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At least 253 records · Page 14
All sourcesNASA NTRSDOE OSTI
DOE OSTI2023

Assembly and method for interchangeably holding an electrochemical substrate

An electrochemical substrate holder assembly, including: a first housing and a second housing; wherein the first housing and the second housing collectively define an interior space; a first gasket coupled to the first housing and adapted to contact a first side of a substrate; and a second gasket coupled to the second housing and adapted to contact a second side of the substrate; wherein the first gasket and the second gasket are collectively adapted to hold the substrate within the interior space. The first housing defines a first port adapted to receive a first electrical lead to electrically contact the first side of the substrate. The second housing defines a second port adapted to allow a fluid to pass there through to fluidly contact at least the second side of the substrate. The second housing is adapted to receive one or more of a second and third electrical lead.

Freiderich, John W.↗
DOE OSTI2023

Understanding the negative triangularity ELM trigger and ELM free state on DIII-D with ECE-imaging

Here, the Electron Cyclotron Emission Imaging (ECEI) diagnostic was used to observe a finite-n interchange mode structure in the edge of negative triangularity shaped plasmas on DIII-D. At a small negative triangularity (δ u = -0.2), the plasma is in the H-mode with ELMs that are triggered by a low-n interchange mode. At a larger negative triangularity (δ u = -0.4) and low NBI power (2 MW), a dithering oscillation is observed that is triggered by a low-n interchange mode, whereas at higher NBI power (>2 MW), the edge reverts to L-mode and the low-n interchange mode is present continuously. In all cases, the edge pressure gradient is clamped when the interchange mode is present. It is concluded that the low-n interchange mode prevents the plasma from transitioning to H-mode at a large negative triangularity. This agrees with linear BOUT++ simulations which suggest that the interchange-type MHD can be a resistive ballooning mode whereby resistivity can significantly increase the finite-n ballooning mode growth rate. The absence of H-mode at large negative triangularity can, thus, be explained by the excitation of low-n pressure driven resistive ballooning modes in the plasma edge.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS2010

The Evolution of Open Magnetic Flux Driven by Photospheric Dynamics

The coronal magnetic field is of paramount importance in solar and heliospheric physics. Two profoundly different views of the coronal magnetic field have emerged. In quasi-steady models, the predominant source of open magnetic field is in coronal holes. In contrast, in the interchange model, the open magnetic flux is conserved, and the coronal magnetic field can only respond to the photospheric evolution via interchange reconnection. In this view the open magnetic flux diffuses through the closed, streamer belt fields, and substantial open flux is present in the streamer belt during solar minimum. However, Antiochos and co-workers, in the form of a conjecture, argued that truly isolated open flux cannot exist in a configuration with one heliospheric current sheet (HCS) - it will connect via narrow corridors to the polar coronal hole of the same polarity. This contradicts the requirements of the interchange model. We have performed an MHD simulation of the solar corona up to 20R solar to test both the interchange model and the Antiochos conjecture. We use a synoptic map for Carrington Rotation 1913 as the boundary condition for the model, with two small bipoles introduced into the region where a positive polarity extended coronal hole forms. We introduce flows at the photospheric boundary surface to see if open flux associated with the bipoles can be moved into the closed-field region. Interchange reconnection does occur in response to these motions. However, we find that the open magnetic flux cannot be simply injected into closed-field regions - the flux eventually closes down and disconnected flux is created. Flux either opens or closes, as required, to maintain topologically distinct open and closed field regions, with no indiscriminate mixing of the two. The early evolution conforms to the Antiochos conjecture in that a narrow corridor of open flux connects the portion of the coronal hole that is nearly detached by one of the bipoles. In the later evolution, a detached coronal hole forms, in apparent violation of the Antiochos conjecture. Further investigation reveals that this detached coronal hole is actually linked to the extended coronal hole by a separatrix footprint on the photosphere of zero width. Therefore, the essential idea of the conjecture is preserved, if we modify it to state that coronal holes in the same polarity region are always linked, either by finite width corridors or separatrix footprints. The implications of these results for interchange reconnection and the sources of the slow solar wind are briefly discussed.

Linker, Jon A.↗
DOE OSTI2015

Structural Insight into How Bacteria Prevent Interference between Multiple Divergent Type IV Secretion Systems

ABSTRACT Prokaryotes use type IV secretion systems (T4SSs) to translocate substrates (e.g., nucleoprotein, DNA, and protein) and/or elaborate surface structures (i.e., pili or adhesins). Bacterial genomes may encode multiple T4SSs, e.g., there are three functionally divergent T4SSs in some Bartonella species ( vir , vbh , and trw ). In a unique case, most rickettsial species encode a T4SS ( rvh ) enriched with gene duplication. Within single genomes, the evolutionary and functional implications of cross-system interchangeability of analogous T4SS protein components remains poorly understood. To lend insight into cross-system interchangeability, we analyzed the VirB8 family of T4SS channel proteins. Crystal structures of three VirB8 and two TrwG Bartonella proteins revealed highly conserved C-terminal periplasmic domain folds and dimerization interfaces, despite tremendous sequence divergence. This implies remarkable structural constraints for VirB8 components in the assembly of a functional T4SS. VirB8/TrwG heterodimers, determined via bacterial two-hybrid assays and molecular modeling, indicate that differential expression of trw and vir systems is the likely barrier to VirB8-TrwG interchangeability. We also determined the crystal structure of Rickettsia typhi RvhB8-II and modeled its coexpressed divergent paralog RvhB8-I. Remarkably, while RvhB8-I dimerizes and is structurally similar to other VirB8 proteins, the RvhB8-II dimer interface deviates substantially from other VirB8 structures, potentially preventing RvhB8-I/RvhB8-II heterodimerization. For the rvh T4SS, the evolution of divergent VirB8 paralogs implies a functional diversification that is unknown in other T4SSs. Collectively, our data identify two different constraints (spatiotemporal for Bartonella trw and vir T4SSs and structural for rvh T4SSs) that mediate the functionality of multiple divergent T4SSs within a single bacterium. IMPORTANCE Assembly of multiprotein complexes at the right time and at the right cellular location is a fundamentally important task for any organism. In this respect, bacteria that express multiple analogous type IV secretion systems (T4SSs), each composed of around 12 different components, face an overwhelming complexity. Our work here presents the first structural investigation on factors regulating the maintenance of multiple T4SSs within a single bacterium. The structural data imply that the T4SS-expressing bacteria rely on two strategies to prevent cross-system interchangeability: (i) tight temporal regulation of expression or (ii) rapid diversification of the T4SS components. T4SSs are ideal drug targets provided that no analogous counterparts are known from eukaryotes. Drugs targeting the barriers to cross-system interchangeability (i.e., regulators) could dysregulate the structural and functional independence of discrete systems, potentially creating interference that prevents their efficient coordination throughout bacterial infection.

Gillespie, Joseph J.↗
DOE OSTI2017

Dynamics of Coronal Hole Boundaries

Remote and in situ observations strongly imply that the slow solar wind consists of plasma from the hot, closed-field corona that is released onto open magnetic field lines. The Separatrix Web theory for the slow wind proposes that photospheric motions at the scale of supergranules are responsible for generating dynamics at coronal-hole boundaries, which result in the closed plasma release. We use three-dimensional magnetohydrodynamic simulations to determine the effect of photospheric flows on the open and closed magnetic flux of a model corona with a dipole magnetic field and an isothermal solar wind. A rotational surface motion is used to approximate photospheric supergranular driving and is applied at the boundary between the coronal hole and helmet streamer. The resulting dynamics consist primarily of prolific and efficient interchange reconnection between open and closed flux. The magnetic flux near the coronal-hole boundary experiences multiple interchange events, with some flux interchanging over 50 times in one day. Additionally, we find that the interchange reconnection occurs all along the coronal-hole boundary and even produces a lasting change in magnetic-field connectivity in regions that were not driven by the applied motions. Our results show that these dynamics should be ubiquitous in the Sun and heliosphere. We discuss the implications of our simulations for understanding the observed properties of the slow solar wind, with particular focus on the global-scale consequences of interchange reconnection.

79 ASTRONOMY AND ASTROPHYSICS↗
NASA NTRS2017

Dynamics of Coronal Hole Boundaries

Remote and in situ observations strongly imply that the slow solar wind consists of plasma from the hot, closed-field corona that is released onto open magnetic field lines. The Separatrix Web theory for the slow wind proposesthat photospheric motions at the scale of supergranules are responsible for generating dynamics at coronal-holeboundaries, which result in the closed plasma release. We use three-dimensional magnetohydrodynamicsimulations to determine the effect of photospheric flows on the open and closed magnetic flux of a model coronawith a dipole magnetic field and an isothermal solar wind. A rotational surface motion is used to approximatephotospheric supergranular driving and is applied at the boundary between the coronal hole and helmet streamer.The resulting dynamics consist primarily of prolific and efficient interchange reconnection between open andclosed flux. The magnetic flux near the coronal-hole boundary experiences multiple interchange events, with someflux interchanging over 50 times in one day. Additionally, we find that the interchange reconnection occurs allalong the coronal-hole boundary and even produces a lasting change in magnetic-field connectivity in regions thatwere not driven by the applied motions. Our results show that these dynamics should be ubiquitous in the Sun andheliosphere. We discuss the implications of our simulations for understanding the observed properties of the slowsolar wind, with particular focus on the global-scale consequences of interchange reconnection.

Higginson, A. K.↗
DOE OSTI2021

Edge counts for the auxiliary pair graph within the graphical unitary group approach

Closed-form expressions are presented for the numbers of edges in the auxiliary pair graphs (APGs) associated with non spin-orbit and spin-orbit Shavitt graphs for full configuration interaction expansions. A Shavitt graph is a visual representation of a configuration state function expansion space constructed via the graphical unitary group approach (GUGA). An APG is an organisational aid and a programmatic tool generated from a Shavitt graph. The number of edges in an APG determines bounds on the computational scaling as a function of the total numbers of electrons, orbitals, and spin multiplicities. The edge counts extend a suite of Shavitt graph statistics based on these functional parameters. The derivation and the presentation of the formulas for the edge counts has been assisted by the bra-ket interchange symmetry and the particle-hole interchange symmetry in the GUGA formalism. Furthermore, these symmetry operators produce one-to-one correspondences between various sets of edges, and this yields identities among some edge count formulas. There are 208 possible edge types. Of these, some do not contribute to two-electron operators, some are related by bra-ket interchange symmetry, and some are related by particle-hole interchange symmetry. For the remaining unique edge types, explicit expressions are derived for the numbers of edges.

74 ATOMIC AND MOLECULAR PHYSICS↗
NASA NTRS2021

Effects of Pseudostreamer Boundary Dynamics on Heliospheric Field and Wind

Interchange reconnection has been proposed as a mechanism for the generation of the slow solar wind, and a key contributor to determining its characteristic qualities. In this paper we study the implications of interchange reconnection for the structure of the plasma and field in the heliosphere. We use the Adaptively Refined Magnetohydrodynamic Solver (ARMS) to simulate the coronal magnetic evolution in a coronal topology containing both a pseudostreamer and helmet streamer. We begin with a geometry containing a low-latitude coronal hole that is separated from the main polar coronal hole by a pseudostreamer. We drive the system by imposing rotating flows at the Solar surface within and around the low-latitude coronal hole, which leads to a corrugation (at low altitudes) of the separatrix surfaces that separate open from closed magnetic flux. Interchange reconnection is induced both at the null points and separators of the pseudostreamer, and at the global helmet streamer. Wedemonstrate that a preferential occurrence of interchange reconnection in the “lanes” between our driving cells leads to a filamentary pattern of newly-opened flux in the heliosphere. These flux bundles connect to but extend far from the separatrix-web (S-Web) arcs at the source surface. We propose that the pattern of granular and supergranular flows on the photosphere should leave an observable imprint in the heliosphere.

V Aslanyan↗
DOE OSTI2020

Magnetic well and Mercier stability of stellarators near the magnetic axis

We have recently demonstrated that by expanding in small distance from the magnetic axis compared with the major radius, stellarator shapes with low neoclassical transport can be generated efficiently. To extend the utility of this new design approach, here we evaluate measures of magnetohydrodynamic interchange stability within the same expansion. In particular, we evaluate the magnetic well, Mercier's criterion, and resistive interchange stability near a magnetic axis of arbitrary shape. In contrast to previous work on interchange stability near the magnetic axis, which used an expansion of the flux coordinates, here we use the ‘inverse expansion’ in which the flux coordinates are the independent variables. Reduced expressions are presented for the magnetic well and stability criterion in the case of quasisymmetry. The analytic results are shown to agree with calculations from the VMEC equilibrium code. Finally, we show that near the axis, Glasser, Greene and Johnson's stability criterion for resistive modes approximately coincides with Mercier's ideal condition.

Physics↗
DOE OSTI2025

Turbulence and transport in mirror geometries in the Large Plasma Device

Thanks to advances in plasma science and enabling technology, mirror machines are being reconsidered for fusion power plants and as possible fusion volumetric neutron sources. However, cross-field transport and turbulence in mirrors remains relatively understudied compared with toroidal devices. Turbulence and transport in mirror configurations were studied utilizing the flexible magnetic geometry of the Large Plasma Device (LAPD). Multiple mirror ratios from $M=1$ to $M=2.68$ and three mirror-cell lengths from $L=3.51$ to $L=10.86$ m were examined. Langmuir and magnetic probes were used to measure profiles of density, temperature, potential and magnetic field. The electric field-fluctuation-driven ${\tilde {\boldsymbol{E}}} \times {\boldsymbol{B}}$ particle flux, where $\boldsymbol{B}$ is the background field, was calculated from these quantities. Two probe correlation techniques were used to infer wavenumbers and two-dimensional structure. Cross-field particle flux and density fluctuation power decreased with increased mirror ratio. Core density and temperatures remain similar with mirror ratio, but radial line-integrated density increased. The physical expansion of the plasma in the mirror cell by using a higher field in the source region may have led to reduced density fluctuation power through the increased gradient scale length. This increased scale length reduced the growth rate and saturation level of rotational interchange and drift-like instabilities. Despite the introduction of magnetic curvature, no evidence of mirror-driven instabilities – interchange, velocity space or otherwise – were observed. For curvature-induced interchange, many possible stabilization mechanisms were present, suppressing the visibility of the instability.

Travis, Phil (ORCID:0000000343170432)↗
DOE OSTI2014

Development and operation of a high-throughput accurate-wavelength lens-based spectrometer

A high-throughput spectrometer for the 400–820 nm wavelength range has been developed for charge exchange recombination spectroscopy or general spectroscopy. A large 2160 mm{sup −1} grating is matched with fast f/1.8 200 mm lenses, which provide stigmatic imaging. A precision optical encoder measures the grating angle with an accuracy ≤0.075 arc sec. A high quantum efficiency low-etaloning CCD detector allows operation at longer wavelengths. A patch panel allows input fibers to interface with interchangeable fiber holders that attach to a kinematic mount at the entrance slit. Computer-controlled hardware allows automated control of wavelength, timing, f-number, automated data collection, and wavelength calibration.

46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
DOE OSTI2020

Different responses of the Rayleigh–Taylor type and resistive drift wave instabilities to the velocity shear

The effects of velocity shear on the unstable modes driven by the effective gravity (Rayleigh–Taylor and interchange) and resistive drift wave instabilities for inhomogeneous equilibrium fluid/plasma density are analyzed for the localized eigenmode problems. It is shown that the effect of the velocity shear drastically depends on the type of instability. Whereas the velocity shear can significantly suppress both Rayleigh–Taylor and interchange instabilities, it has only a weak impact on the growth rate of the resistive drift wave. This is directly related to the physical nature of these instabilities. For the Rayleigh–Taylor and interchange instabilities, the shear flow tilts the eddies of the stream functions, while for the resistive drift wave instability the shear flow simply shifts the eddies in the radial direction with no tilting. However, for a large velocity shear, the eigenmode solutions for resistive drift waves cease to exist.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2021

Self-organized confinement in whole-device modeling of laboratory magnetospheres

Turbulent self-organization driven by global ideal interchange modes in a dipole-confined plasma is explored with self-consistent, whole-plasma simulations using a flux-tube averaged magnetohydrodynamic model in dipole magnetic geometry. We show the existence of robust particle pinch driven by ideal interchange-mode fluctuations, in which the particles are transported up the density gradient. It is found that the plasma profiles in a dipole field spontaneously relax to a marginally stable state as centrally peaked pressure and density are created by global interchange-mode transport.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Dynamics of the transition corona

Magnetic reconnection between the open and closed magnetic fields in the corona is believed to play a crucial role in the corona/heliosphere coupling. At large scale, the exchange of open/closed connectivity is expected to occur in pseudo-streamer (PS) structures. However, there is neither clear observational evidence of how such coupling occurs in PSs, nor evidence for how the magnetic reconnection evolves. Using a newly developed technique, we enhance the off-limb magnetic fine structures observed with the Atmospheric Imaging Assembly and identify a PS-like feature located close to the northern coronal hole. We first identify that the magnetic topology associated with the observation is a PS, null-point (NP) related topology bounded by the open field. By comparing the magnetic field configuration with the EUV emission regions, we determined that most of the magnetic flux associated with plasma emission are small loops below the PS basic NP and open field bounding the PS topology. In order to interpret the evolution of the PS, we referred to a three-dimensional MHD interchange reconnection modeling the exchange of connectivity between small closed loops and the open field. The observed PS fine structures follow the dynamics of the magnetic field before and after reconnecting at the NP obtained by the interchange model. Moreover, the pattern of the EUV plasma emission is the same as the shape of the expected plasma emission location derived from the simulation. These morphological and dynamical similarities between the PS observations and the results from the simulation strongly suggest that the evolution of the PS, and in particular the opening/closing of the field, occurs via interchange/slipping reconnection at the basic NP of the PS. Besides identifying the mechanism at work in the large-scale coupling between the open and closed fields, our results highlight that interchange reconnection in PSs is a gradual physical process that differs from the impulsive reconnection of the solar-jet model.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2016

Transport and Dynamics in Toroidal Fusion Systems

The study entitled, "Transport and Dynamics in Toroidal Fusion Systems," (TDTFS) applied analytical theory and numerical computation to investigate topics of importance to confining plasma, the fourth state of matter, with magnetic fields. A central focus of the work is how non-thermal components of the ion particle distribution affect the "sawtooth" collective oscillation in the core of the tokamak magnetic configuration. Previous experimental and analytical research had shown and described how the oscillation frequency decreases and amplitude increases, leading to "monster" or "giant" sawteeth, when the non-thermal component is increased by injecting particle beams or by exciting ions with imposed electromagnetic waves. The TDTFS study applied numerical computation to self-consistently simulate the interaction between macroscopic collective plasma dynamics and the non-thermal particles. The modeling used the NIMROD code [Sovinec, Glasser, Gianakon, et al., J. Comput. Phys. 195, 355 (2004)] with the energetic component represented by simulation particles [Kim, Parker, Sovinec, and the NIMROD Team, Comput. Phys. Commun. 164, 448 (2004)]. The computations found decreasing growth rates for the instability that drives the oscillations, but they were ultimately limited from achieving experimentally relevant parameters due to computational practicalities. Nonetheless, this effort provided valuable lessons for integrated simulation of macroscopic plasma dynamics. It also motivated an investigation of the applicability of fluid-based modeling to the ion temperature gradient instability, leading to the journal publication [Schnack, Cheng, Barnes, and Parker, Phys. Plasmas 20, 062106 (2013)]. Apart from the tokamak-specific topics, the TDTFS study also addressed topics in the basic physics of magnetized plasma and in the dynamics of the reversed-field pinch (RFP) configuration. The basic physics work contributed to a study of two-fluid effects on interchange dynamics, where "two-fluid" refers to modeling independent dynamics of electron and ion species without full kinetic effects. In collaboration with scientist Ping Zhu, who received separate support, it was found that the rule-of-thumb criteria on stabilizing interchange has caveats that depend on the plasma density and temperature profiles. This work was published in [Zhu, Schnack, Ebrahimi, et al., Phys. Rev. Lett. 101, 085005 (2008)]. An investigation of general nonlinear relaxation with fluid models was partially supported by the TDTFS study and led to the publication [Khalzov, Ebrahimi, Schnack, and Mirnov, Phys. Plasmas 19, 012111 (2012)]. Work specific to the RFP included an investigation of interchange at large plasma pressure and support for applications [for example, Scheffel, Schnack, and Mirza, Nucl. Fusion 53, 113007 (2013)] of the DEBS code [Schnack, Barnes, Mikic, Harned, and Caramana, J. Comput. Phys. 70, 330 (1987)]. Finally, the principal investigator over most of the award period, Dalton Schnack, supervised a numerical study of modeling magnetic island suppression [Jenkins, Kruger, Hegna, Schnack, and Sovinec, Phys. Plasmas 17, 12502 (2010)].

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Multi-scale experimental study of creep-fatigue failure initiation in a 709 Stainless Steel alloy using high resolution digital image correlation. Final report

This report discusses the efforts made towards the multi-scale study of the creep-fatigue response of stainless steel 709. Multiple experimental techniques, including digital image correlation (DIC) and electron backscatter diffraction (EBSD), were used in assessing the evolution of damage accumulation during fatigue, creep-fatigue, and thermomechanical fatigue of alloy 709. The role of microstructural features, hold times, temperature, and loading profiles were quantified and the interchangeability of temperature and time was investigated. Finally, a thermomechanical fatigue model was shown to predict well the failure of samples with varying loading profiles. Results indicate that strain accumulation in 709 steel happens primarily near grain boundaries (GBs) with the strains around GBs being inversely proportional to their measured residual burgers vector. Furthermore, hot-spots for strain accumulation were shown to be the locations of eventual microcrack nucleation. The introduction of hold times to the periodic loading cycle increases the damage accumulation rate and thus shortens the fatigue life of samples–this was true for both room temperature and high temperature. Thermomechanical cycling was shown to have little effect on the end life of samples, with the damage accumulation rate for in-phase and out-of-phase cycling being similar to isothermal fatigue. The interchangeability of time and temperature was shown to be possible within the studied load, time, and temperature ranges, since deformation mechanisms driving strain accumulation did not change with temperature (up to 650°C). The Neu-Sehitoglu thermomechanical fatigue model, with constants obtained from the literature and in part from our experimental results, was applied to predict failure of isothermal creep-fatigue samples. The main objectives of this work were to:(a) Perform high-resolution digital image correlation measurements (HiDIC) for alloy 709; (b) Quantify and assess damage accumulation at the microstructure under fatigue, TMF, and creep-fatigue conditions of 709; (c) Study of the role of hold times, i.e., adding a creep component, to: Room temperature cycling, High temperature cycling, Failure; (d) Study the mechanisms of thermomechanical fatigue in this alloy; (e) Investigate the existence of a time-temperature interchangeability criterion to aid in accelerated creep-fatigue testing; (f) Establish the validity of a combined creep-fatigue model for life prediction for 709. This report is structured with each subsequent section detailing the efforts, results, and conclusions related to each of the objectives described, mostly in chronological order.

36 MATERIALS SCIENCE↗
NASA NTRS1985

A simulation of high latitude F-layer instabilities in the presence of magnetosphere-ionosphere coupling

A simulation of inertial high-latitude ionospheric interchange instabilities, including magnetospheric coupling effects is presented. It is shown that the primary magnetosphere-ionosphere coupling effect is to incorporate the inertia of the magnetospheric plasma in the analysis. The following conclusions are drawn from the simulation: (1) magnetospheric coupling effects reduce the growth rate of the interchange instability, (2) striations produced by the inertial interchange instability develop in a different manner than in the noninertial regime, and (3) striations produced in the inertial regime are more isotropic and spread out, resulting in irregularities oriented perpendicular to those produced in the noninertial case.

Mitchell, H. G., Jr.↗
NASA NTRS1986

Efficient decentralized consensus protocols

Decentralized consensus protocols are characterized by successive rounds of message interchanges. Protocols which achieve a consensus in one round of message interchange require O(N-squared) messages, where N is the number of participants. In this paper, a communication scheme, based on finite projective planes, which requires only O(N sq rt N) messages for each round is presented. Using this communication scheme, decentralized consensus protocols which achieve a consensus within two rounds of message interchange are developed. The protocols are symmetric, and the communication scheme does not impose any hierarchical structure. The scheme is illustrated using blocking and nonblocking commit protocols, decentralized extrema finding, and computation of the sum function.

Lakshman, T. V.↗