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The Ligon-Schaaf regularization (LS mapping) was introduced in 1976 and has been used several times. However, we are not aware of any direct usage of the inverse mapping, perhaps since it appears at first sight to be quite complex, involves the use of a transcendental equation (referred to as the generalized Kepler equation) that cannot be solved in closed form, and lacks smoothness near the collision point. Here, we provide some insight into the significance of this equation, along with a very simple derivation and confirmation of the inverse LS mapping. Then we use numerical methods to investigate three applications: 1) solutions of the Kepler function, 2) calculation of orbits including time-of-flight data based on the Delaunay Hamiltonian, and 3) numerical evidence for the Birkhoff conjecture for the circular restricted 3-body problem. [ABSTRACT FROM AUTHOR]

The communication processes of knowledge creation represent a particular class of human dynamics where the expertise of individuals plays a substantial role, thus offering a unique possibility to study the structure of knowledge networks from online data. Here, we use the empirical evidence from questions-and-answers in mathematics to analyse the emergence of the network of knowledge contents (or tags) as the individual experts use them in the process. After removing extra edges from the network-associated graph, we apply the methods of algebraic topology of graphs to examine the structure of higher-order combinatorial spaces in networks for four consecutive time intervals. We find that the ranking distributions of the suitably scaled topological dimensions of nodes fall into a unique curve for all time intervals and filtering levels, suggesting a robust architecture of knowledge networks. Moreover, these networks preserve the logical structure of knowledge within emergent communities of nodes, labeled according to a standard mathematical classification scheme. Further, we investigate the appearance of new contents over time and their innovative combinations, which expand the knowledge network. In each network, we identify an innovation channel as a subgraph of triangles and larger simplices to which new tags attach. Our results show that the increasing topological complexity of the innovation channels contributes to network’s architecture over different time periods, and is consistent with temporal correlations of the occurrence of new tags. The methodology applies to a wide class of data with the suitable temporal resolution and clearly identified knowledge-content units. [ABSTRACT FROM AUTHOR]

Given the amount of knowledge and data accruing in the neurosciences, is it time to formulate a general principle for neuronal dynamics that holds at evolutionary, developmental, and perceptual timescales? In this paper, we propose that the brain (and other self-organised biological systems) can be characterised via the mathematical apparatus of a gauge theory. The picture that emerges from this approach suggests that any biological system (from a neuron to an organism) can be cast as resolving uncertainty about its external milieu, either by changing its internal states or its relationship to the environment. Using formal arguments, we show that a gauge theory for neuronal dynamics—based on approximate Bayesian inference—has the potential to shed new light on phenomena that have thus far eluded a formal description, such as attention and the link between action and perception. [ABSTRACT FROM AUTHOR]

The automatic detection and tracking of human eyes and, in particular, the precise localization of their centers (pupils), is a widely debated topic in the international scientific community. In fact, the extracted information can be effectively used in a large number of applications ranging from advanced interfaces to biometrics and including also the estimation of the gaze direction, the control of human attention and the early screening of neurological pathologies. Independently of the application domain, the detection and tracking of the eye centers are, currently, performed mainly using invasive devices. Cheaper and more versatile systems have been only recently introduced: they make use of image processing techniques working on periocular patches which can be specifically acquired or preliminarily cropped from facial images. In the latter cases the involved algorithms must work even in cases of non-ideal acquiring conditions (e.g in presence of noise, low spatial resolution, non-uniform lighting conditions, etc.) and without user's awareness (thus with possible variations of the eye in scale, rotation and/or translation). Getting satisfying results in pupils' localization in such a challenging operating conditions is still an open scientific topic in Computer Vision. Actually, the most performing solutions in the literature are, unfortunately, based on supervised machine learning algorithms which require initial sessions to set the working parameters and to train the embedded learning models of the eye: this way, experienced operators have to work on the system each time it is moved from an operational context to another. It follows that the use of unsupervised approaches is more and more desirable but, unfortunately, their performances are not still satisfactory and more investigations are required. To this end, this paper proposes a new unsupervised approach to automatically detect the center of the eye: its algorithmic core is a representation of the eye's shape that is obtained through a differential analysis of image intensities and the subsequent combination with the local variability of the appearance represented by self-similarity coefficients. The experimental evidence of the effectiveness of the method was demonstrated on challenging databases containing facial images. Moreover, its capabilities to accurately detect the centers of the eyes were also favourably compared with those of the leading state-of-the-art methods. [ABSTRACT FROM AUTHOR]