Euclidean Minimum Spanning Tree Research Articles

Hopcroft’s Problem, Log* Shaving, Two-dimensional Fractional Cascading, and Decision Trees

We revisit Hopcroft’s problem and related fundamental problems about geometric range searching. Given n points and n lines in the plane, we show how to count the number of point-line incidence pairs or the number of point-above-line pairs in O ( n 4/3 ) time, which matches the conjectured lower bound and improves the best previous time bound of \(n^{4/3}2^{O(\log ^*n)}\) obtained almost 30 years ago by Matoušek [ 58 ]. We describe two interesting and different ways to achieve the result: The first is randomized and uses a new two-dimensional version of fractional cascading for arrangements of lines; the second is deterministic and uses decision trees in a manner inspired by the sorting technique of Fredman [42]. The second approach extends to any constant dimension. Many consequences follow from these new ideas: For example, we obtain an O ( n 4/3 )-time algorithm for line segment intersection counting in the plane, O ( n 4/3 )-time randomized algorithms for distance selection in the plane and bichromatic closest pair and Euclidean minimum spanning tree in three or four dimensions, and a randomized data structure for halfplane range counting in the plane with O ( n 4/3 ) preprocessing time and space and \(O(n^{1/3})\) query time.

Central limit theorem for euclidean minimal spanning acycles

In this paper, we investigate asymptotics for the minimal spanning acycles (MSAs) of the (Alpha)-Delaunay complex on a stationary Poisson process on [Formula: see text]. MSAs are topological (or higher-dimensional) generalizations of minimal spanning trees. We establish a central limit theorem (CLT) for total weight of the MSA on a Poisson Alpha-Delaunay complex. Our approach also allows us to establish CLTs for the sum of birth times and lifetimes in the persistent diagram of the Delaunay complex. The key to our proof is in showing the so-called weak stabilization of MSAs which proceeds by establishing suitable chain maps and uses matroidal properties of MSAs. In contrast to the proof of weak-stabilization for Euclidean minimal spanning trees via percolation-theoretic estimates, our weak-stabilization proof is algebraic in nature and provides an alternative proof even in the case of minimal spanning trees.

Time and memory scalable algorithms for clustering tendency assessment of big data

Large-volume and high-dimensional big datasets are being generated quickly. They are expected to provide data-driven solutions for various pressing challenges such as cybersecurity, credit card fraud, network intrusion detection, etc. The visual assessment of clustering tendency (VAT) family of algorithms provides an excellent tool for assessing clustering tendency and subsequent clustering of these novel datasets to extract groups and anomalies to better understand the data generation phenomenon. However, VAT and improved VAT (iVAT) algorithms are plagued with O(n2) time and space complexity as they use Prim's algorithm for building Euclidean minimum spanning tree (EMST) of the data points, leaving it inapplicable to large datasets. This paper develops three novel time- and memory-scalable algorithms for fast computation of VAT EMST by reducing the search space of the following possible EMST edge and using an efficient data structure, the k-d tree. Next, we develop a novel approach to compute iVAT reordered dissimilarity image (RDI) in a novel memory-preserving manner using the EMST edge lengths calculated previously without having to compute memory intensive n×n VAT RDI. We experimented on several synthetic and real-life datasets to showcase that the proposed approaches can help quickly assess their clustering tendency in a memory-saving manner. We also demonstrate the applicability of the proposed methods for the anomaly detection task in large volumes of high-dimensional data.

Minimum weight Euclidean [formula omitted]-spanners

Given a set S of n points in the plane and a parameter ɛ>0, a Euclidean (1+ɛ)-spanner is a geometric graph G=(S,E) that contains, for all p,q∈S, a pq-path of weight at most (1+ɛ)‖pq‖. We show that the minimum weight of a Euclidean (1+ɛ)-spanner for n points in the unit square [0,1]2 is O(ɛ−3/2n), and this bound is the best possible. The upper bound is based on a new spanner algorithm in the plane. It improves upon the baseline O(ɛ−2n), obtained by combining a tight bound for the weight of a Euclidean minimum spanning tree (MST) on n points in [0,1]2, and a tight bound for the lightness of Euclidean (1+ɛ)-spanners, which is the ratio of the spanner weight to the weight of the MST. Our result generalizes to Euclidean d-space for every constant dimension d∈N: The minimum weight of a Euclidean (1+ɛ)-spanner for n points in the unit cube [0,1]d is Od(ɛ(1−d2)/dn(d−1)/d), and this bound is the best possible.For the n×n section of the integer lattice in the plane, we show that the minimum weight of a Euclidean (1+ɛ)-spanner is between Ω(ɛ−3/4⋅n2) and O(ɛ−1log(ɛ−1)⋅n2). These bounds become Ω(ɛ−3/4⋅n) and O(ɛ−1log(ɛ−1)⋅n) when scaled to a grid of n points in the unit square. In particular, this shows that the integer grid is not an extremal configuration for minimum weight Euclidean (1+ɛ)-spanners.

A DNS evaluation of three MMC-like mixing models for transported PDF modelling of turbulent nonpremixed flames

Transported probability density function (TPDF) methods are suitable for modelling turbulent reactive flows. One of the main challenges is to accurately model the molecular mixing terms. In TPDF mixing models, it is desired that the principle of localness is satisfied so that the molecular mixing is performed locally in both physical and composition spaces. The multiple mapping conditioning (MMC) mixing model can ensure mixing localness without violating other desired principles. The present study examines three MMC-like mixing models, including the original MMC (OMMC) mixing model, shadow-position mixing model (SPMM) and a conceptually simplified multiple mapping conditioning (SMMC) mixing model. Three direct numerical simulation (DNS) datasets modelling turbulent nonpremixed ethylene flames with increasing levels of extinction are used for model evaluation. The DNS datasets are also used to provide both initial conditions and inputs needed over the course of TPDF runs to remove the uncertainties caused by turbulence closure, allowing the study to focus on the molecular mixing model. The mixing model coefficients are specified analytically by reference to a canonical mean scalar gradient (MSG) flow in order to achieve specifiable dissipation rate and user-controllable localness. The results show that the MMC-like mixing models yield similar prediction of flame extinction and reignition if the coefficients are properly specified. The MMC-like mixing models can also be tuned to achieve a desired level of localness. For conditional statistics, the MMC-like mixing models can yield correct level of conditional variances. By changing localness, the MMC-like models can yield solutions resembling the interaction by exchange with the mean (IEM) or Euclidean minimum spanning tree (EMST) mixing model for extreme parameter choices. The models differ in their abilities to specify unconditional dissipation rates, at least in the flow considered, and in their ease of implementation.

A pairwise mixing model with kernel constraint and its appraisal in transported PDF simulations of ethylene flames

Modeling micro-mixing remains a primary challenge for the transported probability density function (TPDF) method. In this study, the pairwise mixing with kernel constraint (KerM) model is proposed and evaluated in TPDF simulations of a turbulent non-premixed ethylene flame. The new model features controllable localness by selecting mixing particle pairs with a probability that depends on their distance in composition space, normalized by a characteristic kernel size. To evaluate the model in a controlled setting, TPDF simulations were carried out with the mean velocity, turbulent diffusivity and scalar mixing frequencies extracted from a direct numerical simulation (DNS) dataset. The predictions of the mean statistics and the conditional PDFs illustrate that the KerM model exhibits asymptotic behavior to the modified Curl (MC) and the Euclidean minimum spanning trees (EMST) model, respectively, depending on the kernel size. For the flame considered, results for KerM approach those for EMST when the kernel size is less than 0.01, while they approach those for MC when the kernel size is larger than 1.0. Given a kernel size of 0.1, KerM yields a notably better prediction of the overall combustion process than the conventional MC model. The predictions of the mean temperature and species mass fractions by KerM are similar to those by EMST, while KerM better predicts the conditional PDF of temperature than EMST that overdamps the conditional fluctuations. In addition, its complexity with the number of particles per cell is instead of O(Np2) for EMST. By incorporating different mixing timescales among species, the KerM-DD model better predicts the mean temperature field at the time instant close to local extinction than the KerM and EMST models. This work demonstrates that the KerM model, apart from being simple and robust in terms of implementation, features the advantages of being flexible in localness and being capable of incorporating differential diffusion.

Quantitative two-scale stabilization on the Poisson space

We establish inequalities for assessing the distance between the distribution of a (possibly multidimensional) functional of a Poisson random measure and that of a Gaussian element. Our bounds only involve add-one cost operators at the order one—that we evaluate and compare at two different scales—and are specifically tailored for studying the Gaussian fluctuations of sequences of geometric functionals displaying a form of weak stabilization—see Penrose and Yukich (Ann. Appl. Probab. 11 (2001) 1005–1041) and Penrose (Ann. Probab. 33 (2005) 1945–1991). Our main bounds extend the estimates recently exploited by Chatterjee and Sen (Ann. Appl. Probab. 27 (2017) 1588–1645) in the proof of a quantitative version of the central limit theorem (CLT) for the length of the Poisson-based Euclidean minimal spanning tree (MST). We develop in full detail three applications of our bounds, namely: (i) to a quantitative multidimensional spatial CLT for functionals of the on-line nearest neighbour graph, (ii) to a quantitative multidimensional CLT involving functionals of the empirical measure associated with the edge-length of the Euclidean MST, and (iii) to a collection of multidimensional CLTs for geometric functionals of the excursion set of heavy-tailed shot noise random fields. Application (i) is based on a collection of general probabilistic approximations for strongly stabilizing functionals, that is of independent interest.

Inference of trajectory presence by tree dimension and subset specificity by subtree cover.

The complexity of biological processes such as cell differentiation is reflected in dynamic transitions between cellular states. Trajectory inference arranges the states into a progression using methodologies propelled by single-cell biology. However, current methods, all returning a best trajectory, do not adequately assess statistical significance of noisy patterns, leading to uncertainty in inferred trajectories. We introduce a tree dimension test for trajectory presence in multivariate data by a dimension measure of Euclidean minimum spanning tree, a test statistic, and a null distribution. Computable in linear time to tree size, the tree dimension measure summarizes the extent of branching more effectively than globally insensitive number of leaves or tree diameter indifferent to secondary branches. The test statistic quantifies trajectory presence and its null distribution is estimated under the null hypothesis of no trajectory in data. On simulated and real single-cell datasets, the test outperformed the intuitive number of leaves and tree diameter statistics. Next, we developed a measure for the tissue specificity of the dynamics of a subset, based on the minimum subtree cover of the subset in a minimum spanning tree. We found that tissue specificity of pathway gene expression dynamics is conserved in human and mouse development: several signal transduction pathways including calcium and Wnt signaling are most tissue specific, while genetic information processing pathways such as ribosome and mismatch repair are least so. Neither the tree dimension test nor the subset specificity measure has any user parameter to tune. Our work opens a window to prioritize cellular dynamics and pathways in development and other multivariate dynamical systems.

Assessment of Finite-Rate Chemistry Effects in a Turbulent Dilute Ethanol Spray Flame

Simulations of a benchmark ethanol dilute spray flame are performed with four combustion models at different levels of closure for turbulence–chemistry interaction (TCI) to provide a head-to-head comparative analysis on the closure effects of TCI. The evaluation is performed in the context of Reynolds-averaged Navier–Stokes simulations taking advantage of the simple jet configuration, together with a Lagrangian discrete phase model for tracking the spray droplets. Two well-validated ethanol mechanisms consisting of 40 and 50 species, respectively, are employed to describe the chemical kinetics and reveal the impact of chemistry. Results are compared against measurement, including gas-phase velocity, temperature, and droplet velocity, demonstrating the improvement with the increasing accuracy in TCI modeling from characteristic time scale model, laminar finite rate model, eddy dissipation concept model, and transported probability density function (PDF) model. Effects of micromixing in PDF simulation are further investigated considering different model formulations and mechanical-to-scalar timescale ratios (). The Euclidean minimum spanning tree model with yields the best prediction of the mean temperature. The analysis on flame structure suggests that the flame exhibits a mixed mode of combustion, i.e., a stratified premixed flame in the upstream and a typical non-premixed flame in the downstream.

A Numerical Investigation of Mixing Models in LES-FMDF for Compressible Reactive Flows

The filtered mass density function (FMDF) model has been employed for large-eddy simulations (LES) of compressible high-speed turbulent mixing and reacting flows. However, the mixing model remains a pressing challenge for FMDF methods, especially for compressible reactive flows. In this work, a temporal development mixing layer with two different convective Mach numbers, Mc=0.4 and Mc=0.8, is used to investigate the mixing models. A simplified one-step reaction and a real hydrogen/air reaction are employed to study the mixing and turbulence-chemistry interaction. Two widely used mixing models, interaction by exchange with the mean (IEM) and Euclidean minimum spanning tree (EMST), are studied. Numerical results indicate that no difference is observed between the IEM and EMST models in simple reaction flows. However, for hydrogen/air reactions, the EMST model can predict the reaction more accurately in high-speed flow. For mixing models in compressible reactive flows, the requirement of localness preservation tends to be more essential as the convective Mach number increases. With the increase of compressibility, the sensitivity of the mixing model coefficient is reduced significantly. Therefore, the appropriate mixing model coefficient has a wider range. Results also indicate that a large error may result when using a fixed mixing model coefficient in compressible flows.

Euclidean Bottleneck Bounded-Degree Spanning Tree Ratios

Inspired by the seminal works of Khuller et al. (SIAM J. Comput. 25(2), 355–368 (1996)) and Chan (Discrete Comput. Geom. 32(2), 177–194 (2004)) we study the bottleneck version of the Euclidean bounded-degree spanning tree problem. A bottleneck spanning tree is a spanning tree whose largest edge-length is minimum, and a bottleneck degree-K spanning tree is a degree-K spanning tree whose largest edge-length is minimum. Let $$\beta _K$$ be the supremum ratio of the largest edge-length of the bottleneck degree-K spanning tree to the largest edge-length of the bottleneck spanning tree, over all finite point sets in the Euclidean plane. It is known that $$\beta _5=1$$ , and it is easy to verify that $$\beta _2\geqslant 2$$ , $$\beta _3\geqslant \sqrt{2}$$ , and $$\beta _4>1.175$$ . It is implied by the Hamiltonicity of the cube of the bottleneck spanning tree that $$\beta _2\leqslant 3$$ . The degree-3 spanning tree algorithm of Ravi et al. (25th Annual ACM Symposium on Theory of Computing, pp. 438–447. ACM, New York (1993)) implies that $$\beta _3\leqslant 2$$ . Andersen and Ras (Networks 68(4), 302–314 (2016)) showed that $$\beta _4\leqslant \sqrt{3}$$ . We present the following improved bounds: $$\beta _2\geqslant \sqrt{7}$$ , $$\beta _3\leqslant \sqrt{3}$$ , and $$\beta _4\leqslant \sqrt{2}$$ . As a result, we obtain better approximation algorithms for Euclidean bottleneck degree-3 and degree-4 spanning trees. As parts of our proofs of these bounds we present some structural properties of the Euclidean minimum spanning tree which are of independent interest.

Euclidean minimum spanning trees with independent and dependent geometric uncertainties

We address the problems of constructing the Euclidean Minimum Spanning Tree (EMST) of points in the plane with mutually dependent location uncertainties, testing its stability, and computing its total weight. Point coordinate uncertainties are modeled with the Linear Parametric Geometric Uncertainty Model (LPGUM), an expressive and computationally efficient worst-case, first order linear approximation of geometric uncertainty that supports parametric dependencies between point locations. We define uncertain EMST stability of n LPGUM points modeled with k real valued uncertainty parameters. We prove that when the uncertain EMST is unstable, it may have an exponential number of topologically different instances, thus precluding its polynomial-time computation. We present algorithms for comparing two edge weights defined by the distance between the edge endpoints for the independent and dependent cases with time complexity of O(klog⁡k) and O(T(k)) respectively, where T(k) is the time required to solve a quadratic optimization problem with k parameters. We describe an uncertain EMST stability test algorithm whose time complexity is O(n3klog⁡k) and O(nk+T(k)) for the independent and dependent case, respectively. We then present a more efficient O(nklog⁡nklog⁡n) time algorithm for the independent case and a method for computing the minimum and maximum total weight whose complexity is O(N3) time, where N=max⁡{k,n}.

Component-based 2-/3-dimensional nearest neighbor search based on Elias method to GPU parallel 2D/3D Euclidean Minimum Spanning Tree Problem

We present improved data parallel approaches working on graphics processing unit (GPU) compute unified device architecture (CUDA) platform to build hierarchical Euclidean minimum spanning forest or tree (EMSF/EMST) for applications whose input only contains N points with arbitrary data distribution in 2D/3D Euclidean space. Characteristic of the proposed parallel algorithms follows “data parallelism, decentralized control and O(1) local memory occupied by each GPU thread”. This research has to solve GPU parallelism of component-based nearest neighbor search (component-based NNS), tree traversal, and other graph operations like union-find. For exact NNS, instead of using classical K-d tree search or brute-force computing method, we propose a K-d search method working based on dividing the Euclidean K-dimensional space into congruent and non-overlapping square/cubic cells where size of points in each cell is bounded. For component-based NNS, with the uniqueness property based on 2D/3D square/cubic space partition, we propose dynamic and static pruning techniques to prune unnecessary neighbor cells’ search. For tree traversal, instead of using breadth-first-search, this paper proposes CUDA kernels working with a distributed dynamic link list for selecting a local spanning tree’s shortest outgoing edge since size of local EMSTs in EMSF cannot be predicted. Source code is provided online and experimental comparisons are conducted on both 2D and 3D benchmarks with up to 107 points to build final EMST. Results show that applying K-d search with static pruning technique and the proposed operators totally working in parallel on GPU, our current implementation runs faster than our previous work and current optimal sequential dual-tree mlpack EMST library.

Shape Estimation for Elongated Deformable Object using B-spline Chained Multiple Random Matrices Model.

In this paper, a B-spline chained multiple random matrix models (RMMs) representation is proposed to model geometric characteristics of an elongated deformable object. The hyper degrees of freedom structure of the elongated deformable object make its shape estimation challenging. Based on the likelihood function of the proposed B-spline chained multiple RMMs, an expectation-maximization (EM) method is derived to estimate the shape of the elongated deformable object. A split and merge method based on the Euclidean minimum spanning tree (EMST) is proposed to provide initialization for the EM algorithm. The proposed algorithm is evaluated for the shape estimation of the elongated deformable objects in scenarios, such as the static rope with various configurations (including configurations with intersection), the continuous manipulation of a rope and a plastic tube, and the assembly of two plastic tubes. The execution time is computed and the accuracy of the shape estimation results is evaluated based on the comparisons between the estimated width values and its ground-truth, and the intersection over union (IoU) metric.

An evaluation of gas-phase micro-mixing models with differential mixing timescales in transported PDF simulations of sooting flame DNS

The use of transported probability density function (TPDF) models to predict soot has the strong advantage that the effects of turbulent fluctuations on soot source terms can be rigorously accounted for. However, soot processes are closely coupled to gas-phase composition. Among the open issues for gas-phase micro-mixing is the species-dependence of mixing timescales. The objective is to carry out an evaluation on the effect of incorporating differential mixing timescales among gas-phase species in a TPDF simulation for soot prediction. A DNS having the configuration of a temporally evolving, non-premixed ethylene flame with a four-step, three-moment soot model is considered as the target for evaluation. The DNS dataset is applied to provide key inputs for TPDF simulations to limit the sources of error to micro-mixing. TPDF simulations with the interaction by exchange with the mean (IEM) and modified Curl (MC) models, which impose the same mixing timescale to all species, underpredict soot mass fraction and overpredict extinction levels regardless of the prescribed mixing frequency. By incorporating differential mixing timescales among gas-phase species, IEM-DD and MC-DD models yield notable improvement in predictions of the overall extinction and soot levels, highlighting the benefit of accounting for differential mixing timescales. A TPDF simulation with the Euclidean minimum spanning tree (EMST) model yields even better predictions, illustrating that the localness in composition space remains a critical issue. The indicated species mixing frequencies by the EMST model are shown to follow the DNS results qualitatively, illustrating that the micro-mixing process based on the Euclidean distance in composition space reproduces to a certain extent the differential mixing timescales due to reaction. Finally, it is shown that incorporating differential mixing timescales of soot moments is expected to have limited value as the mixing timescales of soot moments are sufficiently large to safely neglect soot mixing.

A Zero-Dimensional Velocity-Composition-Frequency Probability Density Function Model for Compression-Ignition Engine Simulation

<div class="section abstract"><div class="htmlview paragraph">Numerical simulation of in-cylinder processes can significantly reduce the development and refinement costs of engines. While it can be argued that higher fidelity models improve accuracy of prediction, it comes at the expense of high computational cost. In this respect, a 3D analysis of in-cylinder processes may not be feasible for evaluating large number of design and operating conditions. The situation can be more foreboding for transient simulations. In the current work a phenomenological combustion modeling approach is explored that can be implemented in a lower fidelity modeling framework and can approach the accuracy of higher dimensional models with significant reduction in computational cost. The proposed model uses transported probability density function (tPDF) method within a 0D framework to provide a computationally efficient solution while capturing the essential physics of in-cylinder combustion. Scalar mixing is accounted for by a modified Euclidean Minimum Spanning Tree (EMST) mixing model. A new model has been formulated to calculate the mixing timescale using the velocity-composition-frequency (VCF) tPDF method. Numerical comparisons have been performed for two engines: one with a 3D-CFD simulated diesel engine to compare the in-cylinder composition and thermal stratification, and one with a diesel engine where measured data is available over a range of operating conditions encompassing a range of speed, load, and exhaust-gas recirculation (EGR). A 42-species chemical mechanism including thermal NO is used to represent the gas-phase chemistry. The proposed 0D-VCF model predicted pressure, heat-release-rate, equivalence-ratio vs temperature distribution, and NOx are compared to the 3D-CFD data, as well as to the experimentally measured data. The proposed model was able to capture the 3D-CFD trend of equivalence-ratio vs temperature distribution at different crank-angle degrees quite accurately. The 0D-VCF model computed pressure and heat-release-rate along with the NOx showed reasonable agreement with the experiment. In summary, the proposed 0D numerical model has the potential to provide a computationally less expensive alternative to a 3D numerical simulation for estimating engine performance and engine-out emission quantities when transient or large number of operating points are to be simulated.</div></div>

Filtered Density Function Simulations of a Near-Limit Turbulent Lean Premixed Flame

The predictive numerical simulation of near-limit turbulent premixed combustion, in which the turbulent intensity is high and the fuel/air mixture is near the flammability limit, remains challenging. In this study, large-eddy simulation (LES)/filtered density function (FDF) simulations of a high-speed piloted premixed jet burner flame are performed to illustrate the importance of the resolved level of reactive scalars and micromixing modeling on predicting the overall combustion process. The simulations with increased resolved levels of reactive scalars yield notably improved predictions throughout the extinction–reignition region of this flame. The sources of grid sensitivity are thoroughly examined through the quantification of the resolved levels of the velocity and scalar fields, as well as the mixing-reaction budgets. Subgrid mixing is identified as the most prominent factor for grid sensitivity. The effects of mixing timescale modeling are investigated and quantified through a parametric study of the mixing rate parameter and the development of an augmented hybrid mixing timescale model in the LES context. As far as the mixing formulation is concerned, the Euclidean minimum spanning tree model shows a lower level of grid dependence than the interaction by exchange with the mean model due to the enforcement of localness in composition space.

Convergence Rates for Empirical Estimation of Binary Classification Bounds

Bounding the best achievable error probability for binary classification problems is relevant to many applications including machine learning, signal processing, and information theory. Many bounds on the Bayes binary classification error rate depend on information divergences between the pair of class distributions. Recently, the Henze–Penrose (HP) divergence has been proposed for bounding classification error probability. We consider the problem of empirically estimating the HP-divergence from random samples. We derive a bound on the convergence rate for the Friedman–Rafsky (FR) estimator of the HP-divergence, which is related to a multivariate runs statistic for testing between two distributions. The FR estimator is derived from a multicolored Euclidean minimal spanning tree (MST) that spans the merged samples. We obtain a concentration inequality for the Friedman–Rafsky estimator of the Henze–Penrose divergence. We validate our results experimentally and illustrate their application to real datasets.

Impact of molecular mixing and scalar dissipation rate closures on turbulent bluff-body flames with increasing local extinction

Bluff-body turbulent CH4: H2 (1:1) flames at 50% (HM1), 75% (HM2) and 91% (HM3) of the blow-off velocity (235 m s−1) were studied experimentally by Masri and co-workers and found to exhibit gradually increasing periodic and shear layer instabilities. The latter are coupled with increasing levels of local extinction with subsequent re-ignition further downstream. This study provides a systematic evaluation of the sensitivity of predictions to molecular mixing and scalar dissipation rate closures. The latter include extended forms of the Euclidean Minimum Spanning Tree (EMST) and modified Curl’s (MC) models, applicable to premixed turbulent flames via a closure that accounts for local Damköhler number effects (EEMST and EMC), and a conceptually related blended scalar time-scale approach (BEMST and BMC). Computations are performed using a hybrid finite volume (FV) – transported Joint Probability Density Function (JPDF) algorithm featuring stochastic Lagrangian particles, a comprehensive 48-scalar systematically reduced C/H/N/O mechanism, and a second moment method based on the Generalised Langevin Model that provides a partial resolution of the unsteady fluid motion. The sensitivity to solution parameters affecting the temporal resolution is quantified using Fourier transforms of the time histories of velocity and scalar traces. Radial profiles, conditional means and scatter plots are compared to the experimental data along with burning indices based on the conditional mean temperature. Vortex related instabilities ∼ 1 kHz in the outer shear layer appear for all closures with EMC showing periodic local extinction and re-ignition in the neck region for HM3 and flame turbules (i.e., discrete pockets of hot gas) separating periodically at frequencies ∼ 85 Hz. Results are similar to well–resolved JPDF/LES simulations for HM1. It is shown that the EMC and (E)EMST models essentially enclose the experimental data for HM2 and HM3. For HM3, emissions of NO are controlled by local extinction events that become increasingly sensitive to the molecular mixing closure as blow-off is approached.

Minimizing the maximum sender interference by deploying additional nodes in a Wireless Sensor Network

Interference is one of the major challenges faced by the communication networks. Since the interference leads to packet loss, packet collision and data re-transmission, higher the interference, higher is the energy consumption. Several algorithms were proposed for reducing the interference in a Wireless sensor network (WSN). By deploying additional nodes at an appropriate position in a WSN, it is possible to reduce the interference. We propose an algorithm in which, the main objective is to reduce the maximum Sender interference by deploying the additional nodes in the network, while connectivity of the network is preserved. We use the properties of Gabriel graph to achieve the reduction in interference. We present the simulation results which show the number of additional nodes to be deployed. The comparison of the maximum Sender interference obtained by the proposed algorithm with that of the Euclidean minimum spanning tree (MST) of the given network is presented through simulation. We show that the additional number of nodes required for deployment has an upper bound of n/2 , where n is the number of nodes. We also compute the average reduction in Sender interference of the network for a various number of nodes.