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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References in periodicals archive ?
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.
Since the distribution of the winding number [omega] of the small perturbed system is slightly different from the frequency-ratio distribution [Omega] of unperturbed system, one can expect the perturbed system to have a resonance band of period three and the Poincar[acute{e}] sections will indicate three islands of stability (unmixed zones, or dead spots) neat the [Omega] = 1/3 orbit.
10, the winding number associated with the resonance band will be [omega] m/n with m = 1 and n = 3.
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.