Randomness and differentiability in higher dimensions

We present two theorems concerned with algorithmic randomness and differentiability of functions of several variables. Firstly, we prove an effective form of the Rademacher's Theorem: we show that computable randomness implies differentiability of computable Lipschitz functions of several variables. Secondly, we show that weak 2-randomness is equivalent to differentiability of computable a.e. differentiable functions of several variables.Comment: 19 page

LINEAR ORDERS REALIZED BY CE EQUIVALENCE RELATIONS

AbstractLetEbe a computably enumerable (c.e.) equivalence relation on the setωof natural numbers. We say that the quotient set$\omega /E$(or equivalently, the relationE)realizesa linearly ordered set${\cal L}$if there exists a c.e. relation ⊴ respectingEsuch that the induced structure ($\omega /E$; ⊴) is isomorphic to${\cal L}$. Thus, one can consider the class of all linearly ordered sets that are realized by$\omega /E$; formally,${\cal K}\left( E \right) = \left\{ {{\cal L}\,|\,{\rm{the}}\,{\rm{order}}\, - \,{\rm{type}}\,{\cal L}\,{\rm{is}}\,{\rm{realized}}\,{\rm{by}}\,E} \right\}$. In this paper we study the relationship between computability-theoretic properties ofEand algebraic properties of linearly ordered sets realized byE. One can also define the following pre-order$\le _{lo}$on the class of all c.e. equivalence relations:$E_1 \le _{lo} E_2$if every linear order realized byE1is also realized byE2. Following the tradition of computability theory, thelo-degrees are the classes of equivalence relations induced by the pre-order$\le _{lo}$. We study the partially ordered set oflo-degrees. For instance, we construct various chains and anti-chains and show the existence of a maximal element among thelo-degrees.</jats:p

Annotating Table Data Using Separate Tables

There are no mechanisms for users to provide comments on graphs and charts with aggregated data or at data line level generated from a database. Solutions such as changing schemas of the original database tables or making duplicates of the original tables are sub-optimal and require substantial user effort. This disclosure describes techniques that enable users to add annotations to data records in a database and to retrieve such annotations using data queries, including while extracting aggregations of data. While aggregating the data, annotations from reference data can be included when data is grouped by a particular field. Annotations can be stored in a separate metadata table or as a view within the database. Merging annotations with the data can be accomplished by macros or a table-valued function (TVF). The TVF can be tailored to user specifications to read from specific annotation tables with appropriate selection criteria. Annotations can be displayed anywhere the data is shown. TVFs for public annotation tables can be implemented to select annotations based on moderator-vetting of annotations on these tables, without touching the data being annotated, and multiple annotation tables can be queried if necessary

Emotional Graphic Cigarette Warning Labels Reduce the Electrophysiological Brain Response to Smoking Cues

There is an ongoing public debate about the new graphic warning labels (GWLs) that the Food and Drug Administration (FDA) proposes to place on cigarette packs. Tobacco companies argued that the strongly emotional images FDA proposed to include in the GWLs encroached on their constitutional rights. The court ruled that FDA did not provide sufficient scientific evidence of compelling public interest in such encroachment. This study\u27s objectives were to examine the effects of the GWLs on the electrophysiological and behavioral correlates of smoking addiction and to determine whether labels rated higher on the emotional reaction (ER) scale are associated with greater effects. We studied 25 non-treatment-seeking smokers. Event-related potentials (ERPs) were recorded while participants viewed a random sequence of paired images, in which visual smoking (Cues) or non-smoking (non-Cues) images were preceded by GWLs or neutral images. Participants reported their cigarette craving after viewing each pair. Dependent variables were magnitude of P300 ERPs and self-reported cigarette craving in response to Cues. We found that subjective craving response to Cues was significantly reduced by preceding GWLs, whereas the P300 amplitude response to Cues was reduced only by preceding GWLs rated high on the ER scale. In conclusion, our study provides experimental neuroscience evidence that weighs in on the ongoing public and legal debate about how to balance the constitutional and public health aspects of the FDA-proposed GWLs. The high toll of smoking-related illness and death adds urgency to the debate and prompts consideration of our findings while longitudinal studies of GWLs are underway