Nonlinear Local Invertibility Preservers

AbstractA new criterion for local invertibility of slice regular quaternionic functions is obtained. This paper is motivated by the need to find a geometrical interpretation for analytic conditions on the real Jacobian associated with a slice regular function f. The criterion involves spherical and Cullen derivatives of f and gives rise to several geometric implications, including an application to related conformality properties.

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An Elementary Proof of Local Invertibility for Generalized and Attenuated Radon Transforms

SIAM Journal on Mathematical Analysis ◽

10.1137/0516082 ◽

1985 ◽

Vol 16 (5) ◽

pp. 1114-1119 ◽

Cited By ~ 20

Author(s):

Andrew Markoe ◽

Eric Todd Quinto

Keyword(s):

Elementary Proof ◽

Radon Transforms ◽

Local Invertibility

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Local invertibility in subrings of C*(X)

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700012119 ◽

1992 ◽

Vol 46 (3) ◽

pp. 449-458 ◽

Cited By ~ 2

Author(s):

H. Linda Byun ◽

Lothar Redlin ◽

Saleem Watson

Keyword(s):

Continuous Functions ◽

Closed Sets ◽

Complete Ring ◽

Maximal Ideals ◽

One To One ◽

Local Invertibility ◽

Bounded Continuous Functions ◽

Ring Of Functions ◽

Uniformly Closed

It is known that the maximal ideals in the rings C(X) and C*(X) of continuous and bounded continuous functions on X, respectively, are in one-to-one correspondence with βX. We have shown previously that the same is true for any ring A(X) between C(X) and C*(X). Here we consider the problem for rings A(X) contained in C*(X) which are complete rings of functions (that is, they contain the constants, separate points and closed sets, and are uniformly closed). For every noninvertible f ∈ A(X), we define a z–filter ZA(f) on X which, in a sense, provides a measure of where f is ‘locally invertible’. We show that the map ZA generates a correspondence between ideals of A(X) and z–filters on X. Using this correspondence, we construct a unique compactification of X for every complete ring of functions. Each such compactification is explicitly identified as a quotient of βX. In fact, every compactification of X arises from some complete ring of functions A(X) via this construction. We also describe the intersections of the free ideals and of the free maximal ideals in complete rings of functions.

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