Zeros and poles

In complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is holomorphic (i.e. complex differentiable) in some neighbourhood of z0. A function f is meromorphic in an open set U if for every point z of U there is a neighborhood of z in which either f or 1/f is holomorphic. If f is meromorphic in U, then a zero of f is a pole of 1/f, and a pole of f is a zero of 1/f. This induces a duality between zeros and poles, that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then

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In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation

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Research on modal vibration suppression of maglev steel strip based on phase compensator

Yuanping Xu

In the process of vibration suppression of hot galvanized steel strip by using maglev technology, the flexible modal vibration control of steel strip is a challenging problem. Aiming at this problem, this paper proposes a phase compensation control algorithm to control the modal vibration in the maglev steel strip vibration suppression system. The modal vibration suppression process is as follows. First, the model of the steel strip and the mode shape and frequency response curve of the strip are established by using the finite element analysis software. Second, the electromagnet-steel strip model is simplified as a single degree of freedom maglev system. Then, the equivalent damping is obtained according to the theory. And the vibration is suppressed according to the principle of the phase compensator. In addition, the effect of the phase lead angle, compensating field and scale factor on the damping performance of the phase compensator from the zero-pole of the closed-loop system is studied. Finally, the experimental study is carried out on the vibration suppression system of maglev steel strip based on phase compensator. The experimental results show that the phase compensator with appropriate parameters can be selected to reduce the amplitude of the first order bending mode by 89.2%. (c) 2018 Elsevier Ltd. All rights reserved.

2018

Precision electroweak measurements on the Z resonance☆☆☆

Georgios Anagnostou, Aurelio Bay, François Bianchi, Ivan Bozovic, Simone Campana, Francesco Cerutti, Peter Clarke, Greig Alan Cowan, Hans Dijkstra, Javier García Hernández, Csaba Hajdu, Peter Hansen, Shan Jin, Donghyun Kim, Ji Hyun Kim, Doohyun Kim, Zhen Liu, Werner Lustermann, Bénédicte Marie Maréchal, Peter Molnar, Raluca Anca Muresan, Niko Neufeld, Darren Robinson, Paolo Ronchese, Stefano Rosati, André Rouge, Eduardo Sanchez, Francisco Sanchez, Olivier Schneider, Sourav Sen, Varun Sharma, Pushpendra Singh, Michael Strauss, Liang Sun, Frédéric Teubert, Rémy Teuscher, Mark Tobin, Andromachi Tsirou, Horst Vogel, Qian Wang, Xiao Wang, Zheng Wang, Jian Wang, Matthias Weber, John Wilson, Songmei Wu, Zhirui Xu, Yong Yang, Igor Zacharov, Lei Zhang, Jian Zhao

We report on the final electroweak measurements performed with data taken at the Z resonance by the experiments operating at the electron-positron colliders SLC and LEP. The data consist of 17 million Z decays accumulated by the ALEPH, DELPHI, L3 and OPAL experiments at LEP, and 600 thousand Z decays by the SLD experiment using a polarised beam at SLC. The measurements include cross-sections, forward-backward asymmetries and polarised asymmetries. The mass and width of the Z boson, mZ Z, and ΓZ, its couplings to fermions, for example the ρ parameter and the efffective electroweak mixing angle for leptons, are precisely measured: mZ = 91.1875 ± 0.0021 GeV, ΓZ = 2.4952 ± 0.0023 GeV, ρℓ = 1.0050 ± 0.0010, sin2 θeff lept = 0.23153 ± 0.00016. The number of light neutrino species is determined to be 2.9840 ± 0.0082, in agreement with the three observed generations of fundamental fermions. The results are compared to the predictions of the Standard Model (SM). At the Z-pole, electroweak radiative corrections beyond the running of the QED and QCD coupling constants are observed with a significance of five standard deviations, and in agreement with the Standard Model. Of the many Z-pole measurements, the forward-backward asymmetry in b-quark production shows the largest difference with respect to its SM expectation, at the level of 2.8 standard deviations. Through radiative corrections evaluated in the framework of the Standard Model, the Z-pole data are also used to predict the mass of the top quark, mt = 173-10 +13 GeV, and the mass of the W boson, mW = 80.363 ± 0.032 GeV. These indirect constraints are compared to the direct measurements, providing a stringent test of the SM. Using in addition the direct measurements of mt and mW, the mass of the as yet unobserved SM Higgs boson is predicted with a relative uncertainty of about 50% and found to be less than 285 GeV at 95% confidence level. © 2006 Elsevier B.V. All rights reserved.

2006

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In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers