# lemniscate

(redirected from Lemniscates)

## lemniscate

(ˈlɛmnɪskɪt)
n
(Mathematics) a closed plane curve consisting of two symmetrical loops meeting at a node. Equation: (x2 + y2)2 = a2(x2y2), where a is the greatest distance from the curve to the origin. The symbol for infinity (∞) is an example
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The most experienced transition curves among these recommended curves in the open literature are clot-hoid, lemniscates, cubic parabola, cubic spiral sinusoidal, bloss and 2R radius circles (Can et al.
Denoting the foci of the lemniscates [GAMMA][delta] (5.
dagger]) Lemniscates of polynomials of degree 2 are also known as Ovals of Cassini.
0], whereas for d > d0 the lemniscate is a single analytic Jordan curve.
because otherwise the lemniscate passing through ([[lambda].
1) are lemniscates of order 1, while the Brauer ovals of Cassini [K.
This brings us to the penetrating work of Brualdi [2], which shows how the union of higher-order lemniscates for a general matrix A (these lemniscates not necessarily being of the same order) can give compact sets in the complex plane C which contain _(A), thereby circumventing the counterexample of (2.
7) the quantity B(A), which we call the Brualdi lemniscate set for A, as it is the union of lemniscates (see (2.
11] that a boundary point z of B(F) can be an eigenvalue of F only if z is a boundary point of each of the lemniscates [B.
Here, we investigate the relationships between G(A), the union of Gersgorin disks, K(A), the union of Brauer ovals of Cassini, and B(A), the union of Brualdi lemniscate sets, for eigenvalue inclusions of an n x n complex matrix A.
24697, the associated Brualdi set, consisting of the union of two Brualdi lemniscate sets, is such that the boundary of each Brualdi lemniscate passes through [z.
19806, the associated Brualdi set has its two lemniscate sets passing through [z.

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