Richard L. Bishop

A tumor promoter stimulates phosphorylation on tyrosine

The tumor promoter 12-O-tetradecanoyl-phorbol-13-acetate is mitogenic for normal chicken embryo fibroblasts and also causes these cells to express transiently many properties of cells transformed by Rous sarcoma virus. Since some mitogenic hormones stimulate a tyrosine-specific protein kinase activity, and since the transforming protein of RSV is a tyrosine-specific protein kinase, we have examined whether TPA also stimulates protein phosphorylation on tyrosine. We report here that TPA treatment of normal cells resulted in a very rapid phosphorylation on tyrosine of a protein peak of Mr 40 to 43 kilodaltons. Thus, a similar biochemical activity (tyrosine phosphorylation) is associated with the action of polypeptide mitogenic hormones, Rous sarcoma virus and a tumor promoter. In addition, TPA treatment resulted in rapid changes in phosphorylation of proteins on serine and threonine.

Geometric curvature bounds in Riemannian manifolds with boundary

An Alexandrov upper bound on curvature for a Riemannian manifold with boundary is proved to be the same as an upper bound on sectional curvature of interior sections and of sections of the boundary which bend away from the interior. As corollaries those same sectional curvatures are related to estimates for convexity and conjugate radii; the Hadamard-Cartan theorem and Yaus isoperimetric inequality for spaces with negative curvature are generalized

Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above

Comparison and rigidity theorems are proved for curves of bounded geodesic curvature in singular spaes of curvature bounded above. Most of these estimates do not appear in the literature even for smooth curves in Riemannian manifolds. Geodesic curvature (which agrees with the usual one in the smooth case) is defined by comparison to curves of constant curvature in a model space. Two methods of comparison are used, preserving either sidelengths of inscribed triangles or arclength and chordlength. Using a majorization theorem of Reshetnyak, we obtain best possible global comparisons for arclength, chordlength, width and base angles in a CAT(K) space. A criterion for a metric ball to be a CAT(K) space is also given, in terms of the radius and the radial uniqueness property.

Capture pursuit games on unbounded domains

We introduce simple tools from geometric convexity to analyze capturetype (or “Lion and Man”) pursuit problems in unbounded domains. The main result is a necessary and sufficient condition for eventual capture in equal-speed discrete-time multi-pursuer capture games on convex Euclidean domains of arbitrary dimension and shape. This condition is presented in terms of recession sets in unit tangent spheres. The chief difficulties lie in utilizing the boundary of the domain as a constraint on the evader’s escape route. We also show that these convex-geometric techniques provide sufficient criteria for pursuit problems in non-convex domains with a convex decomposition.

The Fary-Milnor theorem in Hadamard manifolds

The Fary-Milnor theorem is generalized: Let γ be a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. If γ has total curvature less than or equal to 4π, then γ is the boundary of an embedded disk. The example of a trefoil knot which moves back and forth abritrarily close to a geodesic segment shows that the bound 4π is sharp in any such space. The original theorem was for closed curves in Euclidean 3-space and the proof by integral geometry did not apply to spaces of variable curvature. Now, instead, a combinatorial proof has been devised.

Spines and topology of thin Riemannian manifolds

Consider Riemannian manifolds M for which the sectional curvature of M and second fundamental form of the boundary B are bounded above by one in absolute value. Previously we proved that if M has sufficiently small inradius (i.e. all points are sufficiently close to the boundary), then the cut locus of B exhibits canonical branching behavior of arbitrarily low branching number. In particular, if M is thin in the sense that its inradius is less than a certain universal constant (known to lie between.108 and.203), then M collapses to a triply branched simple polyhedral spine. We use a graphical representation of the stratification structure of such a collapse, and relate numerical invariants of the graph to topological invariants of M when B is simply connected. In particular, the number of connected strata of the cut locus is a topological invariant. When M is 3-dimensional and compact, M has complexity 0 in the sense of Matveev, and is a connected sum of p copies of the real projective space P 3 , t copies chosen from the lens spaces L(3,±1), and l handles chosen from S 2 x S 1 or S 2 XS 1 , with β 3-balls removed, where p + t + l + β > 2. Moreover, we construct a thin metric for every graph, and hence for every homeomorphism type on the list.

Racetracks and Extermal Problems for Curves of Bounded Curvature in Metric Spaces

For a closed curve in a CAT(K) space with given circumradius and upper bound on curvature, a basic lower bound on the length is established. The inequality is sharp, assumed only when the curve is the boundary of an isometric copy of a racetrack (the convex hull of two congruent circles) from a plane of constant curvature K. Previously such a theorem was proved for Euclidean plane curves by G.D.Chakerian, H.H. Johnson, and A. Vogt, and for curves in higher dimensional Euclidean spaces by A.D. Milka. A similar theorem is proved for nonclosed curves, with a notion of breadth replacing circumradius. Thus we illustrate how singular methods can extend classical Euclidean theorems to a large class of new spaces (including Riemannian manifolds of curvature bounded above) and also give significant strengthenings even in Euclidean space.