Rational Samuelson maps are univalent

Campbell, L. Andrew

doi:10.1016/0022-4049(94)90096-5

Cited by 8 publications

(9 citation statements)

References 16 publications

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“…This will be the case for instance when each component f i of F is the quotient of two polynomials in n variables. In that case, a result of Campbell [2] leads to a stronger conclusion than Theorem 5 1). According to [2], if d k (a) > 0 for all k = 1, .…”

Section: The Algebraic Casementioning

confidence: 90%

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Graphic Requirements for Multistationarity

Soulé¹

2003

ComPlexUs

207

183

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We discuss properties which must be satisfied by a genetic network in order for it to allow differentiation. These conditions are expressed as follows in mathematical terms. Let F be a differentiable mapping from a finite dimensional real vector space to itself. The signs of the entries of the Jacobian matrix of F at a given point a define an interaction graph, i.e. a finite oriented finite graph G(a) where each edge is equipped with a sign. René Thomas conjectured 20 years ago that if F has at least two nondegenerate zeroes, there exists a such that G(a) contains a positive circuit. Different authors proved this in special cases, and we give here a general proof of the conjecture. In particular, in this way we get a necessary condition for genetic networks to lead to multistationarity, and therefore to differentiation. We use for our proof the mathematical literature on global univalence, and we show how to derive from it several variants of Thomas rule, some of which had been anticipated by Kaufman and Thomas.

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Section: The Algebraic Casementioning

confidence: 90%

“…In that case, a result of Campbell [2] leads to a stronger conclusion than Theorem 5 1). According to [2], if d k (a) > 0 for all k = 1, . .…”

Section: The Algebraic Casementioning

confidence: 90%

Graphic Requirements for Multistationarity

Soulé¹

2003

ComPlexUs

207

183

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“…The proof [5] is a variant, using bounds rather than rationality, of the analogous result for (real) rational maps [4]. Actually, more is established.…”

Section: Pivotsmentioning

confidence: 99%

“…Namely, if one considers the pivots in their original order, rather than as arranged in nonascending order of size, then if the first n − 1 of them are bounded away from 0 the map is injective, and if the last one is as well, then the map is a homeomorphism and hence has a global inverse. The conclusion can also be strengthened; the map can be factored as the composition of n C 1 maps, each of which changes only a single coordinate; see [4] for details.…”

Section: Pivotsmentioning

confidence: 99%

“…Pinchuk constructs a class of polynomial maps of R 2 to itself which have a nowhere vanishing (but nonconstant) Jacobian determinant, and which are not injective. There are also plentiful examples of polynomial Samuelson maps with nonconstant Jacobian determinant that are (necessarily, by [4]) bijective, but whose inverses are not polynomial; for instance, the map F (x) = x + x 3 from R to itself.…”

Section: Polynomial Mapsmentioning

confidence: 99%

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Characteristic values of the Jacobian matrix and global invertibility

Campbell

2001

Ann. Polon. Math.

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Characteristic matrix values (singular values, eigenvalues, and pivots arising from Gaussian elimination) for the Jacobian matrix and its inverse are considered for maps of real n-space to itself with a nowhere vanishing Jacobian determinant. Bounds on these are related to global invertibility of the map. Polynomial maps with a constant nonzero Jacobian determinant are a special case that allows for sharper characterizations.

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