On Approximating Node-Disjoint Paths in Grids

In the Node-Disjoint Paths (NDP) problem, the input is an undirected n-vertex graph G, and a collection {(s_1,t_1),...,(s_k,t_k)} of pairs of vertices called demand pairs. The goal is to route the largest possible number of the demand pairs (s_i,t_i), by selecting a path connecting each such pair, so that the resulting paths are node-disjoint. NDP is one of the most basic and extensively studied routing problems. Unfortunately, its approximability is far from being well-understood: the best current upper bound of O(sqrt(n)) is achieved via a simple greedy algorithm, while the best current lower bound on its approximability is Omega(log^{1/2-delta}(n)) for any constant delta. Even for seemingly simpler special cases, such as planar graphs, and even grid graphs, no better approximation algorithms are currently known. A major reason for this impasse is that the standard technique for designing approximation algorithms for routing problems is LP-rounding of the standard multicommodity flow relaxation of the problem, whose integrality gap for NDP is Omega(sqrt(n)) even on grid graphs. Our main result is an O(n^{1/4} * log(n))-approximation algorithm for NDP on grids. We distinguish between demand pairs with both vertices close to the grid boundary, and pairs where at least one of the two vertices is far from the grid boundary. Our algorithm shows that when all demand pairs are of the latter type, the integrality gap of the multicommodity flow LP-relaxation is at most O(n^{1/4} * log(n)), and we deal with demand pairs of the former type by other methods. We complement our upper bounds by proving that NDP is APX-hard on grid graphs

Improved Approximation for Node-Disjoint Paths in Grids with Sources on the Boundary

We study the classical Node-Disjoint Paths (NDP) problem: given an undirected n-vertex graph G, together with a set {(s_1,t_1),...,(s_k,t_k)} of pairs of its vertices, called source-destination, or demand pairs, find a maximum-cardinality set {P} of mutually node-disjoint paths that connect the demand pairs. The best current approximation for the problem is achieved by a simple greedy O(sqrt{n})-approximation algorithm. Until recently, the best negative result was an Omega(log^{1/2-epsilon}n)-hardness of approximation, for any fixed epsilon, under standard complexity assumptions. A special case of the problem, where the underlying graph is a grid, has been studied extensively. The best current approximation algorithm for this special case achieves an O~(n^{1/4})-approximation factor. On the negative side, a recent result by the authors shows that NDP is hard to approximate to within factor 2^{Omega(sqrt{log n})}, even if the underlying graph is a subgraph of a grid, and all source vertices lie on the grid boundary. In a very recent follow-up work, the authors further show that NDP in grid graphs is hard to approximate to within factor Omega(2^{log^{1-epsilon}n}) for any constant epsilon under standard complexity assumptions, and to within factor n^{Omega(1/(log log n)^2)} under randomized ETH. In this paper we study the NDP problem in grid graphs, where all source vertices {s_1,...,s_k} appear on the grid boundary. Our main result is an efficient randomized 2^{O(sqrt{log n}* log log n)}-approximation algorithm for this problem. Our result in a sense complements the 2^{Omega(sqrt{log n})}-hardness of approximation for sub-graphs of grids with sources lying on the grid boundary, and should be contrasted with the above-mentioned almost polynomial hardness of approximation of NDP in grid graphs (where the sources and the destinations may lie anywhere in the grid). Much of the work on approximation algorithms for NDP relies on the multicommodity flow relaxation of the problem, which is known to have an Omega(sqrt n) integrality gap, even in grid graphs, with all source and destination vertices lying on the grid boundary. Our work departs from this paradigm, and uses a (completely different) linear program only to select the pairs to be routed, while the routing itself is computed by other methods

Hyperbolic Kac-Moody Algebra from Intersecting p-branes

A subclass of recently discovered class of solutions in multidimensional gravity with intersecting p-branes related to Lie algebras and governed by a set of harmonic functions is considered. This subclass in case of three Euclidean p-branes (one electric and two magnetic) contains a cosmological-type solution (in 11-dimensional model with two 4-forms) related to hyperbolic Kac-Moody algebra ${\cal F}_3$ (of rank 3). This solution describes the non-Kasner power-law inflation.Comment: 15 pages, Latex. A talk presented at the Second Winter School on Branes, Fields and Math. Phys. (Seoul, Korea). Corrected version. Journ. ref.: J. Math. Phys., 40, (1999) 4072-4083; Corrigenda to appea

Sense of Purpose in Life and Five Health Behaviors in Older Adults

Accumulating evidence shows that a higher sense of purpose in life is associated with lower risk of chronic conditions and premature mortality. Health behaviors might partially explain these findings, however, the prospective association between sense of purpose and health behaviors is understudied. We tested whether a higher sense of purpose at baseline was associated with lower likelihood of developing unhealthy behaviors over time. Prospective data were from the Health and Retirement Study, a national sample of U.S. older adults. Our sample included 13,770 adults assessed up to five times across eight years. Among people who met recommended guidelines for a given health behavior outcome at baseline, those in the top versus lowest quartile of purpose in life had 24% lower likelihood of becoming physically inactive (95% CI: 0.68–0.85), 33% lower likelihood of developing sleep problems (95% CI: 0.58–0.79), and 22% lower likelihood of developing unhealthy body mass index (BMI) (95% CI: 0.69–0.87) in sociodemographic-adjusted models. Further there was a marginal reduction in smoking relapse (HR = 0.65, 95% CI: 0.41–1.03) and no association with heavy alcohol use (HR = 1.02, 95% CI: 0.81–1.29). Findings for physical inactivity, sleep problems, and unhealthy BMI remained evident after further adjusting for baseline health status and depression. Our results, suggest that a sense of purpose in life might emerge (with further research) as a valuable target to consider for interventions aimed at helping older adults maintain some health behaviors

Regulation of stanniocalcin-1 secretion by BeWo cells and first trimester human placental tissue from normal pregnancies and those at increased risk of developing preeclampsia.

Stanniocalcin-1 (STC-1) is a multi-functional glycosylated peptide present in the plasma of healthy women postpartum and increased further in pregnancies complicated by preeclampsia. Although the STC-1 gene is expressed by the placenta what regulates its secretion and from which cells at the feto-maternal interface is unknown. Here, we demonstrate for the first time that the syncytiotrophoblast and cytotrophoblast are a major site of STC-1 protein expression in first trimester placental tissue. Further, in response to low oxygen, first trimester chorionic villous tissue from pregnancies at increased risk of developing preeclampsia secreted significantly more STC-1 than normal tissue under the same conditions. Using the human trophoblast cell line BeWo we have shown that low oxygen increased the secretion of STC-1 but it required co-stimulation with the Adenosine-3', 5'-cyclic monophosphate (cAMP) analogue, 8-Bromo adenosine-3', 5'-cyclic monophosphate cAMP (8 Br-cAMP) to reach significance. Inhibition of Hypoxia inducible factor 2α (HIF-2α) and the Phosphatidylinositol-3 kinase (PI3 -Kinase)/AKT/Serum and glucocorticoid-induced kinase-1(SGK-1) pathway resulted in significant inhibition of STC-1 secretion. As both low oxygen and cAMP are known to play a central role in placental function, their regulation of STC-1 points to a potentially important role in the maintenance of a normal healthy pregnancy and we would hypothesize that it may act to protect against prolonged placental hypoxia seen in preeclampsia

Longitudinal Associations Between Psychological Well-Being and the Consumption of Fruits and Vegetables

Background: Psychological well-being is associated with longevity and reduced risk of disease, but possible mechanisms are understudied. Health behaviors like eating fruits and vegetables may link psychological well-being with better health; however, most evidence is cross-sectional. Purpose: This study investigated psychological well-being’s longitudinal association with fruit and vegetable consumption across as many as 7 years. Method: Participants were 6,565 older adults from the English Longitudinal Study of Ageing, which includes men and women aged 50 years or older. Psychological well-being was assessed with 17 items from the Control, Autonomy, Satisfaction, Pleasure Scale. Fruit and vegetable consumption was initially assessed during 2006–2007 and then approximately every 2 years through 2012–2013. Covariates included sociodemographic factors, health status, and other health behaviors. Results: Mixed linear models showed that higher baseline levels of psychological well-being were associated with more fruit and vegetable consumption at baseline (β = 0.05, 95% confidence interval [CI] [0.02, 0.08]) and that fruit and vegetable consumption declined across time (β = −0.01, 95% CI [−0.02, −0.004]). Psychological well-being interacted significantly with time such that individuals with higher baseline psychological well-being had slower declines in fruit and vegetable consumption (β = 0.01, 95% CI [0.01, 0.02]). Among individuals who initially met recommendations to consume 5 or more servings of fruits and vegetables (N = 1,719), higher baseline psychological well-being was associated with 11% reduced risk of falling below recommended levels during follow-up (hazard ratio = 0.89, 95% CI [0.83, 0.95]). Conclusions: Findings suggest that psychological well-being may be a precursor to healthy behaviors such as eating a diet rich in fruits and vegetables. (PsycInfo Database Record (c) 2020 APA, all rights reserved

Intrinsic Absorption in the Spectrum of NGC 7469: Simultaneous Chandra, FUSE, and STIS Observations

We present simultaneous X-ray, far-ultraviolet, and near-ultraviolet spectra of the Seyfert 1 galaxy NGC 7469 obtained with the Chandra X-Ray Observatory, the Far Ultraviolet Spectroscopic Explorer, and the Space Telescope Imaging Spectrograph on the Hubble Space Telescope. Previous non-simultaneous observations of this galaxy found two distinct UV absorption components, at -560 and -1900 km/s, with the former as the likely counterpart of the X-ray absorber. We confirm these two absorption components in our new UV observations, in which we detect prominent O VI, Ly alpha, N V, and C IV absorption. In our Chandra spectrum we detect O VIII emission, but no significant O VIII or O VII absorption. We also detect a prominent Fe K alpha emission line in the Chandra spectrum, as well as absorption due to hydrogen-like and helium-like neon, magnesium, and silicon at velocities consistent with the -560 km/s UV absorber. The FUSE and STIS data reveal that the H I and C IV column densities in this UV- and X-ray- absorbing component have increased over time, as the UV continuum flux decreased. We use measured H I, N V, C IV, and O VI column densities to model the photoionization state of both absorbers self-consistently. We confirm the general physical picture of the outflow in which the low velocity component is a highly ionized, high density absorber with a total column density of 10^20 cm^-2, located near the broad emission line region, although due to measurable columns of N V and C IV, we assign it a somewhat smaller ionization parameter than found previously, U~1. The high velocity UV component is of lower density, log N=18.6, and likely resides farther from the central engine as we find its ionization parameter to be U=0.08.Comment: Minor correction to abstract; STScI eprint #1683; 50 pages, incl. 19 figures, 4 tables; Accepted to Ap

Genuine Dyons in Born-Infeld Electrodynamics

Study of magnetic monopoles in the original version of Born-Infeld (BI) electrodynamics is performed. It then is realized that interesting new physics emerge and they include exotic behavior of radial electric monopole field such as its regularity as $r\to 0$ and its changing behavior with the absence or presence of the radial magnetic monopole field. This last point has been interpreted as the manifestation of the existence of point-like dyons in abelian BI theory. Two pieces of clear evidences in favor of this dyon interpretation are provided. It is also demonstrated that despite these unique features having no analogues in standard Maxwell theory, the cherished Dirac quantisation condition remains unchanged. Lastly, comments are given concerning that dyons found here in the original version of BI electrodynamics should be distinguished from the ones with the same name or BIons being studied in the recent literature on D-brane physics.Comment: 19 pages, Revtex, references added, no other change