On Self-Mullineux and Self-Conjugate Partitions

The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this work, we give an explicit bijection between the two families of partitions in terms of the Mullineux symbol.

An Interpolation-based Compiler and Optimizer for Relational Queries (System design Report)

We outline the implementation of a query compiler for relational queries that generates query plans with respect to a database schema, that is, a set of arbitrary first-order constraints, and a distinguished subset of predicate symbols from the underlying signature that correspond to access paths. The compiler is based on a variant of the Craig interpolation theorem, with reasoning realized via a modified analytic tableau proof procedure. This procedure decouples the generation of candidate plans that are interpolants from the tableau proof procedure, and applies A*-based search with respect to an external cost model to arbitrate among the alternative candidate plans. The tableau procedure itself is implemented as a virtual machine that operates on a compiled and optimized byte-code that faithfully implements reasoning with respect to the database schema constraints and a user query.

Human attention filters for single colors

The visual images in the eyes contain much more information than the brain can process. An important selection mechanism is feature-based attention (FBA). FBA is best described by attention filters that specify precisely the extent to which items containing attended features are selectively processed and the extent to which items that do not contain the attended features are attenuated. The centroid-judgment paradigm enables quick, precise measurements of such human perceptual attention filters, analogous to transmission measurements of photographic color filters. Subjects use a mouse to locate the centroid—the center of gravity—of a briefly displayed cloud of dots and receive precise feedback. A subset of dots is distinguished by some characteristic, such as a different color, and subjects judge the centroid of only the distinguished subset (e.g., dots of a particular color). The analysis efficiently determines the precise weight in the judged centroid of dots of every color in the display (i.e., the attention filter for the particular attended color in that context). We report 32 attention filters for single colors. Attention filters that discriminate one saturated hue from among seven other equiluminant distractor hues are extraordinarily selective, achieving attended/unattended weight ratios >20:1. Attention filters for selecting a color that differs in saturation or lightness from distractors are much less selective than attention filters for hue (given equal discriminability of the colors), and their filter selectivities are proportional to the discriminability distance of neighboring colors, whereas in the same range hue attention-filter selectivity is virtually independent of discriminabilty.

ISOMETRY GROUPS OF WEIGHTED SPACES OF HOLOMORPHIC FUNCTIONS: TRANSITIVITY AND UNIQUENESS

We survey some recent results on the isometries of weighted spaces of holomorphic functions defined on an open subset of ℂn. We will see that these isometries are determined by a subgroup of the automorphisms on a distinguished subset of the domain. We will look for weights with 'large' groups of isometries and observe that in certain circumstances the group of isometries determines the weight.

AN OPTIMAL REBUILDING STRATEGY FOR AN INCREMENTAL TREE PROBLEM

This paper is devoted to the following incremental problem. Initially, a graph and a distinguished subset of vertices, called initial group, are given. This group is connected by an initial tree. The incremental part of the input is given by an on-line sequence of vertices of the graph, not yet in the current group, revealed on-line one after one. The goal is to connect each new member to the current tree, while satisfying a quality constraint: the average distance between members in each constructed tree must be kept in a given range compared to the best possible one. Under this quality constraint, our objectives are to minimize the number of critical stages and the number of elementary changes of the sequence of constructed trees. We call "critical" a stage where the inclusion of a new member implies heavy changes in the current tree. Otherwise, the new member is just added by connecting it with a (well chosen) path to the current tree. In both cases, updating a tree implies a certain number of elementary changes (that we define). We propose a strategy leading to at most O(log i) critical stages (i is the number of new members) and to at most a constant average number of elementary changes per stage. We also prove that there exists situations where Ω(log i) critical stages are necessary to any algorithm to maintain the quality constraint. Our strategy is then worst case optimal in order of magnitude for the number of critical stages criterion and induces a constant number of elementary changes in average per stage.

Bi-coloured fields on the complex numbers

We consider two theories of “bad fields” constructed by B.Poizat using Hrushovski's amalgamation and show that these theories have natural models representable as the field of complex numbers with a distinguished subset given as a union of countably many real analytic curves. One of the two examples is based on the complex exponentiation and the proof assumes Schanuel's conjecture.

On the Classification of Simple Stably Projectionless C*-Algebras

It is shown that simple stably projectionless C*-algebras which are inductive limits of certain specified building blocks with trivial K-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.