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Magnetic Skyrmions on Discrete Lattices
Published in Evgeny Y. Tsymbal, Igor Žutić, Spintronics Handbook: Spin Transport and Magnetism, Second Edition, 2019
Elena Y. Vedmedenko, Roland Wiesendanger
The magnetization defines a topological space. Once a topological space has been defined, one can define a path through it as a mapping f(ρ,φ) of the real-line segment [(ρ1,φ1),(ρ2,φ2)] ((ρ1,φ1)<(ρ,φ)<(ρ2,φ2)) to the topological space. If (ρ1,φ1)=(ρ2,φ2), the path becomes closed. If the space has holes in it, these loops can be divided into classes, each one characterized by the number of times the loop winds round the hole. This quantity is known as winding number. The winding number in all cases of Figure 10.1 is unity, because θ(ρ,φ) and ψ(ρ,φ) change only once by 2π in one circuit along any contour enclosing the core [26]. Such configurations are important, because they cannot disappear by any continuous deformation of the order parameter (in our case, the magnetization) because of the singularity at the origin. In contrast, if such a loop does not enclose a core, it can disappear and is topologically trivial with a winding number equal to zero.
A reliable graphical criterion for TDS stability analysis
Published in International Journal of Systems Science, 2020
Tiao Yang Cai, Hui Long Jin, Xiang Peng Xie
As is indicated in argument principle that, the number of zeros of in the interior of Γ, is equal to the winding number of the imaginary curve of with respect to the origin, Division of by an auxiliary polynomial, will determine a curve whose rotations will be given by: And is chosen to be a nth order polynomials with n known roots with negative real parts as in (4), thus we have Therefore, this choice of gives the result that the zeros of inside Γ equals to the number of clockwise encirclements of the origin by the locus of the function for λ varies along the Jordan curve Γ.
Invariant output feedback stabilisability: the scalar case
Published in International Journal of Control, 2022
Aristotelis Yannakoudakis, Michael Sfakiotakis
The Gain plot , unlike the Nyquist plot , is not a closed curve. As a consequence, the winding number is not defined mathematically. However, the function is well-defined. Following the approach of the proof of Theorem 3.2, we have: where m is the degree of the numerator, r is the relative degree, and is the number of unstable zeros. The above expression indicates that the winding number w is a non-integer number if the relative degree is odd. Recalling Equation (31), we notice that for small frequencies is essentially equal to , while for large frequencies, it is essentially equal to . But is a polynomial of degree r, and its is equal to . To the open curve representing the plot of , we can assign the winding number (recall the Michailov stability criterion Netushil 1978). We, therefore, claim that it is also possible to assign a non-integer winding number to . Moreover, the difference of the above winding numbers from the one around the origin gives:
Using design to develop an in-depth understanding of mathematical surfaces: The hairy Klein bottle
Published in International Journal of Mathematical Education in Science and Technology, 2021
The winding number of a closed curve in the plane around a given point is an integer representing the total number of times the curve travels counterclockwise around the point. The winding number depends on the orientation of the curve and is negative if the curve travels around the point clockwise (Figure 3).