Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Data Driven Analysis”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,135 records · Page 63

Porous Flow Modeling of Axial Gas Redistribution in Fragmented LWR Fuel Rods using MOOSE

Understanding how gas axially redistributes within fragmented fuel pellets is crucial for predicting the behavior of Light Water Reactor (LWR) fuel rods, particularly during transient and accidental scenarios. The time scale of this phenomenon plays a fundamental role in determining the progression and hazard of a Loss Of Coolant Accident (LOCA), especially when high burn-up fuel in a severe state of fragmentation is involved. Here, this study presents a Computational Fluid Dynamics (CFD) model developed within the Multiphysics Object-Oriented Simulation Environment (MOOSE) to predict the time-scale of plenum depressurization in Light-Water Reactor (LWR) fuel rods driven by axial gas transport through fragmented pellets. The model examines the effects of incorporating non-linearities in the friction term by comparing the results with experimental data. The experiment employed surrogate fuel rods containing pellets subjected to mechanical and/or thermal loadings to simulate various severity of cracking, and aimed at studying the influence of fuel conditions on axial gas redistribution. The results of this analysis indicate that under certain flow regime conditions - determined by the value of an equivalent Reynolds number - accounting for the non-linear friction term in Navier-Stokes equations guarantees better predictions for the time-scale of plenum depressurization. Also, the model enabled the simulation of the pressure decay by assigning distinct permeability values to each pellet instead of a single uniform value. Multiple simulations were run across all possible pellet position combinations, having each pellet assigned with values of permeability extracted from the experimental data. This allows to quantify the impact of the considering various non-uniform distributions of permeability on the dynamics of axial gas redistribution. The present work findings enhance the understanding of axial gas transport, and provide valuable insights for the integration of a model for predicting the axial gas redistribution during a LOCA scenario into the BISON fuel performance code.

11 - NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗

Structural Evidence of Interanionic Hydrogen Bonding in Phosphoric Acid Solutions

Interanionic hydrogen bonding (IAHB) is a noncovalent interaction between like-charged ions that challenges conventional electrostatic understanding. This study provides direct structural evidence of IAHB in concentrated aqueous phosphoric acid (PA) solutions, which exhibit >60% dissociation under these conditions. Oxygen K-edge X-ray absorption fine structure spectroscopy, combined with electron affinity time-dependent density functional theory calculations, reveals the formation of stable, cyclic phosphate-phosphate IAHB dimers at PA concentrations ≥7 M. Extended X-ray absorption fine structure data show distinct long-range ordering consistent with these dimers, and near-edge X-ray absorption fine structure spectra confirm a concentration-dependent transition from monomeric to dimeric species. Energy decomposition analysis through density functional theory shows that the formation of solution-phase IAHB is energetically favored and is attributed to polarization of and the charge transfer between the two fragments driven by the surrounding solvent molecules, in addition to the permanent electrostatics. These findings offers crucial structural insights into the H-bonded networks in concentrated PA, highlighting the critical role of solvent in facilitating anion–anion association.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

To Fail or not to Fail: An Exploration of Machine Learning Techniques for Predictive Maintenance

Predictive maintenance refers to the ability to predict when machinery or systems need to be maintained. Making an accurate prediction is quite challenging given the costs for both over-estimating (unnecessary maintenance and reduction in availability of assets) and under-estimating (untimely breakdowns and possible loss of equipment or lives). To address these challenges researchers were able to develop new approaches for analyzing oil samples taken extracting samples from oil-wetted machinery that may provide information critical to developing predictive capabilities. We consider the problem from both supervised (though data limited) and unsupervised approaches and provide a first look into a data driven approach for identification of condition indicators. Through this work we identify a collection of candidate features that can form the basis of condition indicators for both a high level discrimination of failure vs. normal operation as well as a set for potential failure mode identification. Finally, we present an anomaly detection framework for detecting failures which can be a viable solution for an onboard analysis tool in deployed systems.

predictive maintenance, anomaly detection, Laserne↗

Turbulence and Cloudiness in Cumulus-Topped Marine Boundary Layers

Turbulence and cloudiness in cumulus-topped marine boundary layers are studied using data from the Atmospheric Radiation Measurement (ARM) Eastern North Atlantic (ENA) site. The analysis includes eight periods of nonprecipitating shallow cumulus clouds, spanning 141 h and encompassing 603 individual clouds. On average, the cumulus had bases at 558 m, were 99 m thick, had a chord length of 500 m, and exhibited an hourly base-layer cloudiness of 12%. Changes in cloud fraction were primarily driven by variations in cloud number not chord length. High-resolution Doppler lidar and cloud radar observations were used to estimate updraft and downdraft mass fluxes in the cumulus base layer under both clear and cloudy regions. Cloudy updrafts contributed only ∼23% of the total updraft mass flux at cloud base, indicating that most upward mass flux originated from clear-air eddies. Updraft strength, rather than updraft fraction, was found to predominantly control both clear and cloudy updraft mass fluxes. Cloud-base cloudiness strongly correlated with cloudy updraft mass flux but showed negligible correlation with clear-air updraft mass flux. Clear-air downdraft mass flux exhibited a strong relationship with the ratio of surface buoyancy to inversion strength. Mesoscale analysis revealed that moist patches had greater cloudiness, updraft mass flux, and vertical velocity variance compared to dry patches. Additionally, mesoscale base-layer cloud fraction was highly and significantly correlated with cloudy updraft mass flux in both dry and moist environments. Results presented herein have implications for cumulus parameterization development along with climatological and model evaluation studies conducted at the ENA site.

cumulus clouds↗

Hourly Load Profile Dataset for Federal, State, and Municipal Electric Vehicle Fleets in the United States

The electrification of U.S. federal, state, and municipal fleets is accelerating rapidly, driven by an increased availability of competitive electric vehicle (EV) options and supportive policies and targets. The dataset described in this report, accessible at data.nrel.gov/submissions/280, provides a critical foundation for identifying fleet electricity demand, projecting these future demands, and developing actionable strategies to support the widespread electrification of government fleets. The dataset incorporates available fleet data, including 54% of federal agency vehicles approved for analysis (notably, the U.S. Postal Service is absent). Additionally, it includes data from 50,000 state government vehicles and 94,000 local government vehicles. While this represents a small fraction of the 4.4 million vehicles owned by state and local governments reported by the Federal Highway Administration (2022), the framework supports future expansion as more fleet inventory data become available.

33 ADVANCED PROPULSION SYSTEMS↗

LDRD Abbreviated report: High-Order General-Discrete-Ordinates Method Enabling Efficient Deterministic Transport in Hydrodynamic Simulations

Deterministic transport simulations for national-security and energy applications often operate in high-dimensional phase-space, where accuracy and cost both become major challenges. A common numerical artifact in such problems is the “ray-effect,” which appears as unphysical streaks. Beyond misinterpretation, these artifacts can contaminate tightly coupled physics, such as fluid dynamics, radiation-hydrodynamics, and laser-plasma interactions, eroding the predictive capability of entire multiphysics workflows. Our objective was to make high-dimension studies practical on modern hardware while mitigating the ray-effect without relying on prohibitively expensive sampling approaches such as Monte Carlo methods. We developed the Generic Discretization Library (GenDiL), a Graphics Processing Unit (GPU)-first framework that uses high-order Discontinuous Galerkin (DG) methods and matrix-free algorithms to reduce memory usage and improve computational efficiency, critical for phase-space simulations. GenDiL supports phase-space adaptivity in both mesh size and polynomial order (hp-adaptivity) to place resolution only where it is needed. A central capability is Local Dimensional Refinement (LDR), which couples lower-dimension continuum models to higher-dimension kinetic models through stable and conservative interfaces, so that high-fidelity physics is applied only in regions where it is essential. Building on the GenDiL framework, we developed the General SN (GSN) family of algorithms as a true generalization of the polar SN approach (discrete ordinates, often denoted SN). Rather than tying discrete ordinates to a specific polar change of coordinates, GSN formulates transport on an arbitrary change of coordinates chosen to reduce ray-effect. We studied two complementary variants: an analytic variant, where the coordinate map is prescribed in advance by a closed-form function; and a data-driven variant, where a quantity of interest, such as the net flux, guides the coordinate system. GenDiL provides the library infrastructure for efficient GPU execution, but the GSN concept is algorithmic and independent of any one library. Across representative high-dimension tests, including non-symmetric solutions, both variants delivered strong ray-effect mitigation at practical cost, moving four- to six-dimensional analysis toward repeatable, routine studies.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

Space shuttle main engine definition (phase B). Volume 5: Valves and interconnects

The steady state thermodynamic cycle balance of the single preburner staged combustion engine, coupled with dynamic transient analyses, dictated in detail the location and requirements for each valve defined in this volume. Valve configuration selections were influenced by overall engine and vehicle system weight and failure mode determinations. Modulating valve actuators are external to the valve and are line replaceable. Development and satisfactory demonstration of a high pressure dynamic shaft seal has made this configuration practical. Pneumatic motor driven actuators that use engine pumped hydrogen gas as the working fluid are used. The helium control system is proposed as a module containing a cluster of solenoid actuated valves. The separable couplings and flanges are designed to assure minimum leakage with minimum coupling weight. The deflection of the seal surface in the flange is defined by finite element analysis that has been confirmed with test data. The seal design proposed has passed preliminary pressure cycling and thermal cycling tests.

Schultz, D. F.↗

The Interaction Between Dynamics and Chemistry of Ozone in the Set-up Phase of the Northern Hemisphere Polar Vortex

The morphology and evolution of the stratospheric ozone (O3) distribution at high latitudes in the Northern Hemisphere (NH) are examined for the late summer and fall seasons of 1999. This time period sets the O3 initial condition for the SOLVE/THESEO field mission performed during winter 1999-2000. In situ and satellite data are used along with a three-dimensional model of chemistry and transport (CTM) to determine the key processes that control the distribution of O3 in the lower-to-middle stratosphere. O3 in the vortex at the beginning of the winter season is found to be nearly constant from 500 to above 800 K with a value at 3 ppmv +/- approx. 10%. Values outside the vortex are up to a factor of 2 higher and increase significantly with potential temperature. The seasonal time series of data from POAM shows that relatively low O3 mixing ratios, which characterize the vortex in late fall, are already present at high latitudes at the end of summer before the vortex circulation sets up. Analysis of the CTM output shows that the minimum O3 and increase in variance in late summer are the result of: 1) stirring of polar concentric O3 gradients by nascent wave-driven transport, and 2) an acceleration of net photochemical loss with decreasing solar illumination. The segregation of low O3 mixing ratios into the vortex as the circulation strengthens through the fall suggests a possible feedback role between O3 chemistry and the vortex formation dynamics. Trajectory calculations from O3 sample points early in the fall, however, show only a weak correlation between initial O3 mixing ratio and potential vorticity later in the season consistent with order-of-magnitude calculations for the relative importance of O3 in the fall radiative balance at high latitudes. The possible connection between O3 chemistry and the dynamics of vortex formation does suggest that these feedbacks and sensitivities need to be better understood in order to make confident predictions of the recovery of NH O3.

Kawa, S. R.↗

Comparisons of Observations with Results from 3D Simulations and Implications for Predictions of Ozone Recovery

Although chemistry and transport models (CTMs) include the same basic elements (photo- chemical mechanism and solver, photolysis scheme, meteorological fields, numerical transport scheme), they produce different results for the future recovery of stratospheric ozone as chlorofluorcarbons decrease. Three simulations will be contrasted: the Global Modeling Initiative (GMI) CTM driven by a single year\'s winds from a general circulation model; the GMI CTM driven by a single year\'s winds from a data assimilation system; the NASA GSFC CTM driven by a winds from a multi-year GCM simulation. CTM results for ozone and other constituents will be compared with each other and with observations from ground-based and satellite platforms to address the following: Does the simulated ozone tendency and its latitude, altitude and seasonal dependence match that derived from observations? Does the balance from analysis of observations? Does the balance among photochemical processes match that expected from observations? Can the differences in prediction for ozone recovery be anticipated from these comparisons?

Douglass, Anne R.↗

Cloud Computing Applications in Support of Earth Science Activities at Marshall Space Flight Center

Currently, the NASA Nebula Cloud Computing Platform is available to Agency personnel in a pre-release status as the system undergoes a formal operational readiness review. Over the past year, two projects within the Earth Science Office at NASA Marshall Space Flight Center have been investigating the performance and value of Nebula s "Infrastructure as a Service", or "IaaS" concept and applying cloud computing concepts to advance their respective mission goals. The Short-term Prediction Research and Transition (SPoRT) Center focuses on the transition of unique NASA satellite observations and weather forecasting capabilities for use within the operational forecasting community through partnerships with NOAA s National Weather Service (NWS). SPoRT has evaluated the performance of the Weather Research and Forecasting (WRF) model on virtual machines deployed within Nebula and used Nebula instances to simulate local forecasts in support of regional forecast studies of interest to select NWS forecast offices. In addition to weather forecasting applications, rapidly deployable Nebula virtual machines have supported the processing of high resolution NASA satellite imagery to support disaster assessment following the historic severe weather and tornado outbreak of April 27, 2011. Other modeling and satellite analysis activities are underway in support of NASA s SERVIR program, which integrates satellite observations, ground-based data and forecast models to monitor environmental change and improve disaster response in Central America, the Caribbean, Africa, and the Himalayas. Leveraging SPoRT s experience, SERVIR is working to establish a real-time weather forecasting model for Central America. Other modeling efforts include hydrologic forecasts for Kenya, driven by NASA satellite observations and reanalysis data sets provided by the broader meteorological community. Forecast modeling efforts are supplemented by short-term forecasts of convective initiation, determined by geostationary satellite observations processed on virtual machines powered by Nebula.

Molthan, Andrew L.↗

Computational Tools and Workflows for Quantitative Risk Assessment and Decision Support for Geologic Carbon Storage Sites: Progress and Insights from the U.S. DOE’s National Risk Assessment Partnership

The 2005 Intergovernmental Panel on Climate Change (IPCC) Special Report on CCS raised the profile of CO2 capture and storage (CCS) as an important technology for reducing greenhouse gas (GHG) emissions. CCS is now recognized as a key component of most climate change mitigation scenarios. Since publication of that report the international research, development, and deployment (RD&D) community has advanced key technical aspects, clarified regulatory requirements, explored value chain and infrastructure solutions, and developed incentive paradigms to enable and promote large-scale deployment of CCS. These efforts have included research to better characterize geologic storage resources, to improve injection performance and storage efficiency, to assess and manage subsurface environmental risks, and to advance monitoring technologies to assure system conformance. These efforts have helped to build confidence in the viability of geologic carbon storage (GCS), but stakeholder concerns about long-term risks and liability associated with GCS remain a hurdle to broad acceptance and large-scale deployment of CCS. Since 2010, the U.S. DOE’s National Risk Assessment Partnership (NRAP) – a research collaboration between five contributing national laboratories – has worked to establish and demonstrate methods and tools to quantify and manage the subsurface environmental risks associated with GCS, amidst uncertainty. This work supports the Office of Fossil Energy and Carbon Management Carbon Transport and Storage Program’s goal of advancing safe and secure commercial-scale GCS deployment. To address the technical challenge of simulating the physical response of the GCS site to large-scale CO2 injection, NRAP has adopted an approach that relies on coupling computationally efficient reduced-order and/or data-driven proxy models of important system components (i.e., storage reservoir, sealing caprock, leakage pathways, intermediate formations, overlying groundwater aquifers, and the atmosphere) in integrated assessment framework. That integrated model of the physical system is complemented with fit-for purpose functionality to support site characterization and risk-related decisions. The recently released NRAP Phase II toolset includes the Open-Source Integrated Assessment Model (NRAP-Open-IAM) for evaluation of trends in leakage risk and potential impact, tools to support monitoring design optimization (Designs for Risk Evaluation and Management – DREAM v3.0 and Passive Seismic Monitoring Tool - PSMT), and tools for state of stress evaluation (State-of-Stress Analysis Tool - SOSAT) and forecasting induced seismicity risk. The NRAP team has also released a pair of reports describing conceptual workflows to incorporate physics-based, quantitative risk assessment into many of the design, planning, operation, and closure decisions for GCS projects. An online catalogue highlights published studies where these tools and methods are demonstrated. In this presentation, the utility of these products to assess risks and address key stakeholder questions will be highlighted through examples, and related insights about the safety and security of geologic carbon storage in qualified storage sites will be discussed. The prospect of rapid, large-scale deployment of GCS technology to aggressively reduce anthropogenic CO2 emissions requires careful consideration of interference between multiple commercial-scale storage projects within a basin. Going forward, NRAP is expanding and adapting site-scale risk quantification tools and methods to enable assessment of risks and inform management decisions for basin-scale deployment. Increasingly, this work will leverage next-generation approaches for surrogate modelling, fast prediction, and advanced visualization enabled by machine learning and artificial intelligence to promote virtual learning, scenario evaluation, and augment risk-based decision making.

quantitative risk assessment, geologic carbon stor↗

Multi-Omics Driven Metabolic Network Reconstruction and Analysis of Lignocellulosic Carbon Utilization in Rhodosporidium toruloides

An oleaginous yeast Rhodosporidium toruloides is a promising host for converting lignocellulosic biomass to bioproducts and biofuels. In this work, we performed multi-omics analysis of lignocellulosic carbon utilization in R. toruloides and reconstructed the genome-scale metabolic network of R. toruloides . High-quality metabolic network models for model organisms and orthologous protein mapping were used to build a draft metabolic network reconstruction. The reconstruction was manually curated to build a metabolic model using functional annotation and multi-omics data including transcriptomics, proteomics, metabolomics, and RB-TDNA sequencing. The multi-omics data and metabolic model were used to investigate R. toruloides metabolism including lipid accumulation and lignocellulosic carbon utilization. The developed metabolic model was validated against high-throughput growth phenotyping and gene fitness data, and further refined to resolve the inconsistencies between prediction and data. We believe that this is the most complete and accurate metabolic network model available for R. toruloides to date.

09 BIOMASS FUELS↗

New Perspective on Phase Space Density Analysis for Outer Radiation Belt Enhancements: The Influence of MeV Electron Injections

Observation of growing phase space density (PSD) peak in the outer electron radiation belt has been considered evidence for local wave-driven acceleration as a primary cause of radiation belt enhancement. However, recent studies showed that strong substorm-associated MeV electron injections can also cause significant radiation belt enhancements on fast timescales (~10s min). Such rapid enhancements pose challenges for determining true spatial PSD profiles. To address this, we conduct a detailed spatiotemporal analysis of electron flux and PSD during an enhancement event, using Van Allen Probes data. Our analysis reveals rapid and intermittent flux enhancements. During these rapid enhancements, inbound spacecraft observed false PSD peaks, due to spacecraft's relatively slow movement. However, we identify time intervals of stable fluxes between enhancements, enabling us to determine quasi-stationary PSD profiles with no noticeable peaks. This study provides new insights into accurate PSD analysis, critical for understanding the mechanisms underlying radiation belt enhancements.

79 ASTRONOMY AND ASTROPHYSICS↗

Investigation of core transport changes in DIII-D discharges with off-axis T e profile peaks

Abstract DIII-D discharges that transition to H-mode solely with off-axis electron cyclotron heating (ECH) often exhibit strong off-axis peaking of electron temperature profiles at the heating location. Electron heat transport properties near these off-axis temperature peaks have been studied using modulated ECH. The Fourier analyzed electron temperature data have been used to infer electron thermal diffusivity. Comparisons with numerical solutions of the time-dependent electron thermal equation find that the data are consistent with a narrow region with electron diffusivity χ e an order of magnitude lower than the average value across the plasma, suggesting an electron internal transport barrier (ITB) near the ECH heating location. Detailed profile analysis and equilibrium reconstructions suggest that the formation of these ITBs are correlated with off-axis values of the safety factor q being near 1. Furthermore, the ECH driven H-mode discharges demonstrate more rapid electron heating rate near the ECH deposition location than L-mode discharges with higher auxiliary ECH heating power. Additional modeling attributes this difference to the modification of electron heat transport in the core at the L-H transition, which also sustains the off-axis electron temperature peaks.

Physics↗

Data System Implications Derived from User Application Requirements for Satellite Data

An investigation of the data system needs as driven by users of space acquired Earth observation data is documented. Two major categories of users, operational and research, are identified. Limiting data acquisition alleviates some of the delays in processing thus improving timeliness of the delivered product. Trade offs occur between timeliness and data distribution costs, and between data storage and reprocessing. The complexity of the data system requirements to apply space data to users' needs is such that no single analysis suffices to design and implement the optimum system. A series of iterations is required with analyses of the salient problems in a general way, followed by a limited implementation of benefit to some users with a continual upgrade in system capacity, functions, and applications served. The resulting most important requirement for the data system is flexibility to accommodate changing requirements as the system is implemented.

Neiers, J.↗

The Growth of Protein Crystals Using McDUCK

Most of the current microgravity crystal growth hardware is optimized to produce crystals within the limited time available on orbit. This often results in the actual nucleation and growth process being rushed or the system not coming to equilibrium within the limited time available. Longer duration hardware exists, but one cannot readily pick out crystals grown early versus those which nucleated and grew more slowly. We have devised a long duration apparatus, the Multi-chamber Dialysis Unit for Crystallization Kinetics, or McDUCK. This apparatus-is a series of protein chambers, stacked upon a precipitant reservoir chamber. All chambers are separated by a dialysis membrane, which serves to pass small molecules while retaining the protein. The volume of the Precipitant chamber is equal to the sum of the volumes of the protein chamber. In operation, the appropriate chambers are filled with precipitant solution or protein solution, and the McDUCK is placed standing upright, with the precipitant chamber on the bottom. The precipitant diffuses upwards over time, with the time to reach equilibration a function of the diffusivity of the precipitant and the overall length of the diffusion pathway. Typical equilibration times are approximately 2-4 months, and one can readily separate rapid from slow nucleation and growth crystals. An advantage on Earth is that the vertical precipitant concentration gradient dominates that of the solute, thus dampening out solute density gradient driven convective flows. However, large Earth-grown crystals have so far tended to be more two dimensional. Preliminary X-ray diffraction analysis of lysozyme crystals grown in McDUCK have indicated that the best, and largest, come from the middle chambers, suggesting that there is an optimal growth rate. Further, the improvements in diffraction resolution have been better signal to noise ratios in the low resolution data, not an increase in resolution overall. Due to the persistently large crystals grown we are currently proposing McDUCK for the growth of macromolecule crystals for use in neutron diffraction studies.

Ewing, Felicia↗

Future Model-Based Systems Engineering Vision and Strategy Bridge for NASA

A vision for the future of model-based systems engineering (MBSE) at NASA in 2029 and a strategy bridge towards that future are presented. Strategic thinking and leading change concepts were used to analyze reports and presentations on global trends and visionary thinking about the future of systems and digital engineering. The context, strategic time horizon, stakeholders, strategic challenges, strategic advantages, driving forces, and opportunities were considered. The analysis resulted in a future vision of MBSE that shows what NASA systems engineers and digital machines will do to perform rapid, extraordinary, and unprecedented missions. The NASA systems engineer, in this future vision, works with a global project team in a virtual and collaborative environment, engineers the system, and uses digital approaches as the routine and default way of working. The digital machines provide data-driven and automated mission designs; have a backbone of program and project management, systems engineering, and product life-cycle management; and are a knowledge-sharing infrastructure. The NASA systems engineer and the systems engineering team are envisioned to use digital machines to plan and perform rapid exploration missions, develop a digital twin that lasts across the life cycle, and develop enduring and adaptable systems. NASA has an engineering enterprise and a life-cycle management framework that endure, adapt, and respond. A strategy bridge based on the Baldrige Criteria for Performance Excellence Framework and lessons learned from a recent MBSE initiative illuminates a way forward from today to this desired future. The bridge lays out a strategy for leaders and recommends investments of today for immediate benefits and for benefits in 2029.

model-based systems engineering, digital engineeri↗

Towards a New nCTEQ global nPDF release

We discuss the foundation for a new global nCTEQ nuclear PDF analysis, combining a number of our previous analyses into one consistent framework with updates to the underlying theoretical treatment as well as the addition of new available data. In particular, the new global release will be the first nCTEQ release containing neutrino DIS scattering data in a consistent manner together with JLab high-x DIS data and new LHC p-Pb data. These additions will improve the data-driven description of nuclear PDFs in new regions, especially the strange quark and the gluon PDF at low-x.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗