Principal bundles on two-dimensional CW-complexes with disconnected structure group

Given any topological group G, the topological classification of principal G-bundles over a finite CW-complex X is long known to be given by the set of free homotopy classes of maps from X to the corresponding classifying space BG. This classical result has been long-used to provide such classification in terms of explicit characteristic classes. However, even when X has dimension 2, there is a case in which such explicit classification has not been explicitly considered. This is the case where G is a Lie group, whose group of components acts nontrivially on its fundamental group $\pi_1G$ . Here, we deal with this case and obtain the classification, in terms of characteristic classes, of principal G-bundles over a finite CW-complex of dimension 2, with G is a Lie group such that $\pi_0G$ is abelian.

Where are we? Using Scopus to map the literature at the intersection between artificial intelligence and research on crime

Research on artificial intelligence (AI) applications has spread over many scientific disciplines. Scientists have tested the power of intelligent algorithms developed to predict (or learn from) natural, physical and social phenomena. This also applies to crime-related research problems. Nonetheless, studies that map the current state of the art at the intersection between AI and crime are lacking. What are the current research trends in terms of topics in this area? What is the structure of scientific collaboration when considering works investigating criminal issues using machine learning, deep learning, and AI in general? What are the most active countries in this specific scientific sphere? Using data retrieved from the Scopus database, this work quantitatively analyzes 692 published works at the intersection between AI and crime employing network science to respond to these questions. Results show that researchers are mainly focusing on cyber-related criminal topics and that relevant themes such as algorithmic discrimination, fairness, and ethics are considerably overlooked. Furthermore, data highlight the extremely disconnected structure of co-authorship networks. Such disconnectedness may represent a substantial obstacle to a more solid community of scientists interested in these topics. Additionally, the graph of scientific collaboration indicates that countries that are more prone to engage in international partnerships are generally less central in the network. This means that scholars working in highly productive countries (e.g. the United States, China) tend to mostly collaborate domestically. Finally, current issues and future developments within this scientific area are also discussed.

Where Are We? Using Scopus to Map the Literature at the Intersection Between Artificial Intelligence and Research on Crime

Research on Artificial Intelligence (AI) applications has spread over many scientific disciplines. Scientists have tested the power of intelligent algorithms developed to predict (or learn from) natural, physical and social phenomena. This also applies to crime-related research problems. Nonetheless, studies that map the current state of the art at the intersection between AI and crime are lacking. What are the current research trends in terms of topics in this area? What is the structure of scientific collaboration when considering works investigating criminal issues using machine learning, deep learning, and AI in general? What are the most active countries in this specific scientific sphere? Using data retrieved from the Scopus database, this work quantitatively analyzes 692 published works at the intersection between AI and crime employing network science to respond to these questions. Results show that researchers are mainly focusing on cyber-related criminal topics and that relevant themes such as algorithmic discrimination, fairness, and ethics are considerably overlooked. Furthermore, data highlight the extremely disconnected structure of co-authorship networks. Such disconnectedness may represent a substantial obstacle to a more solid community of scientists interested in these topics. Additionally, the graph of scientific collaboration indicates that countries that are more prone to engage in international partnerships are generally less central in the network. This means that scholars working in highly productive countries (e.g. the United States, China) tend to mostly collaborate domestically. Finally, current issues and future developments within this scientific area are also discussed.

FRACTAL TRANSITION: HIERARCHICAL STRUCTURE AND NOISE EFFECT

A Sierpinski gasket with continuous trajectories is presented as an example of the fractal transition that characterizes the behavior of dissipative dynamical systems excited by external temporal inputs. Using this example, we investigate the fractal transition from two points of views, i.e. a hierarchical structure and a noise effect. Depending on internal and external parameters, the structure can be geometrically classified as one of three types, i.e. totally disconnected, just-touching, and overlapping. For the totally disconnected structure, continuous trajectories and their starting points can be characterized by a definite hierarchical tree structure. Even for the just-touching and overlapping structure, a similar hierarchy exists. White noise contaminating the external inputs breaks the hierarchy. In particular, small clustered structures are sensitive to the noise. In such a case, the difference between trajectories and starting points is remarkable in the hierarchy.

Holonomies of Yang–Mills connections for bundles on a surface with disconnected structure group

Let Σ be a closed surface of genus ≥ 1, G a compact Lie group, not necessarily connected with Lie algebra g, ξ,: P → Σ a principal G-bundle, and suppose Σ equipped with a Riemannian metric and g with an invariant scalar product so that the Yang—Mills equations on ξ are defined. Further, letbe the universal central extension of the fundamental group π of Σ and ΓR the group obtained from Γ when its centre Z is extended to the additive group R of the reals. We show that there are bijective correspondences between various spaces of classes of Yang—Mills connections on ξ and spaces of representations of Γ and ΓR (as appropriate) in G. In particular, we show that the holonomy establishes a homeomorphism between the moduli space N(ξ) of central Yang–Mills connections on ξ and the space Repξ(Γ, G) of representations of Γ in G determined by ξ. Our results rely on a detailed study of the holonomy of a central Yang–Mills connection and extend corresponding ones of Atiyah and Bott for the case where G is connected.