Optimal transportation, topology and uniqueness

The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff [14]: give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure \gamma outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph / antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.Comment: 36 pages, 6 figure

Well-posedness of strong solutions for the Vlasov equation coupled to non-Newtonian fluids in dimension three

We consider the Cauchy problem for coupled system of Vlasov and non-Newtonian fluid equations. We establish local well--posedness of the strong solutions, provided that the initial data are regular enough. Global existence of unique strong solutions for any given time interval is shown as well if the initial data are sufficiently small.Comment: 29 page

Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems

Let M denote the space of probability measures on R^D endowed with the Wasserstein metric. A differential calculus for a certain class of absolutely continuous curves in M was introduced by Ambrosio, Gigli and Savare'. In this paper we develop a calculus for the corresponding class of differential forms on M. In particular we prove an analogue of Green's theorem for 1-forms and show that the corresponding first cohomology group, in the sense of de Rham, vanishes. For D=2d we then define a symplectic distribution on M in terms of this calculus, thus obtaining a rigorous framework for the notion of Hamiltonian systems as introduced by Ambrosio and Gangbo. Throughout the paper we emphasize the geometric viewpoint and the role played by certain diffeomorphism groups of R^D.Comment: Version 2. Improved presentation, slight technical changes. To appear in Memoirs AM

Generating and Adding Flows on Locally Complete Metric Spaces

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in \cite{BC}. In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.Comment: 29 pages,6 figure

Maysin and Its Flavonoid Derivative from Centipedegrass Attenuates Amyloid Plaques by Inducting Humoral Immune Response with Th2 Skewed Cytokine Response in the Tg (APPswe, PS1dE9) Alzheimer\u27s Mouse Model

Alzheimer\u27s disease (AD) is a slow, progressive neurodegenerative disease and the most common type of dementia in the elderly. The etiology of AD and its underlying mechanism are still not clear. In a previous study, we found that an ethyl acetate extract of Centipedegrass (CG) (i.e., EA-CG) contained 4 types of Maysin derivatives, including Luteolin, Isoorientin, Rhamnosylisoorientin, and Derhamnosylmaysin, and showed protective effects against Amyloid beta (Aβ) by inhibiting oligomeric Aβ in cellular and in vitro models. Here, we examined the preventative effects of EA-CG treatment on the Aβ burden in the Tg (Mo/Hu APPswe PS1dE9) AD mouse model. We have investigated the EA-CG efficacy as novel anti-AD likely preventing amyloid plaques using immunofluorescence staining to visually analyze Aβ40/42 and fibril formation with Thioflavin-S or 6E10 which are the profile of immunoreactivity against epitope Aβ1-16 or neuritic plaque, the quantitation of humoral immune response against Aβ, and the inflammatory cytokine responses (Th1 and Th2) using ELISA and QRT-PCR. To minimize the toxicity of the extracted CG, we addressed the liver toxicity in response to the CG extract treatment in Tg mice using relevant markers, such as aspartate aminotransferase (AST)/ alanine aminotransferase (ALT) measurements in serum. The EA-CG extract significantly reduced the Aβ burden, the concentration of soluble Aβ40/42 protein, and fibril formation in the hippocampus and cortex of the Tg mice treated with EA-CG (50 mg/kg BW/day) for 6 months compared with the Tg mice treated with a normal diet. Additionally, the profile of anti-inflammatory cytokines revealed that the levels of Th2 (interleukin-4 (IL-4) and interleukin-10 (IL-10)) cytokines are more significantly increased than Th1 (interferon-γ (IFN-γ), interleukin-2(IL-2)) in the sera. These results suggest that the EA-CG fraction induces IL-4/IL-10-dependent anti-inflammatory cytokines (Th2) rather than pro-inflammatory cytokines (Th1), which are driven by IL-2/IFN-γ. With regard to the immune response, EA-CG induced an immunoglobulin IgG and IgM response against the EA-CG treatment in the Tg mice. Furthermore, EA-CG significantly ameliorated the level of soluble Aβ42 and Aβ40. Similarly, we observed that the fibril formation was also decreased by EA-CG treatment in the hippocampus and cortex after quantitative analysis with Thioflavin-S staining in the Tg brain tissues. Taken together, our findings suggested that Maysin and its derivative flavonoid compounds in the EA-CG fraction might be beneficial therapeutic treatments or alternative preventative measures to adjuvant for boosting humoral and cellular include immune response and anti-inflammation which may lead to amyloid plaque accumulation in Alzheimer\u27s patients\u27 brains