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Assume that <i>F</i> is a nonlinear operator on a real Hilbert space <i>H</i> which is -strongly monotone and -Lipschitzian on a nonempty closed convex subset <i>C</i> of <i>H</i>. Assume also that <i>C</i> is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on <i>H</i>. We construct an iterative algorithm with variable parameters which generates a sequence {<i>x</i> _<i>n</i>} from an arbitrary initial point <i>x</i> ₀ <i>H</i>. The sequence {<i>x</i> _<i>n</i>} is shown to converge in norm to the unique solution <i>u</i>  of the variational inequality

In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective benchmarks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.

In the paper, we consider a continuation approach for the binary quadratic program (BQP) based on a class of NCP-functions. More specifically, we recast the BQP as an equivalent minimization and then seeks its global minimizer via a global continuation method. Such approach had been considered in [11] which is based on the FischerBurmeister function. We investigate this continuation approach again by using a more general function, called the generalized FischerBurmeister function. However, the theoretical background for such extension can not be easily carried over. Indeed, it needs some subtle analysis.

Kronheimer-Mrowka recently proved that the Dehn twist along a 3-sphere in the neck of K3#K3 is not smoothly isotopic to the identity. This provides a new example of self-diffeomorphisms on 4-manifolds that are isotopic to the identity in the topological category but not smoothly so. (The first such examples were given by Ruberman.) In this paper, we use the Pin(2)-equivariant Bauer-Furuta invariant to show that this Dehn twist is not smoothly isotopic to the identity even after a single stabilization (connected summing with the identity map on S2×S2). This gives the first example of exotic phenomena on simply connected smooth 4-manifolds that do not disappear after a single stabilization.