where [psi] is the winding flux linkage; In is the rated current; [r.sub.mid] is the average winding radius; b is the winding width; [h.sub.coil] is the winding height; w is the winding number
of turns; [[mu].sub.a] is the absolute magnetic permeability .
(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
"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.
Here, for simplicity, we only consider those configurations for which the winding number
is equal to the unity (n = 1), although it is also possible to find solutions with higher vorticity, as it will be explained below.
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.
Solitons from different sectors have unequal winding numbers
and the fluctuation spectrum changes significantly from one sector to another.
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.