Black hole state counting in loop quantum gravity

The two ways of counting microscopic states of black holes in the U(1) formulation of loop quantum gravity, one counting all allowed spin network labels j,m and the other only m labels, are discussed in some detail. The constraints on m are clarified and the map between the flux quantum numbers and m discussed. Configurations with |m|=j, which are sometimes sought after, are shown to be important only when large areas are involved. The discussion is extended to the SU(2) formulation.Comment: 5 page

Reply to comment by Zaslavskii on extremal black hole action

It is shown that Zaslavskii's misunderstanding of our published proof of the irrelevance of all extremal black hole configurations (whether with equal charge and mass or not) rests on his refusal to see the essential difference between the correct inequality governing extremal and non-extremal actions and his incorrect version.Comment: 1 page, REVTeX, adapted from reply in PRL 80, 3413 (1998

Evaluation of missing data mechanisms in two and three dimensional incomplete tables

The analysis of incomplete contingency tables is a practical and an interesting problem. In this paper, we provide characterizations for the various missing mechanisms of a variable in terms of response and non-response odds for two and three dimensional incomplete tables. Log-linear parametrization and some distinctive properties of the missing data models for the above tables are discussed. All possible cases in which data on one, two or all variables may be missing are considered. We study the missingness of each variable in a model, which is more insightful for analyzing cross-classified data than the missingness of the outcome vector. For sensitivity analysis of the incomplete tables, we propose easily verifiable procedures to evaluate the missing at random (MAR), missing completely at random (MCAR) and not missing at random (NMAR) assumptions of the missing data models. These methods depend only on joint and marginal odds computed from fully and partially observed counts in the tables, respectively. Finally, some real-life datasets are analyzed to illustrate our results, which are confirmed based on simulation studies