Total Representations

Almost all representations considered in computable analysis are partial. We provide arguments in favor of total representations (by elements of the Baire space). Total representations make the well known analogy between numberings and representations closer, unify some terminology, simplify some technical details, suggest interesting open questions and new invariants of topological spaces relevant to computable analysis.Comment: 30 page

Quantum charges and spacetime topology: The emergence of new superselection sectors

In which is developed a new form of superselection sectors of topological origin. By that it is meant a new investigation that includes several extensions of the traditional framework of Doplicher, Haag and Roberts in local quantum theories. At first we generalize the notion of representations of nets of C*-algebras, then we provide a brand new view on selection criteria by adopting one with a strong topological flavour. We prove that it is coherent with the older point of view, hence a clue to a genuine extension. In this light, we extend Roberts' cohomological analysis to the case where 1--cocycles bear non trivial unitary representations of the fundamental group of the spacetime, equivalently of its Cauchy surface in case of global hyperbolicity. A crucial tool is a notion of group von Neumann algebras generated by the 1-cocycles evaluated on loops over fixed regions. One proves that these group von Neumann algebras are localized at the bounded region where loops start and end and to be factorial of finite type I. All that amounts to a new invariant, in a topological sense, which can be defined as the dimension of the factor. We prove that any 1-cocycle can be factorized into a part that contains only the charge content and another where only the topological information is stored. This second part resembles much what in literature are known as geometric phases. Indeed, by the very geometrical origin of the 1-cocycles that we discuss in the paper, they are essential tools in the theory of net bundles, and the topological part is related to their holonomy content. At the end we prove the existence of net representations

Topological map of the body in post-stroke patients: lesional and hodological aspects

Objective: It has been repeatedly hypothesized that at least 3 distinct types of body representations do exist: body schema, a representation derived from multiple sensory and motor inputs; topological map of the body, a structural description of spatial relations among the body parts; and body semantics, a lexical-semantic representation. Although several studies have assessed neural correlates of the topological map of the body in healthy participants, a systematic investigation of neural underpinnings of the topological map of the body in brain-damaged patients is still lacking. Method: Here we investigated the neural substrates of topological map of the body in 23 brain-damaged patients, both from a topological and an hodological perspectives, using Voxel Lesion Symptom Mapping and atlas-based track-wise statistical analysis. Besides neuroimaging investigation, consisting of T1-weighted and FLAIR sequences, patients underwent the frontal body-evocation subtest (FBE) to assess the topological map of the body. Results: The present results reveal a large-scale brain network involved in the topological map of the body assessed with FBE, encompassing both regions of primary elaboration and multisensory associative areas, in the temporal, parietal, frontal, and insular cortices. Hodological analysis revealed significant association between processing of the body topological map and the disconnection of the frontomarginal tract. Conclusions: These findings suggest that the topological map of the body is built up basing on both external and internal information that comes from the body and are constantly updated and integrated. The theoretical and clinical relevance of these results is discussed

Rewiring World Trade. Part II: A Weighted Network Analysis

In this sequel to a companion paper, we complement our analysis of the binary projections of the International Trade Network (ITN) by considering its weighted representations. We show that, unlike the binary case, all possible weighted representations of the ITN (directed/undirected, aggregated/ disaggregated) cannot be traced back to local structural properties, which are therefore of limited informativeness. Our results highlight that any topological property representing only partial information (e.g., degree sequences) cannot in general be obtained from the corresponding weighted property (e.g., strength sequences). Therefore the expectation that weighted structural properties oer a more complete description than purely topological ones is misleading. Our analysis of the ITN detects indirect eects that are not captured by traditional macroeconomic analyses focused only on weighted rst-order country-specic properties, and highlights the limitations of models and theories that overemphasize the need to reproduce and explain such properties.