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Princeton, United States

Patel A.,Princeton
Discrete and Computational Geometry | Year: 2016

The Reeb graph is a construction which originated in Morse theory to study a real-valued function defined on a topological space. More recently, it has been used in various applications to study noisy data which creates a desire to define a measure of similarity between these structures. Here, we exploit the fact that the category of Reeb graphs is equivalent to the category of a particular class of cosheaf. Using this equivalency, we can define an ‘interleaving’ distance between Reeb graphs which is stable under the perturbation of a function. Along the way, we obtain a natural construction for smoothing a Reeb graph to reduce its topological complexity. The smoothed Reeb graph can be constructed in polynomial time. © 2016 Springer Science+Business Media New York Source

Fox J.,Princeton | Sudakov B.,University of California at Los Angeles
Combinatorics Probability and Computing | Year: 2010

We investigate decompositions of a graph into a small number of low-diameter subgraphs. Let P(n, ∈, d) be the smallest k such that every graph G = (V, E) on n vertices has an edge partition E = E0 ∪ E1 ∪ ⋯ ∪ Ek such that |E0| ≤ ∈n2, and for all 1 ≤ i ≤ k the diameter of the subgraph spanned by Ei is at most d. Using Szemerédi's regularity lemma, Polcyn and Ruciski showed that P(n, ∈, 4) is bounded above by a constant depending only on . This shows that every dense graph can be partitioned into a small number of small worlds provided that a few edges can be ignored. Improving on their result, we determine P(n, ∈, d) within an absolute constant factor, showing that P(n, ∈, 2) = Θ(n) is unbounded for ∈ < 1/4, P(n, ∈, 3) = Θl(1/∈2) for ∈ > n-1/2 and P(n, ∈, 4) = Θ(1/∈) for ∈ > n1. We also prove that if G has large minimum degree, all the edges of G can be covered by a small number of low-diameter subgraphs. Finally, we extend some of these results to hypergraphs, improving earlier work of Polcyn, Rödl, Ruciński and Szemerédi. © 2010 Cambridge University Press. Source

Fox J.,Princeton | Pach J.,City College of New York
Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms | Year: 2011

Computing the maximum number of disjoint elements in a collection C of geometric objects is a classical problem in computational geometry with applications ranging from frequency assignment in cellular networks to map labeling in computational cartography. The problem is equivalent to finding the independence number, α(Gc), of the intersection graph Gc of C, obtained by connecting two elements of C with an edge if and only if their intersection is nonempty. This is known to be an NP-hard task even for systems of segments in the plane with at most two different slopes. The best known polynomial time approximation algorithm for systems of arbitrary segments is due to Agarwal and Mustafa, and returns in the worst case an n 1/2+o(1)- approximation for α. Using extensions of the Lipton-Tarjan separator theorem, we improve this result and present, for every ε > 0, a polynomial time algorithm for computing α(Gc) with approximation ratio at most n ε. In contrast, for general graphs, for any ε > 0 it is NP-hard to approximate the independence number within a factor of n 1-ε. We also give a subexponential time exact algorithm for computing the independence number of intersection graphs of arcwise connected sets in the plane. Source

Most existing work on the demand for health insurance focuses on employees' decisions to enroll in employer-provided plans. Yet any attempt to achieve universal coverage must focus on the uninsured, the vast majority of whom are not offered employer-sponsored insurance. In the summer of 2008, we conducted a survey experiment to assess the willingness to pay for a health plan among a large sample of uninsured Americans. The experiment yields price elasticities of around one, substantially greater than those found in most previous studies. We use these results to estimate coverage expansion under the Affordable Care Act, with and without an individual mandate. We estimate that 35 million uninsured individuals would gain coverage and find limited evidence of adverse selection. © 2013. Source

Direct numerical simulations (DNSs), for a stratified flow in HCCI engine-like conditions, are performed to investigate the effects of exhaust gas recirculation (EGR) by NOx and temperature/mixture stratification on autoignition of dimethyl ether (DME) in the negative temperature coefficient (NTC) region. Detailed chemistry for a DME/air mixture with NOx addition is employed and solved by a hybrid multi-time scale (HMTS) algorithm. Three ignition stages are observed. The results show that adding (1000ppm) NO enhances both low and intermediate temperature ignition delay times by the rapid OH radical pool formation (one to two orders of magnitude higher OH radicals concentrations are observed). In addition, NO from EGR was found to change the heat release rates differently at each ignition stage, where it mainly increases the low temperature ignition heat release rate with minimal effect on the ignition heat release rates at the second and third ignition stages. Sensitivity analysis is performed and the important reactions pathways for low temperature chemistry and ignition enhancement by NO addition are specified. The DNSs for stratified turbulent ignition show that the scales introduced by the mixture and thermal stratifications have a stronger effect on the second and third stage ignitions. Compared to homogenous ignition, stratified ignition shows a similar first autoignition delay time, but about 19% reduction in the second and third ignition delay times. Stratification, however, results in a lower averaged LTC ignition heat release rate and a higher averaged hot ignition heat release rate compared to homogenous ignition. The results also show that molecular transport plays an important role in stratified low temperature ignition, and that the scalar mixing time scale is strongly affected by local ignition. Two ignition-kernel propagation modes are observed: a wave-like, low-speed, deflagrative mode (the D-mode) and a spontaneous, high-speed, kinetically driven ignition mode (the S-mode). Three criteria are introduced to distinguish the two modes by different characteristic time scales and Damkhöler (Da) number using a progress variable conditioned by a proper ignition kernel indicator (IKI). The results show that the spontaneous ignition S-mode is characterized by low scalar dissipation rate, high displacement speed flame front, and high mixing Damkhöler number, while the D-mode is characterized by high scalar dissipation rate, low displacement speeds in the order of the laminar flame speed and a lower than unity Da number. The proposed criteria are applied at the different ignition stages. © 2013 The Combustion Institute. Source

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