Jordan arc

Jordan arc

[zhȯr′dän‚ärk]
(mathematics)
References in periodicals archive ?
(a) [[GAMMA].sub.1] is an analytic Jordan arc connecting the two points [w.sub.1] and [w.sub.4] in {Re(w) > 0}, and it satisfies the assumptions made in Subsection 2.2.
The two domains [D.sup.(1).sub.0] and [D.sup.(1).sub.1] are separated by the Jordan arc [[GAMMA].sub.11] = [[GAMMA].sub.1], the two domains [D.sup.(1).sub.0] and [D.sup.(1).sub.2] by the Jordan arc [[GAMMA].sub.12], and the two domains [D.sup.(1).sub.0] and [D.sup.(1).sub.2] by the Jordan arc [[GAMMA].sub.13].
(i) There exist uniquely two analytic Jordan arcs [[GAMMA].sub.-1] and [GAMMA].sub.1] such that the two function elements [[??].sub.-1] and [[??].sub.1] defined in (2.21) have harmonic continuations throughout the domains C/[[GAMMA].sub.-1] and C/[[GAMMA].sub.-1], respectively, and the extended functions are continuous throughout C.
The continuum [K.sub.0] is the union of five analytic Jordan arcs [[GAMMA].sub.00], ..., [[GAMMA].sub.04], and it connects all four points [w.sub.1], ..., [w.sub.4].
(1) With the determination of the two Jordan arcs [[GAMMA].sub.1] and [[GAMMA].sub.-1] in part (i) of the Lemma 2.4 the shape of the three sheets [B.sub.-1], [B.sub.0], [B.sub.1] of the surface R is finally fixed.
There exist uniquely two analytic Jordan arcs [[GAMMA].sub.-1,2], [[GAMMA].sub.1,2], and a set [K.sub.1] such that
are two analytic Jordan arcs, each connecting the two points [z.sup.+.sub.[lambda]] and [z.sup.+.sub.[lambda]] in D, and further that
consists of two disjoint Jordan arcs each connecting one of the two points [z.sup.+.sub.[lambda]] and [z.sup.-.sub.[lambda]] with infinity.
There exists a system [GAMMA] of analytic Jordan arcs such that [h.sub.max] is harmonic in C\[GAMMA], but not harmonic in any neighborhood of a point z [member of] [GAMMA].
It is immediate from what has been said so far that the system [GAMMA] of analytic Jordan arcs is contained in the larger system of intersection arcs
In close analogy to Subsection 5.2, we define two systems [GAMMA] and [GAMMA] of Jordan arcs with the help of the function h from Subsection 6.1.
As before, the Riemann surface R can be broken down in three sheets [S.sub.-1], [S.sub.0], [S.sub.1], which are glued together along two Jordan arcs [[GAMMA].sub.1] and [[GAMMA].sub.-1] in the same way as this has been described after (5.10).