We propose to incorporate a weighted difference of anisotropic and isotropic total variation (TV) norms into a relaxed formulation of the two phase Mumford-Shah (MS) model for image segmentation. We show results exceeding those obtained by the MS model when using the standard TV norm to regularize partition boundaries. In particular, examples illustrating the qualitative differences between the proposed model and the standard MS one are shown. A fast numerical method is introduced to minimize the proposed model utilizing the difference-of-convex algorithm (DCA) and the primal dual hybrid gradient (PDHG) method.

Let <i>T</i> be a compact disjointness preserving linear operator from <i>C</i>₀(<i>X</i>) into <i>C</i>₀(<i>Y</i>), where <i>X</i> and <i>Y</i> are locally compact Hausdorff spaces. We show that <i>T</i> can be represented as a norm convergent countable sum of disjoint rank one operators. More precisely, <i>T</i> = _<i>n</i> <i></i> <i>h_n</i> for a (possibly finite) sequence {<i>x_n</i> }_<i>n</i> of distinct points in <i>X</i> and a norm null sequence {<i>h_n</i> }_<i>n</i> of mutually disjoint functions in <i>C</i>₀(<i>Y</i>). Moreover, we develop a graph theoretic method to describe the spectrum of such an operator ( 2009 WILEYVCH Verlag GmbH &amp; Co. KGaA, Weinheim)

We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions with<i>SU</i>(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

The Feynman maps introduced first by Truman are examined. The domain considered here consists of the Fresnel-integrable functions in the sense of Albeverio and Hoegh-Krohn; extensions to wider classes of functions will be studied in the following paper. The original definition of the F-maps is slightly modified: we start from the underlying measures on the Hilbert space of paths in order to avoid use of improper integrals. Some new properties of the F-maps are derived. In particular, the dominated convergence theorem is shown to be not valid for the F₁-map (or Feynman integral); this fact is of a certain importance for the classical limit of quantum mechanics.

Let M be a compact Criemannian manifold of nonpositive curvature1 and with fundamental group . It is well known [8, p. 102] that M is a K (, 1) and thus completely determined up to homotopy type by . In light of this fact it is natural to ask to what extent the riemannian structure of M is determined by the structure of , and the intent of this paper is to demonstrate that rather strong implications of this sort exist. In the case that M has strictly negative curvature, the group is known to be highly noncommutative. Every abelian, in fact, every solvable, subgroup of is cyclic [3]. It is therefore a plausible conjecture that in the nonpositive curvature case, will possess large amounts of commutativity only under special geometric circumstances. We shall show that this is true, that indeed those properties of which involve commutativity have a dramatic reflection in the riemannian structure of M.