# winding number

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## winding number

[′wīnd·iŋ ‚nəm·bər]
(mathematics)
The number of times a given closed curve winds in the counterclockwise direction about a designated point in the plane. Also known as index.
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coil] is the winding height; w is the winding number of turns; [[mu].
0] is the winding power losses; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] c are the inductances of the current limiter of the first and the second stages; w is the winding number of turns; [a.
8), if let it be mero-morphic and C be a simple closed contour which does not pass through any poles or zeros of f(x), then according to the argument principle, the winding number is equal to the difference between the number of zeros and poles of fx) that are inside the contour.
8) in the effective resonance region equals to the winding number of the determinant value of the characteristic polynomial on the right hand of the original point.
Winding Around: The Winding Number in Topology, Geometry, and Analysis
Based on lecture notes for advanced undergraduate topics courses taught at Penn State's Mathematics Advanced Study Semesters program in the fall of 2013, this book explains the winding number in topology, geometry, and analysis.
Winding 'n'"--where n represents the winding number is the subsystem obtained from the model presented in figure 5.
The resonance band is the banded region associated with the same rational winding number [omega] that is the frequency ratio of the perturbed system.
Working from rigorous theorems and proofs, and offering a broad array of examples and applications he covers point set topology, combinatorial topology, differential topology, geometric topology and algebraic topology in chapters on continuity, compactness and connectedness, manifolds and complexes, homotopy and the winding number, fundamental group, and homology.
He felt pulled in multiple directions, unable, even helpless, to resist, drawn as if by wind or water flowing through a disquieting sphere towards a belief as powerful and true as some mathematical proof arising from some barely sentient terrain, a belief that somehow a system based on an orderly construct consistent with his experience--mathematics and his tours meshed and unified--would become visible in the insistent recalling of figures appearing and reappearing in the swirling sand; apparitions or not they were there like fractals or winding numbers or multiple infinities on a plane where all lines always intersect.
He starts with Cauchy-Riemann equations in the introduction, then proceeds to power series, results on holomorphic functions, logarithms, winding numbers, Couchy's theorem, counting zeros and the open mapping theorem, Eulers formula for sin(z), inverses of holomorphic maps, conformal mappings, normal families and the Riemann mapping theorem, harmonic functions, simply connected open sets, Runge's theorem and the Mittag-Leffler theorem, the Weierstrass factorization theorem, Caratheodory's theorem, analytic continuation, orientation, the modular function, and the promised Picard theorems.
Topics include winding numbers for networks with weak angular data, diffusion over tensor fields via Lie group PDE flows, homotopy meaningful hybrid model structures, cohomology rings of tree braid groups, geometric descriptions of polygon and chain spaces, and symmetric motion planning.

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