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Published: 11 September 2022

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1

The Duffing equation: Nonlinear oscillators and their phenomena. Chichester, West Sussex, U.K: Wiley, 2011.

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2

Shu, Chi-Wang. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997.

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3

Pogorzelski, Ronald J. Coupled-oscillator based active-array antennas. Hoboken, New Jersey: John Wiley & Sons Inc., 2012.

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4

Brown, Catherine Alicia. Oscillatory behavior in an ocean general circulation model of the North Atlantic. Ottawa: National Library of Canada, 1999.

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5

Waywell, M. N. Predictions of wave and tidally induced oscillatory flows with Reynolds Stress Turbulence Models. Salford: University of Salford, 1995.

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6

Aldridge, J. N. Comparison of turbulence models for oscillatory rough turbulent boundary layer flows with suspended sediments. Salford: University of Salford Centre for Computational Fluid Dynamics and Turbulence, 1993.

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7

Osher, Stanley. Essentially non-oscillatory shock capturing methods applied to turbulence amplification in shock wave calculations. [Washington, DC]: National Aeronautics and Space Administration, 1989.

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8

Bauer, Christopher. Low Reynolds number [kappa]-[epsilon] and empirical transition models for oscillatory pipe flow and heat transfer. [Washington, D.C: National Aeronautics and Space Administration, 1993.

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9

Sansen, Willy. Analog Circuit Design: (X)DSL and other Communication Systems; RF MOST Models; Integrated Filters and Oscillators. Boston, MA: Springer US, 1999.

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10

Schütte, Christof. A quasiresonant smoothing algorithm for solving large highly oscillatory differential equations from quantum chemistry. Aachen: Verlag Shaker, 1994.

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11

McGinness, Ann M. Evaluation of shear and elongational flow regimes on the oscillatory rheological properties of a model of chocolate. Birmingham: University of Birmingham, 1996.

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12

Ganiev, Rivner Fazylovich. Nonlinear wave mechanics and technologies: Wave and oscillatory phenomena on the basis of high technologies. Redding, CT: Begell House, 2012.

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13

Rensing, Ludger. Temporal Disorder in Human Oscillatory Systems: Proceedings of an International Symposium University of Bremen, 8-13 September 1986. Berlin, Heidelberg: Springer Berlin Heidelberg, 1987.

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14

Yoshisuke, Ueda. The road to chaos. Santa Cruz, CA: Aerial Press, 1992.

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15

Ueda, Yoshisuke. The road to chaos. Santa Cruz, CA: Aerial Press, 1992.

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16

Yoshisuke, Ueda. The road to chaos-II. 2nd ed. Santa Cruz, CA: Aerial Press, 1992.

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17

Suárez, Almudena. Analysis and design of autonomous microwave circuits. Hoboken, NJ: Wiley, 2008.

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18

Kosmahl, Henry G. Slow wave vane structure with elliptical cross-section slots, an analysis. [Washington, DC]: National Aeronautics and Space Administration, 1994.

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19

Kosmahl, Henry G. Slow wave vane structure with elliptical cross-section slots, an analysis. [Washington, DC]: National Aeronautics and Space Administration, 1994.

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20

Mann, Peter. The Harmonic Oscillator. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0004.

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This chapter discusses the harmonic oscillator, which is a model ubiquitous to all branches of physics. The harmonic oscillator is a system with well-known solutions and has been fully investigated since it was first developed by Robert Hooke in the seventeenth century. These factors ensure that the harmonic oscillator is as relevant to a swinging pendulum as it is to a quantum field. Due to the importance of this model, the chapter investigates its dynamical properties, including the superposition principle in solutions, and construct a probability density function in a single dimension. The chapter also discusses Hooke’s law, modes and the Morse potential. In addition, in an exercise, the chapter introduces series solutions to ordinary differential equations.
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21

Imran, Jasim. Computation of oscillatory flow using two-equation turbulence models. 1993.

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22

Planat, Michel. Noise, Oscillators and Algebraic Randomness. Springer, 2000.

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23

Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997.

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24

Institute for Computer Applications in Science and Engineering., ed. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997.

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25

Institute for Computer Applications in Science and Engineering. and United States. National Aeronautics and Space Administration., eds. Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Hampton, VA: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997.

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26

Stanley, Osher, and Langley Research Center, eds. Efficient implementation of essentially non-oscillatory shock capturing schemes. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1987.

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27

Pogorzelski, Ronald J., and Apostolos Georgiadis. Coupled-Oscillator Based Active-Array Antennas. Wiley & Sons, Incorporated, John, 2012.

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28

Hassan, Ameer. On the periodic and chaotic responses of Duffing's oscillator. 1989.

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29

Center, Langley Research, ed. Analysis of wind tunnel oscillatory data of the X-31A aircraft. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1999.

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30

mathematician, Roux Bernard, and Gesellschaft für Angewandte Mathematik und Mechanik., eds. Numerical simulation of oscillatory convection in low-Pr fluids: A GAMM workshop. Braunschweig: Vieweg, 1990.

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31

W, Jerome Joseph, Osher Stanley, and Langley Research Center, eds. Solution of the hydrodynamic device model using high-order non-oscillatory shock capturing algorithms. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1989.

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32

Boudreau, Joseph F., and Eric S. Swanson. Nonlinear dynamics and chaos. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198708636.003.0013.

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Simple maps and dynamical systems are used to explore chaos in nature. The discussion starts with a review of the properties of nonlinear ordinary differential equations, including the useful concepts of phase portraits, fixed points, and limit cycles. These notions are developed further in an examination of iterative maps that reveal chaotic behavior. Next, the damped driven oscillator is used to illustrate the Lyapunov exponent that can be used to quantify chaos. The famous KAM theorem on the conditions under which chaotic behavior occurs in physical systems is also presented. The principle is illustrated with the Hénon-Heiles model of a star in a galactic environment and billiard models that describe the motion of balls in closed two-dimensional regions.
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33

Cruickshank, Steven. Mathematical models and anaesthesia. Edited by Jonathan G. Hardman. Oxford University Press, 2017. http://dx.doi.org/10.1093/med/9780199642045.003.0027.

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The use of mathematics in medicine is not as widespread as it might be. While professional engineers are instructed in a wide variety of mathematical techniques during their training in preparation for their daily practice, tradition and the demands of other subjects mean that doctors give little attention to numerical matters in their education. A smattering of statistical concepts is typically the main mathematical field that we apply to medicine. The concept of the mathematical model is important and indeed familiar; personal finance, route planning, home decorating, and domestic projects all require the application of the basic mathematical tools we acquire at school. This utility is why we learn them. The insight that can be gained by applying mathematics to physiological and other problems within medical practice is, however, underexploited. The undoubted complexity of human biology and pathology perhaps leads us to give up too soon. There are useful and practical lessons that can be learned from the use of elementary mathematics in medicine. Anaesthetic training in particular lends itself to such learning with its emphasis on physics and clinical measurement. Much can be achieved with simple linear functions and hyperbolas. Further exploration into exponential and sinusoidal functions, although a little more challenging, is well within our scope and enables us to cope with many time-dependent and oscillatory phenomena that are important in clinical anaesthetic practice. Some fundamental physiological relationships are explained in this chapter using elementary mathematical functions. Building further on the foundation of simple models to cope with more complexity enables us to see the process, examine the predictions, and, most importantly, assess the plausibility of these models in practice. Understanding the structure of the model enables intelligent interpretation of its output. Some may be inspired to investigate some of the mathematical concepts and their applications further. The rewards can be intellectually, aesthetically, and practically fruitful. The subtle, revelatory, and quite beautiful connection between exponential and trigonometric functions through the concept of complex numbers is one example. That this connection has widespread practical importance too is most pleasing.
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34

C, Sansen Willy M., Huijsing Johan H. 1938-, and Plassche, Rudy J. van de., eds. Analog circuit design: (X)DSL and other communication systems, RF MOST models, integrated filters and oscillators. Boston: Kluwer Academic Publishers, 1999.

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35

Low Reynolds number [kappa]-[epsilon] and empirical transition models for oscillatory pipe flow and heat transfer. [Washington, D.C: National Aeronautics and Space Administration, 1993.

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36

Willy M.C. Sansen (Editor), Johan H. Huijsing (Editor), and Rudy J. van de Plassche (Editor), eds. Analog Circuit Design: (X)DSL and other Communication Systems; RF MOST models; Integrated Filters and Oscillators. Springer, 1999.

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37

C, Sansen Willy M., Huijsing Johan H. 1938-, and Plassche, Rudy J. van de., eds. Analog circuit design: (X)DSL and other communication systems; RF MOST models; integrated filters and oscillators. Boston: Kluwer Academic Publishers, 1999.

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38

C, Wynne Eleanor, Mabey Dennis G, and Langley Research Center, eds. Calculation of transonic steady and oscillatory pressures on a low aspect ratio model and comparison with experiment. Hampton, Va: National Aeronautics and Space Administration, Langley Research Center, 1985.

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39

Fox, Grenville, Nicholas Hoque, and Timothy Watts. Respiratory support. Oxford University Press, 2017. http://dx.doi.org/10.1093/med/9780198703952.003.0008.

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This chapter includes sections on various modes of both invasive (i.e. via an endotracheal tube) and non-invasive respiratory support in neonates, including conventional ventilation, volume-targeted ventilation, high-frequency oscillatory ventilation (HFOV), extracorporeal membrane oxygenation (ECMO), nasal continuous positive airways pressure (nCPAP), nasal intermittent positive pressure ventilation (nIPPV), and high and low-flow nasal cannula oxygen. There is also a brief section on the care of babies with a tracheostomy as well as management of babies requiring home oxygen. Reference is made to the most recent European Consensus Guidelines. A separate chapter on neonatal respiratory problems (Chapter 7) gives further detail on common lung pathologies requiring respiratory support in neonates.
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40

Temporal Disorder in Human Oscillatory Systems: Proceedings of an International Symposium University of Bremen, 8-13 September 1986. Springer, 2012.

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41

Schroeder, Charles E., Jose L. Herrero, and Saskia Haegens. Neuronal Dynamics and the Mechanistic Bases of Selective Attention. Edited by Anna C. (Kia) Nobre and Sabine Kastner. Oxford University Press, 2014. http://dx.doi.org/10.1093/oxfordhb/9780199675111.013.031.

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Selective attention is a process by which the brain enhances its representation of task relevant, over irrelevant information. This ‘active control’ is essential to normal perception and cognition because it enables information processing to adapt to the immediate goals of the observer. This chapter places the focuses on recent conceptual/empirical developments in four areas that the authors think have significantly advanced the discussion and debate on the mechanistic underpinnings of selective attention: (1) the role of neuronal oscillations, (2) the distinctions between differing modes of dynamic operation, (3) potentially unique roles of specific oscillatory frequencies, (4) the neurochemistry of attention. The authors end by replacing attention within an ‘active sensing’ framework, and posing a set of prime questions for future study.
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42

Solymar, L., D. Walsh, and R. R. A. Syms. Lasers. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198829942.003.0012.

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Two-state and three-state systems are introduced. The properties of gaseous, solid state, and dye lasers are discussed and particular attention is devoted to semiconductor lasers. Reducing the dimensions leading to wells, wires, and dots is shown to have advantages. Quantum cascade lasers working in the THz region are discussed. The phenomena of Q switching, cavity dumping, and mode locking are explained. Parametric oscillators and optical fibre amplifiers are discussed. Masers are briefly mentioned. Laser noise is discussed. Awide variety of applications are mentioned. The curious phenomenon of laser cooling is explained. The basic principles of holographic recording and display are described.
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43

Ludger, Rensing, Heiden, Uwe an der, 1942-, and Mackey Michael C. 1942-, eds. Temporal disorder in human oscillatory systems: Proceedings of an international symposium, University of Bremen, 8-13 September 1986. Berlin: Springer-Verlag, 1987.

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44

Rensing, L. Temporal Disorder in Human Oscillatory Systems: Proceedings of an International Symposium University of Bremen, 8-13 Sept 1986 (Springer Series in Synergetics). Springer, 1987.

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45

1951-, Planat Michel, ed. Noise, oscillators, and algebraic randomness: From noise in communication systems to number theory : lectures of a school held in Chapelle des Bois, France, April 5-10, 1999. Berlin: Springer, 2000.

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46

Rau, Jochen. Simple Systems. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780199595068.003.0004.

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Even though the general framework of statistical mechanics is ultimately targeted at the description of macroscopic systems, it is illustrative to apply it first to some simple systems: a harmonic oscillator, a rotor, and a spin in a magnetic field. These applications serve to illustrate how a key function associated with the Gibbs state, the so-called partition function, is calculated in practice, how the entropy function is obtained via a Legendre transformation, and how such systems behave in the limits of high and low temperatures. After discussing these simple systems, this chapter considers a first example where multiple constituents are assembled into a macroscopic system: a basic model of a paramagnetic salt. It also investigates the size of energy fluctuations and how—in the case of the paramagnet—these fluctuations scale with the number of constituents.
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47

Holtmann, Martin, Björn Albrecht, and Daniel Brandeis. Neurofeedback. Oxford University Press, 2018. http://dx.doi.org/10.1093/med/9780198739258.003.0039.

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Neurofeedback of specific brain activity patterns allows perceiving and learning to gain control over these otherwise unaware neuronal processes. Neurofeedback may improve underlying neuronal deficits, and/or establish more general self-regulatory skills for compensating behavioural difficulties in other domains. Treating ADHD is the most common clinical neurofeedback application. Standard neurofeedback protocols based on electroencephalography train self-regulation of oscillatory activity in certain frequency bands (targeting theta/beta ratio) or slow cortical potential shifts. Both protocols have demonstrated promising outcomes, particularly in improving inattention symptoms, although controlled effects remain heterogeneous and often attenuated in blinded ratings. Further randomized controlled and (as far as possible) blinded evaluation studies are needed for better understanding of the mode of action and to establish robust standard training protocols for routine care. In the current state of evidence, neurofeedback can be recommended as part of a multimodal treatment of ADHD.
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48

(Firm), Aerojet, and United States. National Aeronautics and Space Administration., eds. Integrated Advanced Microwave Sounding Unit-A (AMSU-A), performance verification report, METSAT Phase Locked Oscillator assembly, P/N 1348360-1, S/N's F07 and F08: Contract no. NAS 5-32314. [Washington, DC: National Aeronautics and Space Administration, 1998.

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49

(Firm), Aerojet, and United States. National Aeronautics and Space Administration., eds. Integrated Advanced Microwave Sounding Unit-A (AMSU-A), performance verification report, METSAT Phase Locked Oscillator assembly, P/N 1348360-1, S/N's F07 and F08: Contract no. NAS 5-32314. [Washington, DC: National Aeronautics and Space Administration, 1998.

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50

Cuartero, Mireia, and Niall D. Ferguson. High-frequency ventilation and oscillation. Oxford University Press, 2016. http://dx.doi.org/10.1093/med/9780199600830.003.0098.

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High-frequency oscillatory ventilation (HFOV) is a key member of the family of modes called high-frequency ventilation and achieves adequate alveolar ventilation despite using very low tidal volumes, often below the dead space volume, at frequencies significantly above normal physiological values. It has been proposed as a potential protective ventilatory strategy, delivering minimal alveolar tidal stretch, while also providing continuous lung recruitment. HFOV has been successfully used in neonatal and paediatric intensive care units over the last 25 years. Since the late 1990s adults with acute respiratory distress syndrome have been treated using HFOV. In adults, several observational studies have shown improved oxygenation in patients with refractory hypoxaemia when HFOV was used as rescue therapy. Several small older trials had also suggested a mortality benefit with HFOV, but two recent randomized control trials in adults with ARDS have shed new light on this area. These trials not show benefit, and in one of them a suggestion of harm was seen with increased mortality for HFOV compared with protective conventional mechanical ventilation strategies (tidal volume target 6 mL/kg with higher positive end-expiratory pressure). While these findings do not necessarily apply to patients with severe hypoxaemia failing conventional ventilation, they increase uncertainty about the role of HFOV even in these patients.
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