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At least 127 records · Page 7

A generalized Lyapunov theory for robust root clustering of linear state space models with real parameter uncertainty

The problem of analyzing and designing controllers for linear systems subject to real parameter uncertainty is considered. An elegant, unified theory for robust eigenvalue placement is presented for a class of D-regions defined by algebraic inequalities by extending the nominal matrix root clustering theory of Gutman and Jury (1981) to linear uncertain time systems. The author presents explicit conditions for matrix root clustering for different D-regions and establishes the relationship between the eigenvalue migration range and the parameter range. The bounds are all obtained by one-shot computation in the matrix domain and do not need any frequency sweeping or parameter gridding. The method uses the generalized Lyapunov theory for getting the bounds.

Yedavalli, R. K.↗

GeoThermalCloud: Cloud Fusion of Big Data and Multi-Physics Models using Machine Learning for Discovery, Exploration, and Development of Hidden Geothermal Resources

The primary goals of this project are exploring hidden geothermal resources in the U.S.A. and designing profitable enhanced geothermal systems (EGS). Many processes and parameters control geothermal exploration and energy production from geothermal fields. Diverse datasets (e.g., geology, geochemistry, geophysics, satellite, airborne geophysics) are available to help characterize subsurface geothermal conditions. Sparse and multi-scale characteristics of these datasets prohibit properly leveraging these datasets for geothermal exploration and profitable EGS design. Recent advancements in machine learning (ML) promise to resolve these issues. The tremendous challenges and risks of geothermal exploration and production bring the demand for novel ML methods and tools that can (1) analyze large field datasets, (2) assimilate model simulations (large inputs and outputs), (3) process sparse datasets, (4) perform transfer learning (between sites with different exploratory levels), (5) extract hidden geothermal signatures in the field and simulation data, (6) label geothermal resources and processes, (7) identify high-value data acquisition targets, and (8) guide geothermal exploration and production by selecting optimal exploration, production, and drilling strategies. To address these necessities, ML-based geothermal resources exploration and enhanced geothermal systems (EGS) design tools have been developed. The exploration tool is called GeoThermalCloud and EGS design tool is called GeoDT-ML. GeoThermalCloud (https://github.com/SmartTensors/GeoThermalCloud.jl) utilizes a LANL unsupervised ML platform called SmartTensors (https://tensors.lanl.gov/) to automate data analyses and interpretations by extracting hidden signatures to identify geothermal prospects. Also, it enables the identification of critical measurements needed to identify geothermal resource signatures. Alternatively, GeoDT-ML (https://github.com/SmartTensors/GeoThermalCloud.jl/tree/master/EGS) is an ML-based alternative to GeoDT (https://github.com/GeoDesignTool/GeoDT.git), a fast, simplified multi-physics solver to evaluate EGS project designs in uncertain geologic systems. GeoDT-ML leverages recent advances in deep learning and high-performance computing. It is a faster and simpler version of GeoDT. To make this project a success, we used capabilities of LANL, PNNL, Google, Stanford, and Julia Computing. We analyzed eight datasets of the U.S.A. using GeothermalCloud and demonstrated potential highly prospective geothermal resources and identified key factors defining highly prospective sites. The first data set includes 44 locations in southwest New Mexico and 18 geological, hydrogeological, geophysical, geothermal, geochemical attributes. We defined low- and medium-temperature hydrothermal systems and discovered a new highly prospective site. The second data set analyzed 18 shallow water chemistry attributes at 14,342 locations in the Great Basin. It demarcated modestly, moderately, and highly prospective sites including key attributes for each type of prospectivity. The third data set analyzed Utah FORGE data including satellite (InSAR), geophysical (gravity, seismic), geochemical, and geothermal attributes. Here, we performed prospectivity analysis to identify future drilling locations using geological, geochemical, and geophysical attributes. Maps of temperature at depth and heat flow are constructed based on the available data. Prospectivity maps were generated, and drilling locations were proposed for future geothermal field exploration. The fourth data set analyzed 21 attributes at 120 locations in Tularosa Basin, New Mexico; data comes from past play fairway analyses in this region. ML analyses identified geothermal signatures associated with modestly, moderately, and highly hydrothermal systems. We also defined dominant attributes and spatial distribution of the geothermal signatures. The fifth, sixth, seventh, and eighth datasets include Tohatchi Springs, New Mexico, Hawaii, Brady site, Nevada, and EGS Collab, respectively. Moreover, we coupled GeothermalCloud and magnetotellurics data to pinpoint drilling locations for developing geothermal projects in the Tularosa Basin, New Mexico. GeothermalCloud found potential prospective locations for geothermal resources near White Sands Missile Range and McGregor Range at Fort Bliss. Magnetotellurics data determined the potential depth (~1800m) of geothermal prospects at McGregor Range based on apparent resistivity structures/layers in the subsurface. The McGregor Range consists of three resistivity layers and two resistivity structures. Magnetotellurics data also helps identify that the western portion of the McGregor Range has thick and low-resistivity earth materials. The low resistivity to the west is most likely for a fault system. Assuming temperature is consistent with a geothermal reservoir, the west-central part of the McGregor Range has the highest geothermal potential because of the increase in porosity and associated permeability attributed to the interpreted fault system. Also, we devised a coupling strategy between a process model and GeothermalCloud to characterize hydrogeological conditions and geothermal conditions, respectively. The process model characterizes hydrogeological and geothermal conditions on highly prospective geothermal sites provided by GeothermalCloud. We developed a physics-informed neural network (PINN) version of the Burns equation that can be easily coupled with GeothermalCloud. Furthermore, we performed an optimal design decision maximizing the economic value of an EGS power plant. This study optimized the range of well spacing between injection and production wells maximizing net present value in dollars (NPV). For this task, we used the GeoDT to simulate the Utah FORGE EGS development cycle from the initial well design to the end of production. Next, we accomplished another crucial task, which is predicting permeability of geothermal reservoirs. Predicting permeability of geothermal reservoirs is a non-trivial task because of huge computational runtime of simulation and lack of measurements. To avoid these limitations, we used easy-to-measure chemical concentrations in the subsurface as measurement data and convolutional neural network based ML model of a high-fidelity model. Next, we predicted permeability using Markov chain Monte Carlo simulation. We found that Markov chain Monte Carlo simulation predicts permeability with a high certainty if the prediction zone in the simulation area has chemical concentration data. Finally, we analyzed the DOE funded INGENIOUS and GeoDAWN projects data. For discovering hidden geothermal systems in the Great Basin, the INGENIOUS project accumulated old data, collected new data, and released them in 2022. The dataset includes a total of 24 geological, geophysical, and geochemical attributes. Data resolution and scale significantly vary prohibiting an appropriate usage. To avoid such limitations, we brought all data in the same resolution and scale by applying the inverse distance weighting interpolation technique for predicting data in unsampled locations. Subsequently, we analyzed LiDAR data of the GeoDAWN project. We received data in tiles format. The DOE’s overarching goal is to use ML on LiDAR data for finding favorable geological structures (e.g., step up faults in Brady, Nevada). To serve the purpose, we need to label favorable geologic structures that correspond to LiDAR data. We wrote an algorithm to label the LiDAR data with the favorable geologic structures.

15 GEOTHERMAL ENERGY↗

Multiobjective optimization in structural design with uncertain parameters and stochastic processes

The application of multiobjective optimization techniques to structural design problems involving uncertain parameters and random processes is studied. The design of a cantilever beam with a tip mass subjected to a stochastic base excitation is considered for illustration. Several of the problem parameters are assumed to be random variables and the structural mass, fatigue damage, and negative of natural frequency of vibration are considered for minimization. The solution of this three-criteria design problem is found by using global criterion, utility function, game theory, goal programming, goal attainment, bounded objective function, and lexicographic methods. It is observed that the game theory approach is superior in finding a better optimum solution, assuming the proper balance of the various objective functions. The procedures used in the present investigation are expected to be useful in the design of general dynamic systems involving uncertain parameters, stochastic process, and multiple objectives.

Rao, S. S.↗

Deep Learning for Modeling Enhanced Geothermal Systems

Enhanced Geothermal Systems (EGS) offer a vast potential to expand the use of geothermal energy. Heat is extracted from this engineered system by injecting cold water into a subsurface fractures, which are in contact with the hot dry rock, and pulled through the production wells. Creating EGS requires improving the natural permeability of hot crystalline rocks. To develop economically viable EGS reservoirs, significant technical barriers (e.g., better stimulation technologies without adequate water and/or permeability) and non-technical barriers (e.g., land access and permitting) must be overcome. In this short conference paper, we present a workflow to address a part of this challenge – “How to develop economically viable EGS using existing technologies?”. Our workflow called the GeoThermalCloud (GTC) for EGS, leverages recent advances in machine learning, deep learning, and cloud computing. This GTC framework is open-source and available at https://github.com/SmartTensors/GeoThermalCloud.jl. The GTC framework provides trained deep learning (DL) models to estimate the net present value of a given EGS design scenario. The Geothermal Design Tool (https://github.com/GeoDesignTool/GeoDT.git), a fast and simplified multi-physics solver, is used to develop a database for training DL models. The database consists of EGS design parameters (inputs to DL model) and their net present value (output of DL model) in uncertain geologic systems. The EGS design parameters for constructing this training database are based on Utah FORGE but include the options of more wells and deeper depths. The DL models are trained by ingesting the EGS design parameters and estimating the corresponding net present value. Such an emulation allows us to screen various EGS designs quickly and identify good development strategies by coupling them with optimization techniques. Our preliminary results show promise in DL emulation of net present value. However, a lot more work is needed to improve the predictive capability of DL models (i.e., extensive hyperparameter tuning is necessary). This will be the primary focus of our future work.

artificial neural networks, geothermal↗

Time domain probabilistic seismic risk analysis using ground motion prediction equations of Fourier amplitude spectra

Modeling of Fourier amplitude spectra (FAS) of seismic motions has gained much attention in engineering seismology. In the past few years, several ground motion prediction equations (GMPEs) and inter-frequency correlation structure of FAS have been established. Due to many preferable characteristics of FAS, probabilistic seismic hazard/risk analysis is rapidly changing from ergodic, spectrum acceleration Sa(T 0 )-based approach to non-ergodic, site-specific, FAS-based approach. This paper presents time domain intrusive framework for probabilistic seismic risk analysis using GMPE of FAS. Herein, methodology for time domain stochastic ground motion modeling based on GMPEs of FAS is presented in some detail. The simulated uncertain motions are modeled as a random process and represented by polynomial chaos Karhunen-Loève expansion. The random process excitations are further propagated into the uncertain structural system using Galerkin stochastic finite element method (SFEM). Probabilistic evolution of structural response is solved, and such solution is used to develop seismic risk for any damage state. The presented framework is illustrated through seismic risk analysis of a four-story building subjected to possible earthquakes from two strike slip faults. The influences of the epistemic uncertainties in source stress drop Δσ and site attenuation κ0 on seismic risk are investigated. The need for non-ergodic seismic risk analysis with source-specific and site specific characterizations is emphasized.

58 GEOSCIENCES↗

Quasi-periodic X-ray eruptions years after a nearby tidal disruption event

Abstract Quasi-periodic eruptions (QPEs) are luminous bursts of soft X-rays from the nuclei of galaxies, repeating on timescales of hours to weeks 1–5 . The mechanism behind these rare systems is uncertain, but most theories involve accretion disks around supermassive black holes (SMBHs) undergoing instabilities 6–8 or interacting with a stellar object in a close orbit 9–11 . It has been suggested that this disk could be created when the SMBH disrupts a passing star 8,11 , implying that many QPEs should be preceded by observable tidal disruption events (TDEs). Two known QPE sources show long-term decays in quiescent luminosity consistent with TDEs 4,12 and two observed TDEs have exhibited X-ray flares consistent with individual eruptions 13,14 . TDEs and QPEs also occur preferentially in similar galaxies 15 . However, no confirmed repeating QPEs have been associated with a spectroscopically confirmed TDE or an optical TDE observed at peak brightness. Here we report the detection of nine X-ray QPEs with a mean recurrence time of approximately 48 h from AT2019qiz, a nearby and extensively studied optically selected TDE 16 . We detect and model the X-ray, ultraviolet (UV) and optical emission from the accretion disk and show that an orbiting body colliding with this disk provides a plausible explanation for the QPEs.

Science & Technology - Other Topics↗

Rapidly changing high-latitude seasonality: implications for the 21st century carbon cycle in Alaska

Seasonal variations in high-latitude terrestrial carbon (C) fluxes are predominantly driven by air temperature and radiation. At present, high-latitude net C uptake is largest during the summer. Recent observations and modeling studies have demonstrated that ongoing and projected climate change will increase plant productivity, microbial respiration, and growing season lengths at high-latitudes, but impacts on high-latitude C cycle seasonality (and potential feedbacks to the climate system) remain uncertain. Here we use ecosys, a well-tested and process-rich mechanistic ecosystem model that we evaluate further in this study, to explore how climate warming under an RCP8.5 scenario will shift C cycle seasonality in Alaska throughout the 21st century. The model successfully reproduced recently reported large high-latitude C losses during the fall and winter and yet still predicts a high-latitude C sink, pointing to a resolution of the current conflict between process-model and observation-based estimates of high-latitude C balance. We find that warming will result in surprisingly large changes in net ecosystem exchange (NEE; defined as negative for uptake) seasonality, with spring net C uptake overtaking summer net C uptake by year 2100. This shift is driven by a factor of 3 relaxation of spring temperature limitation to plant productivity that results in earlier C uptake and a corresponding increase in magnitude of spring NEE from –19 to –144 gC m –2 season –1 by the end of the century. Although a similar relaxation of temperature limitation will occur in the fall, radiation limitation during those months will limit increases in C fixation. Additionally, warmer soil temperatures and increased carbon inputs from plants lead to combined fall and winter C losses (163 gC m –2 ) that are larger than summer net uptake (123 gC m –2 season –1 ) by year 2100. However, this increase in microbial activity leads to more rapid N cycling and increased plant N uptake during the fall and winter months that supports large increases in spring NPP. Due to the large increases in spring net C uptake, the high-latitude atmospheric C sink is projected to sustain throughout this century. Our analysis disentangles the effects of key environmental drivers of high-latitude seasonal C balances as climate changes over the 21st century.

54 ENVIRONMENTAL SCIENCES↗

Reconstructing the Stripping History of the Sagittarius Stream with Neural Networks

The Sagittarius (Sgr) Stream is produced by the ongoing disruption of the Sgr dwarf spheroidal (dSph) galaxy and is thought to contain multiple wraps that were stripped during different pericentric passages. In this study, we introduce a neural-network–based method trained on N-body simulations to infer the stripping time of Sgr Stream stars directly from their phase-space coordinates. We combine spectroscopic data from SEGUE, APOGEE DR17, and LAMOST DR7 low-resolution spectroscopic (LRS) survey with Gaia EDR3 astrometry and distance estimates from the latest StarHorse catalog to identify high-quality Sgr Stream members. Applying our method to these stars, we measure a clear metallicity gradient with stripping time, well described by a linear relation with slope ∼0.3 dex Gyr −1 . We further predict the stripping times of globular clusters previously suggested to originate from the Sgr dSph. M 54, Terzan 7, Terzan 8, and Arp 2 exhibit stripping times consistent with being currently bound to the Sgr remnant. Pal 12, Whiting 1, and NGC 2419 are inferred to have been stripped 0.9 ± 0.1, 1.1 ± 0.2, and 2.1 ± 0.2 Gyr ago, respectively. For NGC 4147 and NGC 5634, whose membership in the Sgr system remains uncertain, our analysis suggests stripping times of 1.1 ± 0.4 and 1.1 ± 0.1 Gyr, respectively, if they are ultimately confirmed as genuine Sgr members. These results demonstrate that data-driven models of dynamical stripping histories offer a promising approach for reconstructing the formation and chemical evolution of the Sgr Stream.

79 ASTRONOMY AND ASTROPHYSICS↗

A two-parameter scintillation spectrometer system for measurement of secondary proton, deuteron, and triton distributions from materials under 558-MeV-proton irradiation

A two-parameter scintillation spectrometer system developed and used to obtain proton, deuteron, and triton double differential cross sections from materials under 558-MeV-proton irradiation is described. The system measures both the time of flight of secondary particles over a 488-cm flight path and the energy deposited in a scintillator, 12.7 cm in diameter and 30.48 cm long. The time resolution of the system is 0.39 nsec. The calculated energy resolution based on this time resolution varies with energy from 1.6 precent to 7.75 percent for 50- and 558-MeV protons. Various systematic and statistical errors are evaluated, and the double differential cross sections for secondary proton and deutron production at 20 deg from a 2.35 g/sq cm thick beryllium target are shown as an example of the results obtainable with this system. The uncertainly in the cross sections for secondary protons varies with particle energy from approximately + or - 9 percent at 50 MeV to approximately + or - 11 percent at 558 MeV.

Beck, S. M.↗

The uncertainty threshold principle - Some fundamental limitations of optimal decision making under dynamic uncertainty

This note shows that the optimal control of dynamic systems with uncertain parameters has certain limitations. In particular, by means of a simple scalar linear-quadratic optimal control example, it is shown that the infinite horizon solution does not exist if the parameter uncertainty exceeds a certain quantifiable threshold; we call this the uncertainty threshold principle. The philosophical and design implications of this result are discussed.

Athans, M.↗

Control technology development

Static and dynamic control design approaches were developed for distributed parameter systems. A hardware flexible beam facility was constructed to demonstrate and verify the theoretical control concepts. Efforts were made in the area of model order estimation for control systems with uncertain or time varying parameters.

Rodriguez, G.↗

Robust control of flexible space vehicles with minimum structural excitation: On-off pulse control of flexible space vehicles

Both feedback and feedforward control approaches for uncertain dynamical systems (in particular, with uncertainty in structural mode frequency) are investigated. The control objective is to achieve a fast settling time (high performance) and robustness (insensitivity) to plant uncertainty. Preshaping of an ideal, time optimal control input using a tapped-delay filter is shown to provide a fast settling time with robust performance. A robust, non-minimum-phase feedback controller is synthesized with particular emphasis on its proper implementation for a non-zero set-point control problem. It is shown that a properly designed, feedback controller performs well, as compared with a time optimal open loop controller with special preshaping for performance robustness. Also included are two separate papers by the same authors on this subject.

Wie, Bong↗

Hubble space telescope Goddard High-Resolution Spectrograph observation of U Geminorum during quiescence: Evidence for a slowly rotating white dwarf

We have obtained a pair of consecutive far-ultraviolet Goddard High-Resolution Spectrograph (GHRS) exposures of the Si IV region of the dwarf nova U Geminorum in early quiescence, 8 days after its return to optical quiescence when the underlying white dwarf dominates the ultraviolet light of the system. Our GHRS observation revealed a fully resolved line profile for the resonance doublet of Si IV. If it is associated with the white dwarf photosphere, then our best synthetic fits are consistent with T (sub eff) = 35,000 K-38,000 K, log g = 8, a rotational velocity of 50 to 100 km per sec, with a modestly enhanced silicon abundance (1.3-2.3 times solar) and our results suggest that at least in U Gem and perhaps in other similar dwarf novae, the missing boundary layer cannot be explained by rapid rotation of the white dwarf. However, the gamma-velocity of the system remains uncertain. If the gamma-velocity is 43 km per sec (Friend et al. 1990), then a gravitational redshift of approximately 50-60 km per sec is implied for the white dwarf. If the gamma-velocity is 84 km per sec (Wade 1981), then a gravitational redshift of only 10-30 km per sec is indicated, which may imply that either the white dwarf has a low (0.5-0.6 of Solar Mass) mass or an extended atmosphere (corona) due to the outburst heating 8 days earlier. The implications of our line-fitting results for the structure and temperature of boundary layers in cataclysmic variables are discussed.

Sion, Edward M.↗

Wavelet transform analysis of the small-scale X-ray structure of the cluster Abell 1367

We have developed a new technique based on a wavelet transform analysis to quantify the small-scale (less than a few arcminutes) X-ray structure of clusters of galaxies. We apply this technique to the ROSAT position sensitive proportional counter (PSPC) and Einstein high-resolution imager (HRI) images of the central region of the cluster Abell 1367 to detect sources embedded within the diffuse intracluster medium. In addition to detecting sources and determining their fluxes and positions, we show that the wavelet analysis allows a characterization of the sources extents. In particular, the wavelet scale at which a given source achieves a maximum signal-to-noise ratio in the wavelet images provides an estimate of the angular extent of the source. To account for the widely varying point response of the ROSAT PSPC as a function of off-axis angle requires a quantitative measurement of the source size and a comparison to a calibration derived from the analysis of a Deep Survey image. Therefore, we assume that each source could be described as an isotropic two-dimensional Gaussian and used the wavelet amplitudes, at different scales, to determine the equivalent Gaussian Full Width Half-Maximum (FWHM) (and its uncertainty) appropriate for each source. In our analysis of the ROSAT PSPC image, we detect 31 X-ray sources above the diffuse cluster emission (within a radius of 24 min), 16 of which are apparently associated with cluster galaxies and two with serendipitous, background quasars. We find that the angular extents of 11 sources exceed the nominal width of the PSPC point-spread function. Four of these extended sources were previously detected by Bechtold et al. (1983) as 1 sec scale features using the Einstein HRI. The same wavelet analysis technique was applied to the Einstein HRI image. We detect 28 sources in the HRI image, of which nine are extended. Eight of the extended sources correspond to sources previously detected by Bechtold et al. Overall, using both the PSPC and the HRI observations, we detect 16 extended features, of which nine have galaxies coincided with the X-ray-measured positions (within the positional error circles). These extended sources have luminosities lying in the range (3 - 30) x 10(exp 40) ergs/s and gas masses of approximately (1 - 30) x 10(exp 9) solar mass, if the X-rays are of thermal origin. We confirm the presence of extended features in A1367 first reported by Bechtold et al. (1983). The nature of these systems remains uncertain. The luminosities are large if the emission is attributed to single galaxies, and several of the extended features have no associated galaxy counterparts. The extended features may be associated with galaxy groups, as suggested by Canizares, Fabbiano, & Trinchieri (1987), although the number required is large.

Grebeney, S. A.↗

A Programmatic and Engineering Approach to the Development of a Nuclear Thermal Rocket for Space Exploration

With the announcement of the Vision for Space Exploration on January 14, 2004, there has been a renewed interest in nuclear thermal propulsion. Nuclear thermal propulsion is a leading candidate for in-space propulsion for human Mars missions; however, the cost to develop a nuclear thermal rocket engine system is uncertain. Key to determining the engine development cost will be the engine requirements, the technology used in the development and the development approach. The engine requirements and technology selection have not been defined and are awaiting definition of the Mars architecture and vehicle definitions. The paper discusses an engine development approach in light of top-level strategic questions and considerations for nuclear thermal propulsion and provides a suggested approach based on work conducted at the NASA Marshall Space Flight Center to support planning and requirements for the Prometheus Power and Propulsion Office. This work is intended to help support the development of a comprehensive strategy for nuclear thermal propulsion, to help reduce the uncertainty in the development cost estimate, and to help assess the potential value of and need for nuclear thermal propulsion for a human Mars mission.

Bordelon, Wayne J., Jr.↗

LMI-Based Generation of Feedback Laws for a Robust Model Predictive Control Algorithm

This technical note provides a mathematical proof of Corollary 1 from the paper 'A Nonlinear Model Predictive Control Algorithm with Proven Robustness and Resolvability' that appeared in the 2006 Proceedings of the American Control Conference. The proof was omitted for brevity in the publication. The paper was based on algorithms developed for the FY2005 R&TD (Research and Technology Development) project for Small-body Guidance, Navigation, and Control [2].The framework established by the Corollary is for a robustly stabilizing MPC (model predictive control) algorithm for uncertain nonlinear systems that guarantees the resolvability of the associated nite-horizon optimal control problem in a receding-horizon implementation. Additional details of the framework are available in the publication.

algorithms↗

A Robustly Stabilizing Model Predictive Control Algorithm

A model predictive control (MPC) algorithm that differs from prior MPC algorithms has been developed for controlling an uncertain nonlinear system. This algorithm guarantees the resolvability of an associated finite-horizon optimal-control problem in a receding-horizon implementation.

Ackmece, A. Behcet↗

Input and Output Performance of M-MRAC in the Presence of Bounded Disturbances

This paper presents performance analysis of novel modified model reference adaptive control (M-MRAC) architecture in conjunction with different robust adaptive laws for the uncertain linear systems subject to unknown but bounded disturbances. It is shown that for any robust adaptive law the tracking error of the M-MRAC system can be arbitrarily decreased in transient and in the steady-state by increasing the adaptation rate, without generating high frequency oscillations in the control signal, which are unavoidable in conventional MRAC systems for large adaptation rates. Moreover, the generated adaptive control signal arbitrary closely tracks the ideal control signal when the error feedback gain is simultaneously increased with the adaptation rate, according to derived rule. The results are demonstrated via simulations.

Stepanyan, Vahram↗