Mean curvature flow in a Ricci flow background

Following work of Ecker, we consider a weighted Gibbons-Hawking-York functional on a Riemannian manifold-with-boundary. We compute its variational properties and its time derivative under Perelman's modified Ricci flow. The answer has a boundary term which involves an extension of Hamilton's Harnack expression for the mean curvature flow in Euclidean space. We also derive the evolution equations for the second fundamental form and the mean curvature, under a mean curvature flow in a Ricci flow background. In the case of a gradient Ricci soliton background, we discuss mean curvature solitons and Huisken monotonicity.Comment: final versio

On Minimum Violations Ranking in Paired Comparisons

Ranking a set of objects from the most dominant one to the least, based on the results of paired comparisons, proves to be useful in many contexts. Using the rankings of teams or individuals players in sports to seed tournaments is an example. The quality of a ranking is often evaluated by the number of violations, cases in which an object is ranked lower than another that it has dominated in a comparison, that it contains. A minimum violations ranking (MVR) method, as its name suggests, searches specifically for rankings that have the minimum possible number of violations which may or may not be zero. In this paper, we present a method based on statistical physics that overcomes conceptual and practical difficulties faced by earlier studies of the problem.Comment: 10 pages, 10 figures; typos corrected (v2

Upper bounds on the first eigenvalue for a diffusion operator via Bakry-\'{E}mery Ricci curvature II

Let $L=\Delta-\nabla\varphi\cdot\nabla$ be a symmetric diffusion operator with an invariant measure $d\mu=e^{-\varphi}dx$ on a complete Riemannian manifold. In this paper we prove Li-Yau gradient estimates for weighted elliptic equations on the complete manifold with $|\nabla \varphi|\leq\theta$ and $\infty$-dimensional Bakry-\'{E}mery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator $L$ on this kind manifold, and thereby generalize a Cheng's result on the Laplacian case (Math. Z., 143 (1975) 289-297).Comment: Final version. The original proof of Theorem 2.1 using Li-Yau gradient estimate method has been moved to the appendix. The new proof is simple and direc

Folding and unfolding phylogenetic trees and networks

Phylogenetic networks are rooted, labelled directed acyclic graphs which are commonly used to represent reticulate evolution. There is a close relationship between phylogenetic networks and multi-labelled trees (MUL-trees). Indeed, any phylogenetic network $N$ can be "unfolded" to obtain a MUL-tree $U(N)$ and, conversely, a MUL-tree $T$ can in certain circumstances be "folded" to obtain a phylogenetic network $F(T)$ that exhibits $T$. In this paper, we study properties of the operations $U$ and $F$ in more detail. In particular, we introduce the class of stable networks, phylogenetic networks $N$ for which $F(U(N))$ is isomorphic to $N$, characterise such networks, and show that they are related to the well-known class of tree-sibling networks.We also explore how the concept of displaying a tree in a network $N$ can be related to displaying the tree in the MUL-tree $U(N)$. To do this, we develop a phylogenetic analogue of graph fibrations. This allows us to view $U(N)$ as the analogue of the universal cover of a digraph, and to establish a close connection between displaying trees in $U(N)$ and reconcilingphylogenetic trees with networks

A Renormalization Group Approach to Relativistic Cosmology

We discuss the averaging hypothesis tacitly assumed in standard cosmology. Our approach is implemented in a "3+1" formalism and invokes the coarse graining arguments, provided and supported by the real-space Renormalization Group (RG) methods. Block variables are introduced and the recursion relations written down explicitly enabling us to characterize the corresponding RG flow. To leading order, the RG flow is provided by the Ricci-Hamilton equations studied in connection with the geometry of three-manifolds. The properties of the Ricci-Hamilton flow make it possible to study a critical behaviour of cosmological models. This criticality is discussed and it is argued that it may be related to the formation of sheet-like structures in the universe. We provide an explicit expression for the renormalized Hubble constant and for the scale dependence of the matter distribution. It is shown that the Hubble constant is affected by non-trivial scale dependent shear terms, while the spatial anisotropy of the metric influences significantly the scale-dependence of the matter distribution.Comment: 57 pages, LaTeX, 15 pictures available on request from the Author

Angular momentum sharing in dissipative collisions

Light charged particles emitted by the projectile-like fragment were measured in the direct and reverse collision of $^{93}$Nb and $^{116}$Sn at 25 AMeV. The experimental multiplicities of Hydrogen and Helium particles as a function of the primary mass of the emitting fragment show evidence for a correlation with net mass transfer. The ratio of Hydrogen and Helium multiplicities points to a dependence of the angular momentum sharing on the net mass transfer.Comment: 8 pages, 2 figure

Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds

Measure contraction property is one of the possible generalizations of Ricci curvature bound to more general metric measure spaces. In this paper, we discover sufficient conditions for a three dimensional contact subriemannian manifold to satisfy this property.Comment: 49 page

Retinal vessel diameters and reactivity in diabetes mellitus and/or cardiovascular disease

Background: Retinal vessel calibre and vascular dilation/constriction in response to flicker light provocation may provide a measure distinguishing patients suffering from diabetes mellitus and/or cardiovascular disease. Methods: One hundred and sixteen age and sex matched patients with diabetes mellitus (DM), cardiovascular disease (CVD) and both DM and CVD (DM+CVD) underwent systemic and intraocular pressure measurements. Retinal vessel calibres were assessed using a validated computer-based program to compute central retinal artery and vein equivalents (CRVE) from monochromatic retinal images. Vessel dilation and constriction responses to flicker light provocation were assessed by continuous retinal vessel diameter recordings. Plasma endothelial markers von Willebrand factor (vWf) and soluble E selectin (sEsel) were measured by ELISA. Results: Retinal vessel calibres were comparable across groups but CRVE correlated significantly with disease duration in DM patients (r=0.57, p<0.001). Patients suffering DM only exhibited reduced arterial vasomotion at rest and reduced arterial constriction following flicker light induced vessel dilation compared to patients with CVD and those suffering both CVD+DM (p=0.030). Patients suffering from CVD+DM exhibited significant differences between each flicker cycle in regards to arterial maximum constriction (p=0.006) and time needed to reach arterial maximum dilation (p=0.004), whereas the other two groups did not show such inconsistencies between individual flicker cycles. vWf was raised in CVD+DM compared to the other two groups (p≤0.02), whilst sEsel was raised in CVD+DM compared to DM alone (p=0.044). Conclusions: Dynamic retinal vascular calibres as obtained by continuous diameter measurements using flicker light provocation can reveal subtle differences between groups suffering from CVD with and without DM. This difference in reaction pattern and lack of arterial constriction in DM may provide a suitable marker to monitor progression

Hemangioma in a pulmonary hilar lymph node: Case report

<p>Abstract</p> <p>Background</p> <p>Different types of vascular proliferation may occur in lymph nodes, but hemangiomas in lymph nodes are extremely rare.</p> <p>Case Presentation</p> <p>A 73-year-old man was found to have a 15-mm nodular shadow in the left lung on computed tomography, and bronchoscopic brush cytology yielded a diagnosis of squamous cell carcinoma. Chest computed tomography showed no evidence of hilar or mediastinal lymphadenopathy. Left lower lobectomy with hilar and mediastinal lymph node dissection was performed. Postoperative histopathological examination revealed squamous cell carcinoma and no lymph node metastasis. On the other hand, a lobar bronchial lymph node presented a small lesion showing the dense proliferation of capillary blood vessels with elastic change. Immunohistochemically, the lesion was positive for factor VIII and CD34, leading to a diagnosis of primary hemangioma of the lymph node.</p> <p>Conclusion</p> <p>To our knowledge, this is the first case reported in the literature of hemangioma in a pulmonary hilar lymph node. Intranodal hemangioma needs to be differentiated from malignant vascular tumors.</p

An intermediate-depth source of hydrothermal 3He and dissolved iron in the North Pacific

We observed large water column anomalies in helium isotopes and trace metal concentrations above the Loihi Seamount. The 3He/4He of the added helium was 27.3 times the atmospheric ratio, clearly marking its origin to a primitive mantle plume. The dissolved iron to 3He ratio (dFe:3He) exported to surrounding waters was 9.3 ± 0.3 × 106. We observed the Loihi 3He and dFe “signal” at a depth of 1100 m at several stations within ∼100 – 1000 km of Loihi, which exhibited a distal dFe:3He ratio of ∼4 × 106, about half the proximal ratio. These ratios were remarkably similar to those observed over and near the Southern East Pacific Rise (SEPR) despite greatly contrasting geochemical and volcanictectonic origins. In contrast, the proximal and distal dMn:3He ratios were both ∼ 1 × 106, less than half of that observed at the SEPR. Dissolved methane was minimally enriched in waters above Loihi Seamount and was distally absent. Using an idealized regional-scale model we replicated the historically observed regional 3He distribution, requiring a hydrothermal 3He source from Loihi of 10.4 ± 4.2 mola−1, ∼2% of the global abyssal hydrothermal 3He flux. From this we compute a corresponding dFe flux of ∼40 Mmola−1. Global circulation model simulations suggest that the Loihi-influenced waters eventually upwell along the west coast of North America, also extending into the shallow northwest Pacific, making it a possibly important determinant of marine primary production in the subpolar North Pacific