Ample Pairs

We show that the ample degree of a stable theory with trivial forking is preserved when we consider the corresponding theory of belles paires, if it exists. This result also applies to the theory of $H$-structures of a trivial theory of rank $1$.Comment: Research partially supported by the program MTM2014-59178-P. The second author conducted research with support of the programme ANR-13-BS01-0006 Valcomo. The third author would like to thank the European Research Council grant 33882

Ample Pairs

We show that the ample degree of a stable theory with trivial forking is preserved when we consider the corresponding theory of belles paires, if it exists. This result also applies to the theory of $H$-structures of a trivial theory of rank $1$.Comment: Research partially supported by the program MTM2014-59178-P. The second author conducted research with support of the programme ANR-13-BS01-0006 Valcomo. The third author would like to thank the European Research Council grant 33882

Common Errors in Digital Panoramic Radiographs of Patients with Mixed Dentition and Patients with Permanent Dentition

Purpose. To compare errors in digital panoramic radiographs of permanent and mixed dentitions. Methods. 143 and 146 digital radiographs of mixed and permanent dentitions were examined. Results. Significantly fewer errors presented in the mixed dentition. Positioning too forward significantly prevalent in the mixed dentition; slumped position and nonpositioning of chin properly were significantly prevailed in the permanent dentition. Blurred or shortened upper incisors were significantly more prevalent in the mixed dentition. Diagnostic ability could be improved by manipulating the brightness or contrast in nearly 45% of all radiographs. In the mixed dentition, tilting the chin down and a slumped position made the lower incisors significantly nondiagnostic. In the permanent dentition, tilting the chin down made the lower incisors to be significantly nondiagnostic. Conclusions. More errors were prevalent in panoramic radiographs of permanent dentitions. Properly positioning the patient is the most important factor in preventing a cascade of errors

Closed quantum subgroups of locally compact quantum groups

We investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group. Two definitions - one due to S.Vaes and one due to S.L.Woronowicz - are analyzed and relations between them discussed. Among many reformulations we prove that the former definition can be phrased in terms of quasi-equivalence of representations of quantum groups while the latter can be related to an old definition of Podle\'s from the theory of compact quantum groups. The cases of classical groups, duals of classical groups, compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context. A deep relationship with the quantum group generalization of Herz restriction theorem from classical harmonic analysis is also established, in particular, in the course of our analysis we give a new proof of Herz restriction theorem.Comment: 24 pages, v3 adds another reference. The paper will appear in Advances in Mathematic

Quanta of Maths

The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics

Quanta of Maths

The work of Alain Connes has cut a wide swath across several areas of math- ematics and physics. Reflecting its broad spectrum and profound impact on the contemporary mathematical landscape, this collection of articles covers a wealth of topics at the forefront of research in operator algebras, analysis, noncommutative geometry, topology, number theory and physics