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Register a new userAccelerated dynamics: Mathematical foundations and algorithmic improvements
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Authors
Tony Leli`evre
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We present a review of recent works on the mathematical analysis of algorithms which have been proposed by A.F. Voter and co-workers in the late nineties in order to efficiently generate long trajectories of metastable processes. These techniques have been successfully applied in many contexts, in particular in the field of materials science. The mathematical analysis we propose relies on the notion of quasi stationary distribution.
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One-shot anonymous unselfishness in economic games is commonly explained by social preferences, which assume that people care about the monetary payoffs of others. However, during the last ten years, research has shown that different types of unselfish behaviour, including cooperation, altruism, truth-telling, altruistic punishment, and trustworthiness are in fact better explained by preferences for following ones own personal norms - internal standards about what is right or wrong in a given situation. Beyond better organising various forms of unselfish behaviour, this moral preference hypothesis has recently also been used to increase charitable donations, simply by means of interventions that make the morality of an action salient. Here we review experimental and theoretical work dedicated to this rapidly growing field of research, and in doing so we outline mathematical foundations for moral preferences that can be used in future models to better understand selfless human actions and to adjust policies accordingly. These foundations can also be used by artificial intelligence to better navigate the complex landscape of human morality.
A geometric setup for control theory is presented. The argument is developed through the study of the extremals of action functionals defined on piecewise differentiable curves, in the presence of differentiable non-holonomic constraints. Special emphasis is put on the tensorial aspects of the theory. To start with, the kinematical foundations, culminating in the so called variational equation, are put on geometrical grounds, via the introduction of the concept of infinitesimal control . On the same basis, the usual classification of the extremals of a variational problem into normal and abnormal ones is also rationalized, showing the existence of a purely kinematical algorithm assigning to each admissible curve a corresponding abnormality index, defined in terms of a suitable linear map. The whole machinery is then applied to constrained variational calculus. The argument provides an interesting revisitation of Pontryagin maximum principle and of the Erdmann-Weierstrass corner conditions, as well as a proof of the classical Lagrange multipliers method and a local interpretation of Pontryagins equations as dynamical equations for a free (singular) Hamiltonian system. As a final, highly non-trivial topic, a sufficient condition for the existence of finite deformations with fixed endpoints is explicitly stated and proved.
I briefly present the foundations of relativistic cosmology, which are, General Relativity Theory and the Cosmological Principle. I discuss some relativistic models, namely, Einstein static universe and Friedmann universes. The classical bibliographic references for the relevant tensorial demonstrations are indicated whenever necessary, although the calculations themselves are not shown.
Symbiotic Autonomous Systems (SAS) are advanced intelligent and cognitive systems exhibiting autonomous collective intelligence enabled by coherent symbiosis of human-machine interactions in hybrid societies. Basic research in the emerging field of SAS has triggered advanced general AI technologies functioning without human intervention or hybrid symbiotic systems synergizing humans and intelligent machines into coherent cognitive systems. This work presents a theoretical framework of SAS underpinned by the latest advances in intelligence, cognition, computer, and system sciences. SAS are characterized by the composition of autonomous and symbiotic systems that adopt bio-brain-social-inspired and heterogeneously synergized structures and autonomous behaviors. This paper explores their cognitive and mathematical foundations. The challenge to seamless human-machine interactions in a hybrid environment is addressed. SAS-based collective intelligence is explored in order to augment human capability by autonomous machine intelligence towards the next generation of general AI, autonomous computers, and trustworthy mission-critical intelligent systems. Emerging paradigms and engineering applications of SAS are elaborated via an autonomous knowledge learning system that symbiotically works between humans and cognitive robots.
Many low-frequency radio interferometers are aiming to detect very faint spectral signatures from structures at cosmological redshifts, particularly of neutral Hydrogen using its characteristic 21 cm spectral line. Due to the very high dynamic range needed to isolate these faint spectral fluctuations from the very bright foregrounds, spectral systematics from the instrument or the analysis, rather than thermal noise, are currently limiting their sensitivity. Failure to achieve a spectral calibration with fractional inaccuracy $lesssim 10^{-5}$ will make the detection of the critical cosmic signal unlikely. The bispectrum phase from interferometric measurements is largely immune to this calibration issue. We present a basis to explore the nature of bispectrum phase in the limit of small spectral fluctuations. We establish that they measure the intrinsic dissimilarity in the transverse structure of the cosmic signal relative to the foregrounds, expressed as rotations in the underlying phase angle. Their magnitude is related to the strength of the cosmic signal relative to the foregrounds. Using a range of sky models, we detail the behavior of bispectrum phase fluctuations using standard Fourier-domain techniques and find it comparable to existing approaches, with a few key differences. Mode-mixed foreground contamination is more pronounced than in existing approaches because the bispectrum phase is a product of three individual interferometric phases. The multiplicative coupling of foregrounds in the bispectrum phase fluctuations results in the mixing of foreground signatures with that of the cosmic signal. We briefly outline a variation of this approach to avoid extensive mode-mixing. Despite its limitations, the interpretation of results using bispectrum phase is possible with forward-modeling. Importantly, it is an independent and a viable alternative to existing approaches.
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