On the Rational Real Jacobian Conjecture

Jacobian conjectures (that nonsingular implies a global inverse) for rational everywhere defined maps of real n-space to itself are considered, with no requirement for a constant Jacobian determinant or a rational inverse. The birational case is proved and the Galois case clarified. Two known special cases of the Strong Real Jacobian Conjecture (SRJC) are generalized to the rational map context. For an invertible map, the associated extension of rational function fields must be of odd degree and must have no nontrivial automorphisms. That disqualifies the Pinchuk counter examples to the SRJC as candidates for invertibility.Comment: 7 pages. arXiv admin note: substantial text overlap with arXiv:1202.294

Reduction Theorems for the Strong Real Jacobian Conjecture

Implementations of known reductions of the Strong Real Jacobian Conjecture (SRJC), to the case of an identity map plus cubic homogeneous or cubic linear terms, and to the case of gradient maps, are shown to preserve significant algebraic and geometric properties of the maps involved. That permits the separate formulation and reduction, though not so far the solution, of the SRJC for classes of nonsingular polynomial endomorphisms of real n-space that exclude the Pinchuk counterexamples to the SRJC, for instance those that induce rational function field extensions of a given fixed odd degree.Comment: 9 page

Who Counsels Parents of Newborns Who Are Carriers of Sickle Cell Anemia or Cystic Fibrosis?

Our objective was to describe: 1) physicians’ knowledge of whether genetic counseling is provided to parents of newborns with sickle cell trait (SCT) or who are cystic fibrosis carriers (CFC), and 2) the prevalence of genetic counseling provided by primary care physicians. We conducted a cross‐sectional descriptive survey of 600 randomly‐sampled Michigan‐based pediatricians and family physicians, assessing physician knowledge of where and whether genetic counseling is received by parents whose newborns are carriers. Chi‐squared testing determined associations between genetic counseling location and physician demographic characteristics. Our response rate was 62 %: 298 (84 %) provided infant well care (183 pediatricians, 115 family physicians). Most respondents were non‐Hispanic White (65 %). Virtually all physicians believed parents whose newborns are carriers of either SCT or CFC should receive some genetic counseling (from the physician and/or another source), yet 20 % reported that parents of newborns with SCT did not receive counseling. Parents of infants with CFC received more counseling overall (92 % vs. 80 %; p < 0.01) and were counseled more frequently by genetic counselors or specialty centers than parents of newborns with SCT (85 % vs. 60 %; p < 0.01). Although physicians agreed that parents whose newborns are carriers should receive genetic counseling, fewer parents of newborns with SCT than with CFC received counseling from any source. This finding strongly suggests the need for further education and investigation of this apparent health disparity.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/147004/1/jgc40218.pd