This essay argues that Nabokov's Lolita is suffused with a rhetoric of death. Humbert Humbert's discursive constructions of Lolita trap her in a semantic web of death that conjures up her literal death in childbed at the age of seventeen. My reading of Lolita traces the fibres of that web in the more sinister implications of Humbert's intertextual references, his persistent gestures of aestheticization and his reflections on the nature of nymphets

A quantum cluster algebra of Kronecker type and the dual canonical basis

The article concerns the dual of Lusztig's canonical basis of a subalgebra of the positive part U_q(n) of the universal enveloping algebra of a Kac-Moody Lie algebra of type A_1^{(1)}. The examined subalgebra is associated with a terminal module M over the path algebra of the Kronecker quiver via an Weyl group element w of length four. Geiss-Leclerc-Schroeer attached to M a category C_M of nilpotent modules over the preprojective algebra of the Kronecker quiver together with an acyclic cluster algebra A(C_M). The dual semicanonical basis contains all cluster monomials. By construction, the cluster algebra A(C_M) is a subalgebra of the graded dual of the (non-quantized) universal enveloping algebra U(n). We transfer to the quantized setup. Following Lusztig we attach to w a subalgebra U_q^+(w) of U_q(n). The subalgebra is generated by four elements that satisfy straightening relations; it degenerates to a commutative algebra in the classical limit q=1. The algebra U_q^+(w) possesses four bases, a PBW basis, a canonical basis, and their duals. We prove recursions for dual canonical basis elements. The recursions imply that every cluster variable in A(C_M) is the specialization of the dual of an appropriate canonical basis element. Therefore, U_q^+(w) is a quantum cluster algebra in the sense of Berenstein-Zelevinsky. Furthermore, we give explicit formulae for the quantized cluster variables and for expansions of products of dual canonical basis elements.Comment: 32 page