In this paper, we show how under the continuum hypothesis one can obtain an integral representation for elements of the topological dual of the space of functions of bounded variation in terms of Lebesgue and Kolmogorov-Burkill integrals.

Inflammatory response following central nervous system (CNS) injury contributes to progressive neuropathology and reduction in functional recovery. Axons are sensitive to mechanical injury and toxic inflammatory mediators, which may lead to demyelination. Although it is well documented that degenerated myelin triggers undesirable inflammatory responses in autoimmune diseases such as multiple sclerosis (MS) and its animal model, experimental autoimmune encephalomyelitis (EAE), there has been very little study of the direct inflammatory consequences of damaged myelin in spinal cord injury (SCI), i.e., there is no direct evidence to show that myelin debris from injured spinal cord can trigger undesirable inflammation in vitro and in vivo. Our data showed that myelin can initiate inflammatory responses in vivo, which is complement receptor 3 (CR3)-dependent via stimulating macrophages to express pro-inflammatory molecules and down-regulates expression of anti-inflammatory cytokines. Mechanism study revealed that myelin-increased cytokine expression is through activation of FAK/PI3K/Akt/NF-B signaling pathways and CR3 contributes to myelin-induced PI3K/Akt/NF-B activation and cytokine production. The myelin induced inflammatory response is myelin specific as sphingomyelin (the major lipid of myelin) and myelin basic protein (MBP, one of the major proteins of myelin) are not able to activate NF-B signaling pathway. In conclusion, our results demonstrate a crucial role of myelin as an endogenous inflammatory stimulus that induces pro-inflammatory responses and suggest that blocking myelin-CR3 interaction and

We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When n=3, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.

We introduce and study an approximate solution of the p-Laplace equation, and a linearlization L_{ε} of a perturbed p-Laplace operator. By deriving an L_{ε}-type Bochner's formula and Kato type inequalities, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Liouville type theorem for strongly p-harmonic functions with finite q-energy on Riemannian manifolds. As an application, we extend this theorem to some p-harmonic maps such as p-harmonic morphisms and conformal maps between Riemannian manifolds. In particular, we obtain a Picard-type Theorem for p-harmonic morphisms.

The main purpose of this paper is to prove a general result about the geometry of holomorphic line bundles over Khler manifolds. This result is essentially a partial verification of a conjecture of Tian [30] and Tian has, over many years, highlighted the importance of the question for the existence theory of KhlerEinstein metrics. We will begin by stating this main result.