A general framework for measuring inconsistency through minimal inconsistent sets

Hunter and Konieczny explored the relationships between measures of inconsistency for a belief base and the minimal inconsistent subsets of that belief base in several of their papers. In particular, an inconsistency value termed MIVC, defined from minimal inconsistent subsets, can be considered as a Shapley Inconsistency Value. Moreover, it can be axiomatized completely in terms of five simple axioms. MinInc, one of the five axioms, states that each minimal inconsistent set has the same amount of conflict. However, it conflicts with the intuition illustrated by the lottery paradox, which states that as the size of a minimal inconsistent belief base increases, the degree of inconsistency of that belief base becomes smaller. To address this, we present two kinds of revised inconsistency measures for a belief base from its minimal inconsistent subsets. Each of these measures considers the size of each minimal inconsistent subset as well as the number of minimal inconsistent subsets of a belief base. More specifically, we first present a vectorial measure to capture the inconsistency for a belief base, which is more discriminative than MIVC. Then we present a family of weighted inconsistency measures based on the vectorial inconsistency measure, which allow us to capture the inconsistency for a belief base in terms of a single numerical value as usual. We also show that each of the two kinds of revised inconsistency measures can be considered as a particular Shapley Inconsistency Value, and can be axiomatically characterized by the corresponding revised axioms presented in this paper.

Tensor products of operator systems

The purpose of the present paper is to lay the foundations for a systematic study of tensor products of operator systems. After giving an axiomatic definition of tensor products in this category, we examine in detail several particular examples of tensor products, including a minimal, maximal, maximal commuting, maximal injective and some asymmetric tensor products. We characterize these tensor products in terms of their universal properties and give descriptions of their positive cones. We also characterize the corresponding tensor products of operator spaces induced by a certain canonical inclusion of an operator space into an operator system. We examine notions of nuclearity for our tensor products which, on the category of C*-algebras, reduce to the classical notion. We exhibit an operator system S which is not completely order isomorphic to a C*-algebra yet has the property that for every C*-algebra A, the minimal and maximal tensor product of S and A are equal.

Operator systems from discrete groups

We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlations. To do this we formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group Fn on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group Z2. We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems.

Application of self-quenched JH consensus primers for real-time quantitative PCR of IGH gene to minimal residual disease evaluation in multiple myeloma.

Monitoring multiple myeloma patients for relapse requires sensitive methods to measure minimal residual disease and to establish a more precise prognosis. The present study aimed to standardize a real-time quantitative polymerase chain reaction (PCR) test for the IgH gene with a JH consensus self-quenched fluorescence reverse primer and a VDJH or DJH allele-specific sense primer (self-quenched PCR). This method was compared with allele-specific real-time quantitative PCR test for the IgH gene using a TaqMan probe and a JH consensus primer (TaqMan PCR). We studied nine multiple myeloma patients from the Spanish group treated with the MM2000 therapeutic protocol. Self-quenched PCR demonstrated sensitivity of >or=10(-4) or 16 genomes in most cases, efficiency was 1.71 to 2.14, and intra-assay and interassay reproducibilities were 1.18 and 0.75%, respectively. Sensitivity, efficiency, and residual disease detection were similar with both PCR methods. TaqMan PCR failed in one case because of a mutation in the JH primer binding site, and self-quenched PCR worked well in this case. In conclusion, self-quenched PCR is a sensitive and reproducible method for quantifying residual disease in multiple myeloma patients; it yields similar results to TaqMan PCR and may be more effective than the latter when somatic mutations are present in the JH intronic primer binding site.