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Conclusion and outlooks
Published in Dževad Belkić, Karen Belkić, Signal Processing in Magnetic Resonance Spectroscopy with Biomedical Applications, 2010
General rational functions R(u) are defined as quotients R(u) = f(u)/g(u) of two other functions f(u) and g(u) of a complex-valued independent variable u. They play by far the most prominent role in the mathematical theory of approximations. Importantly, it is this latter practical theory by which mathematics make their most significant and useful bridges towards other disciplines across different research fields. The key mathematical features that determine any function used in mathematical modeling across inter-disciplinary applications are the possible singularities (poles, cuts, branch points) and zeros. The former and the latter are tightly connected, respectively, with the potential existence of maximae (peaks) and minimae (valleys between adjacent peaks) of the given function. The principal reason for the central role of rational functions of the general type R(u) = f(u)/g(u) in the theory of approximations is in their mathematical form by which the numerator g(u) can provide adequate descriptions of singularities, whereas the denominator f(u) is suitable for description of zeros. Poles and zeros can fully describe any system.
Futures of digital public space
Published in Naomi Jacobs, Rachel Cooper, Living in Digital Worlds, 2018
The ‘technological singularity’ is an idea that was initially discussed in the 1950s. First coined by Jon von Neumann, it was described by Stanislaw Ulam as a point: ‘beyond which human affairs, as we know them, could not continue’ (Vinge, 1993). The name comes from a concept in physics, singularities being an infinitely compact point in space in which the laws of the universe cease to function, and the true nature of which cannot be described or understood; this is the hypothesised nature of a black hole. The technological singularity therefore, is a point in time where models of the future fail to give reliable answers. The idea of the technological singularity was popularised by Verner Vinge in the 1990s, and by Ray Kurzweil (2005) who linked it specifically to the idea of the emergence of superintelligence. This could be achieved by the creation of a ‘strong’ artificial intelligence smarter than humans in a range of different domains (rather than one specific task) possibly with the ability to make improvements to itself. General intelligence, which would be the necessary first step for this, is a goal which artificial intelligence researchers have sought for many years, but so far with limited success. An AI research boom in the 1970s led to predictions by leaders in the field that the development of general intelligence on a par with humans was imminent, however it became apparent soon afterwards that the leading research direction, rule based learning, was not going to achieve this. Current AI research is seeing a new resurgence based on ‘deep learning’ and neural networks, and some think it is more likely to be successful, while others predict a similar disappointment to the previous optimism (Adee, 2016). However, if an artificial intelligence model is successful, and achieves ‘human’ aspects such as creativity and innovation along with rapid processing speeds and vast data processing abilities, we may see it outstrip us as it learns to improve itself.
Global Lipschitz stability for an inverse source problem for the Navier–Stokes equations
Published in Applicable Analysis, 2023
Oleg Y. Imanuvilov, M. Yamamoto
Our key machinery is a Carleman type estimate. There are two kinds of Carleman estimates according to choices of the weight functions: Weight function without singularities in and : Weight function with singularities in t: where , , on
Identifying Stick-Slip Characteristics of a Smart Device on a Seismically Excited Surface Using On-Board Sensors
Published in Journal of Earthquake Engineering, 2022
Yunsu Na, Sherif El-Tawil, Ahmed Ibrahim, Ahmed Eltawil
In regions C and D in Fig. 4e, the transition between sliding and sticking is sharp in the computed acceleration data. Singularities (points at which a function does not possess a derivative) can be observed at these transition points. In signal and image processing field, these singularities are called edges and are the basis of commonly used edge detection methods as discussed earlier. In regions A and B in Fig. 4b, however, transitions between sliding and sticking motions in the experimental acceleration data are rounded. Also, the sliding regime has significant chattering and is not as smooth as the computed data, where the behavior is a plateau with a constant value equal to KCOF. The contrast can be seen in more detail in Fig. 4c,f. The rounded transitions and uneven sliding response in the experimental data hinder differentiating between sliding and sticking motions and complicate finding the KCOF. Given this difficulty, the remainder of the paper is devoted to finding means by which to estimate an appropriate KCOF value from measured data.
On singular solutions of time-periodic and steady Stokes problems in a power cusp domain
Published in Applicable Analysis, 2018
Alicija Eismontaite, Konstantin Pileckas
There is an extensive literature concerning the asymptotic behavior of solutions of different elliptic problems near cusp (peak)-type singularities of the boundary, see, e.g. [5–18]. Constructing the asymptotic representation of the solution to the problem (1.1) near the cusp point O, we follow the procedure proposed in the paper [19] where the asymptotic behavior of the solutions to stationary Stokes and Navier–Stokes problems was studied in unbounded domains with paraboloidal outlets to infinity. In turn, the method used in [19] is a variant of the algorithm of constructing the asymptotics for solutions to elliptic equations in slender domains, see, e.g. [20–23] for arbitrary elliptic problems, [24,25] for the stationary Stokes and Navier–Stokes equations, and [26–28] for the nonstationary and time-periodic Navier–Stokes equations.