Oxidation of an oil rich in docosahexaenoic acid compared to linoleic acid in lactating women

Background: We studied the oxidation of an oil rich in docosahexaenoic acid (DHA; DHASCO(R)) in lactating mothers receiving a dietary DHA supplement or a placebo. The results were compared with the oxidation of linoleic acid. Methods: Breast-feeding mothers received a dietary supplement (DHASCO; 200 mg DHA/day, n = 5) or a placebo (n = 5) for 14 days. Six weeks post partum all 10 mothers received a single dose of 2 mg/kg body weight uniformly C-13-labeled DHASCO. In a previously reported study 6 mothers received 1 mg/kg body weight uniformly C-13-labeled linoleic acid. Breath samples were collected over 48 h after tracer application. The total CO2 production was measured by indirect calorimetry and the C-13 isotopic enrichment of labeled CO2 by isotopic ratio mass spectrometry. Results: The oxidation of C-13-labeled DHASCO in the supplemented and placebo groups was similar. Maximal C-13 enrichment was reached earlier in the group receiving C-13-DHASCO (median 1.0 vs. 3.0 h in the linoleic acid group). The cumulative C-13 recovery in breath was higher in the DHASCO versus the linoleic acid group until 10 h after tracer application and comparable thereafter. Conclusions: The difference in oxidation of DHASCO versus linoleic acid after tracer ingestion might be partly due to a faster absorption and oxidation of shorter chain saturated fatty acids contained in DHASCO. The cumulative oxidation of DHASCO and linoleic acid 24 and 48 h after tracer ingestion is similar. Copyright (C) 2000 S. Karger AG, Basel

Intersection and mixing times for reversible chains

© 2017, University of Washington. All rights reserved. We consider two independent Markov chains on the same finite state space, and study their intersection time, which is the first time that the trajectories of the two chains intersect. We denote by tI the expectation of the intersection time, maximized over the starting states of the two chains. We show that, for any reversible and lazy chain, the total variation mixing time is O(tI). When the chain is reversible and transitive, we give an expression for tI using the eigenvalues of the transition matrix. In this case, we also show that tI is of order √nE[I], where I is the number of intersections of the trajectories of the two chains up to the uniform mixing time, and n is the number of states. For random walks on trees, we show that tI and the total variation mixing time are of the same order. Finally, for random walks on regular expanders, we show that tI is of order √n

Choice and bias in random walks

We analyse the following random walk process inspired by the power-of-two-choice paradigm: starting from a given vertex, at each step, unlike the simple random walk (SRW) that always moves to a randomly chosen neighbour, we have the choice between two uniformly and independently chosen neighbours. We call this process the choice random walk (CRW). We first prove that for any graph, there is a strategy for the CRW that visits any given vertex in expected tim

Bounds on the satisfiability threshold for power law distributed random SAT

Propositional satisfiability (SAT) is one of the most fundamental problems in computer science. The worst-case hardness of SAT lies at the core of computational complexity theory. The averagecase analysis of SAT has triggered the development of sophisticated rigorous and non-rigorous techniques for analyzing random structures. Despite a long line of research and substantial progress, nearly all theoretical work on random SAT assumes a uniform distribution on the variables. In contrast, real-world instances often exhibit large fluctuations in variable occurrence. This can be modeled by a scale-free distribution of the variables, which results in distributions closer to industrial SAT instances. We study random k-SAT on n variables, m = ϵ(n) clauses, and a power law distribution on the variable occurrences with exponent β. We observe a satisfiability threshold at β≤(2k-1)/(k-1). This threshold is tight in the sense that instances with β ≥ (2k-1)/(k-1)-ϵ for any constant ϵ > 0 are unsatisfiable with high probability (w. h. p.). For β > (2k-1)/(k-1)+ ϵ, the picture is reminiscent of the uniform case: instances are satisfiable w. h. p. for sufficiently small constant clause-variable ratios m/n; they are unsatisfiable above a ratio m/n that depends on β

Faster rumor spreading with multiple calls

We consider the random phone call model introduced by Demers et al., which is a well-studied model for information dissemination in networks. One basic protocol in this model is the so-called Push protocol that proceeds in synchronous rounds. Starting with a single node which knows of a rumor, every informed node calls in each round a random neighbor and informs it of the rumor. The Push-Pull protocol works similarly, but additionally every uninformed node calls a random neighbor and may learn the rumor from it. It is well-known that both protocols need Θ(log n) rounds to spread a rumor on a complete network with n nodes. Here we are interested in how much the spread can be speeded up by enabling nodes to make more than one call in each round. We propose a new model where the number of calls of a node is chosen independently according to a probability distribution R. We provide both lower and upper bounds on the rumor spreading time depending on statistical properties of R such as the mean or the variance (if they exist). In particular, if R follows a power law distribution with exponent β ∈ (2, 3), we show that the Push-Pull protocol spreads a rumor in Θ(log log n) rounds. Moreover, when β = 3, the Push- Pull protocol spreads a rumor in Θ(formula presented) rounds

Randomized rumor spreading in dynamic graphs

International audienceWe consider the well-studied rumor spreading model in which nodes contact a random neighbor in each round in order to push or pull the rumor. Unlike most previous works which focus on static topologies, we look at a dynamic graph model where an adversary is allowed to rewire the connections between vertices before each round, giving rise to a sequence of graphs, G1, G2, . . . Our first result is a bound on the rumor spreading time in terms of the conductance of those graphs. We show that if the degree of each node does not change much during the protocol (that is, by at most a constant factor), then the spread completes within t rounds for some t such that the sum of conductances of the graphs G1 up to Gt is O(log n). This result holds even against an adaptive adversary whose decisions in a round may depend on the set of informed vertices before the round, and implies the known tight bound with conductance for static graphs. Next we show that for the alternative expansion measure of vertex expansion, the situation is different. An adaptive adversary can delay the spread of rumor significantly even if graphs are regular and have high expansion, unlike in the static graph case where high expansion is known to guarantee fast rumor spreading. However, if the adversary is oblivious, i.e., the graph sequence is decided before the protocol begins, then we show that a bound close to the one for the static case holds for any sequence of regular graphs

Multiple random walks on paths and grids

We derive several new results on multiple random walks on "low dimensional" graphs. First, inspired by an example of a weighted random walk on a path of three vertices given by Efremenko and Reingold, we prove the following dichotomy: as the path length n tends to infinity, we have a super-linear speed-up w.r.t. the cover time if and only if the number of walks k is equal to 2. An important ingredient of our proofs is the use of a continuous-time analogue of multiple random walks, which might be of independent interest. Finally, we also present the first tight bounds on the speed-up of the cover time for any d-dimensional grid with d >= 2 being an arbitrary constant, and reveal a sharp transition between linear and logarithmic speed-up

Balls into bins via local search: Cover time and maximum load

© 2015 Wiley Periodicals, Inc. Abstract-We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m=n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is vertex transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m≥n.ETH Zurich Postdoctoral Fellowship Program Marie Curie Career Integration. Grant Number: PCIG13‐GA‐2013‐618588 DSRELI

Balls into bins via local search: Cover time and maximum load

We study a natural process for allocating m balls into n bins that are organized as the vertices of an undirected graph G. Balls arrive one at a time. When a ball arrives, it first chooses a vertex u in G uniformly at random. Then the ball performs a local search in G starting from u until it reaches a vertex with local minimum load, where the ball is finally placed on. Then the next ball arrives and this procedure is repeated. For the case m = n, we give an upper bound for the maximum load on graphs with bounded degrees. We also propose the study of the cover time of this process, which is defined as the smallest m so that every bin has at least one ball allocated to it. We establish an upper bound for the cover time on graphs with bounded degrees. Our bounds for the maximum load and the cover time are tight when the graph is transitive or sufficiently homogeneous. We also give upper bounds for the maximum load when m > n.Comment: arXiv admin note: text overlap with arXiv:1207.212