The doodle of a finitely determined map germ from R(2) to R(3)

Let f : U subset of R(2) -> R(3) be a representative of a finitely determined map germ f : (R(2), 0) -> (R(3), 0). Consider the curve obtained as the intersection of the image of the mapping f with a sufficiently small sphere s(epsilon)(2) centered at the origin in R(3), call this curve the associated doodle of the map germ f. For a large class of map germs the associated doodle has many transversal self-intersections. The topological classification of such map germs is considered from the point of view of the associated doodles. (C) 2009 Elsevier Inc. All rights reserved.

Differential kinase requirement for enhancement of Fc gamma R-mediated phagocytosis in alveolar macrophages by leukotriene B(4) vs. D(4)

In alveolar macrophages, leukotriene (IT) B(4) and cysteinyl LTs (LTC(4), LTD(4) and LTE(4)) both enhance Fc gamma receptor (Fc gamma R)-mediated phagocytosis. In the present study we investigated the role of specific PKC isoforms (PKC-alpha and -delta), the MAP kinases p38 and ERK 1/2, and PI3K in mediating the potentiation of Fc gamma R-mediated phagocytosis induced by addition of leukotrienes to the AMs. It was found that exogenously added LTB(4) and LTD(4) both enhanced PKC-delta and -alpha phosphorylation during Fc gamma R engagement. Studies with isoform-selective inhibitors indicated that exogenous LTB(4) effects were dependent on both PKC-alpha and -delta, while LTD(4) effects were exclusively due to PKC-delta activation. Although both exogenous LTB(4) and LTD(4) enhanced p38 and ERK 1/2 activation, LTB(4) required only ERK 1/2, while LTD(4) required only p38 activation. Activation by both LTs was dependent on PI3K activation. Effects of endogenous LTs on kinase activation were also investigated using selective LT receptor antagonists. Endogenous LTB(4) contributed to Fc gamma R-mediated activation of PKC-alpha, ERK 1/2 and PI3K, while endogenous cysLTs contributes to activation of PKC-delta, p38 and PI3K. Taken together, our data show that the capacities of LTB(4) and LTD(4) to enhance Fc gamma R-mediated phagocytosis reflect their differential activation of specific kinase programs. (C) 2008 Elsevier Ltd. All rights reserved.

Foliations and polynomial diffeomorphisms of R(3)

Let Y = (f, g, h): R(3) -> R(3) be a C(2) map and let Spec(Y) denote the set of eigenvalues of the derivative DY(p), when p varies in R(3). We begin proving that if, for some epsilon > 0, Spec(Y) boolean AND (-epsilon, epsilon) = empty set, then the foliation F(k), with k is an element of {f, g, h}, made up by the level surfaces {k = constant}, consists just of planes. As a consequence, we prove a bijectivity result related to the three-dimensional case of Jelonek`s Jacobian Conjecture for polynomial maps of R(n).

M-deformations of A-simple germs from R(n) to R(n+1)

All A-simple corank-1 germs from R(n) to R(n+1), where n not equal 4, have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the image.

Real map germs and higher open book structures

The paper focusses on the existence of higher open book structures defined by real map germs psi : (R(m), 0) -> (R(p), 0) such that Sing psi boolean AND psi(-1)(0) subset of {0}. A general existence criterion is proved, with view to weighted-homogeneous maps.