The operator theory of the pseudo-inverse. i - bounded operators.
Operator theory of pseudo-inverse generalized to encompass linear bounded operators on Hilbert spaces
HILBERT SPACE↗
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Results for “Differential operators”
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.
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Quantum theory of time-dependent phenomena treated by the evolution operator technique
Time dependent phenomena in nonrelativistic quantum theory treated by evolution operator technique
QUANTUM THEORY↗
The application of NAVSTAR Differential GPS to civil helicopter operations
Principles concerning the operation of the NAVSTAR Global Positioning Systems (GPS) are discussed. Selective availability issues concerning NAVSTAR GPS and differential GPS concepts are analyzed. Civil support and market potential for differential GPS are outlined. It is concluded that differential GPS provides a variation on the baseline GPS system, and gives an assured, uninterrupted level of accuracy for the civilian community.
Beser, J.↗
Using X-ray radiography to study oxygen flow in a proton exchange membrane electrolyzer operating under balanced pressure conditions
Of the various water electrolyzer technologies, the proton exchange membrane electrolyzer (PEMWE) is one of the best solutions for producing clean hydrogen without releasing CO 2 . In order to allow for widespread use of clean hydrogen, it is necessary to decrease its cost, which is intrinsically related to system operation. Current PEMWE plants operate in differential mode, directly pressurizing hydrogen and benefiting from thermodynamic compression, which increases overall system efficiency. However, high differential pressure above 30 bar can cause membrane stress, resulting in membrane creeping and failure. Pressurizing the water and operating at balanced pressure allows hydrogen to be produced at higher pressures while preserving the integrity of the membrane and porous layers. Nevertheless, the impact of pressurizing water on PEMWE performance must be better understood to maximize performance under balanced pressure conditions. Here, this study examined the impact of water pressure on two-phase flow. A high-pressure electrolyzer setup was developed to perform operando X-ray radiography and examine oxygen transport with high temporal resolution. The imaging segmentation process, developed to capture bubble properties in the channel, was applied to a specific experiment. The results clearly showed that as pressure increased up to 30 bars, the initial bubbly flow transitioned to slug flow, which led to channel saturation with oxygen. This work demonstrates that two-phase flow in an electrolyzer can be studied using X-ray radiography, which has the advantages of fast measurements and the ability to probe dense materials, such as those required for pressurized electrolyzers.
Balanced high pressure operation↗
Correlation of ISS Electric Potential Variations with Mission Operations
Orbiting approximately 400 km above the Earth, the International Space Station (ISS) is a unique research laboratory used to conduct ground-breaking science experiments in space. The ISS has eight Solar Array Wings (SAW), and each wing is 11.7 meters wide and 35.1 meters long. The SAWs are controlled individually to maximize power output, minimize stress to the ISS structure, and minimize interference with other ISS operations such as vehicle dockings and Extra-Vehicular Activities (EVA). The Solar Arrays are designed to operate at 160 Volts. These large, high power solar arrays are negatively grounded to the ISS and collect charged particles (predominately electrons) as they travel through the space plasma in the Earth's ionosphere. If not controlled, this collected charge causes floating potential variations which can result in arcing, causing injury to the crew during an EVA or damage to hardware [1]. The environmental catalysts for ISS floating potential variations include plasma density and temperature fluctuations and magnetic induction from the Earth's magnetic field. These alone are not enough to cause concern for ISS, but when they are coupled with the large positive potential on the solar arrays, floating potentials up to negative 95 Volts have been observed. Our goal is to differentiate the operationally induced fluctuations in floating potentials from the environmental causes. Differentiating will help to determine what charging can be controlled, and we can then design the proper operations controls for charge collection mitigation. Additionally, the knowledge of how high power solar arrays interact with the environment and what regulations or design techniques can be employed to minimize charging impacts can be applied to future programs.
Willis, Emily M.↗
Investigation into the common mode rejection ratio of the physiological signal conditioner circuit
The common mode rejection ratio (CMRR) of the single operational amplifier (op amp) differential amplifier and of the three operational amplifier differential amplifier was investigated. The three op amp differential amplifier circuit is used in the signal conditioner circuit which amplifies signals such as the electromyograph or electrocardiogram. The investigation confirmed via SPICE modeling what has been observed by others in the recent literature that the CMRR for the circuit can be maximized without precision resistor values or precisely matched op amps. This can be done if one resistor in the final stage can be adjusted either by a potentiometer or by laser trimming in the case of hybrid circuit fabrication.
Obrien, Edward M.↗
Active operator learning with predictive uncertainty quantification for partial differential equations
With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. Here, we propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework’s uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.
97 MATHEMATICS AND COMPUTING↗
Neural operator transformers capture bifurcating drift-wave turbulence in fusion plasma simulations
Self-consistent modeling of turbulence-driven transport is critical for optimizing confinement in magnetically confined fusion plasmas, such as tokamaks and stellarators. In particular, capturing the long-term co-evolution of turbulence, flow, and background plasma profiles remains computationally challenging. Direct numerical simulation of these multiscale, highly nonlinear processes is often demanding and impractical for real-time control or design optimization. To address this bottleneck, we investigate transformer-based neural operator partial differential equation surrogates for emulating the dynamics of drift-wave turbulence bifurcation mediated by zonal flows, using the modified Hasegawa–Wakatani (MHW) model as a prototypical system. We find that the finetuned neural operator model has excellent performance in capturing the multi-spatiotemporal-scales of MHW turbulence bifurcation and is robust to testing on rare and out-of-distribution dynamics. Specifically, we demonstrate that a single unified model accurately predicts both quasi-steady-state turbulence and a wide range of dynamical transition processes, such as nonlinear saturation, spontaneous suppression of turbulence, and the emergence of macroscopic zonal flows, over time horizons vastly exceeding the local turbulence correlation time. This computationally efficient approach establishes a strong foundation for fast, AI-based modeling of complex, multiscale phenomena in magnetized fusion plasmas.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Linear Back-Drive Differentials
Linear back-drive differentials have been proposed as alternatives to conventional gear differentials for applications in which there is only limited rotational motion (e.g., oscillation). The finite nature of the rotation makes it possible to optimize a linear back-drive differential in ways that would not be possible for gear differentials or other differentials that are required to be capable of unlimited rotation. As a result, relative to gear differentials, linear back-drive differentials could be more compact and less massive, could contain fewer complex parts, and could be less sensitive to variations in the viscosities of lubricants. Linear back-drive differentials would operate according to established principles of power ball screws and linear-motion drives, but would utilize these principles in an innovative way. One major characteristic of such mechanisms that would be exploited in linear back-drive differentials is the possibility of designing them to drive or back-drive with similar efficiency and energy input: in other words, such a mechanism can be designed so that a rotating screw can drive a nut linearly or the linear motion of the nut can cause the screw to rotate. A linear back-drive differential (see figure) would include two collinear shafts connected to two parts that are intended to engage in limited opposing rotations. The linear back-drive differential would also include a nut that would be free to translate along its axis but not to rotate. The inner surface of the nut would be right-hand threaded at one end and left-hand threaded at the opposite end to engage corresponding right- and left-handed threads on the shafts. A rotation and torque introduced into the system via one shaft would drive the nut in linear motion. The nut, in turn, would back-drive the other shaft, creating a reaction torque. Balls would reduce friction, making it possible for the shaft/nut coupling on each side to operate with 90 percent efficiency.
Waydo, Peter↗
Projection operators in the unified reaction theory.
Projection operators applicable for rearrangement and exchange processes with full recoil effect for unified reaction theory of Feschbach
PARTICLE COLLISION↗
Design of digital acquisition for beam current monitor
As a part of the Proton Improvement Plan – II (PIP-II) at Fermilab, instrumentation systems are being modernized to take advantage of the higher speeds and ease of use offered by standardized embedded systems like MicroTCA. A rear-transition module (RTM) is being designed to interface with said embedded systems. In each of the four identical channels on the RTM, the differential signal from an alternating-current current transformer (ACCT) transimpedance amplifier will again be amplified by a differential operation-amplifier, then filtered by a low-pass topology. The conditioned signal is then digitized at a maximum of 10MS/s by an analog to digital converter (ADC) integrated circuit. After digitization, the ADC passes the data to an off the shelf AdvancedMC (AMC) Xilinx FPGA module using low voltage differential signals. This paper will describe the simulation of analog circuitry for signal conditioning, simulation of digital signal integrity based on physical design as well as verification of design characteristics critical to signal integrity. This work aims to create a methodology that can be applied to future RTMs requiring application of high-speed digital design principles.
White, R.Turner [Fermilab]↗
Design of Digital Acquisition for Beam Current Monitor
As a part of the Proton Improvement Plan II (PIP-II) at Fermilab, instrumentation systems are being modernized to take advantage of the higher speeds and ease of use offered by standardized embedded systems like MicroTCA. A rear-transition module (RTM) is being designed to interface with said embedded systems. In each of the four identical channels on the RTM, the differential signal from an alternating-current current transformer (ACCT) transimpedance amplifier will again be amplified by a differential operation-amplifier, then filtered by a low-pass topology. The conditioned signal is then digitized at a maximum of 10MS/s by an analog to digital converter (ADC) integrated circuit. After digitization, the ADC passes the data to an off the shelf AdvancedMC (AMC) Xilinx FPGA module using low voltage differential signals. This paper will describe the simulation of analog circuitry for signal conditioning, simulation of digital signal integrity based on physical design as well as verification of design characteristics critical to signal integrity. This work aims to create a methodology that can be applied to future RTMs requiring application of high-speed digital design principles.
White, Turner [Fermilab] (ORCID:0009000348159256)↗
Coupled Nd-142, Nd-143 and Hf-176 Isotopic Data from 3.6-3.9 Ga Rocks: New Constraints on the Timing of Early Terrestrial Chemical Reservoirs
Increasingly precise data from a range of isotopic decay schemes, including now extinct parent isotopes, from samples of the Earth, Mars, Moon and meteorites are rapidly revising our views of early planetary differentiation. Recognising Nd-142 isotopic variations in terrestrial rocks (which can only arise from events occurring during the lifetime of now extinct Sm-146 [t(sub 1/2)=103 myr]) has been an on-going quest starting with Harper and Jacobsen. The significance of Nd-142 variations is that they unequivocally reflect early silicate differentiation processes operating in the first 500 myr of Earth history, the key time period between accretion and the beginning of the rock record. The recent establishment of the existence of Nd-142 variations in ancient Earth materials has opened a new range of questions including, how widespread is the evidence of early differentiation, how do Nd-142 compositions vary with time, rock type and geographic setting, and, combined with other types of isotopic and geochemical data, what can Nd-142 isotopic variations reveal about the timing and mechanisms of early terrestrial differentiation? To explore these questions we are determining high precision Nd-142, Nd-143 and Hf-176 isotopic compositions from the oldest well preserved (3.63- 3.87 Ga), rock suites from the extensive early Archean terranes of southwest Greenland and western Australia.
Bennett, Vickie C.↗
Uniqueness of the Cauchy problem when the initial surface contains characteristic points.
Uniqueness of Cauchy problem when initial surface contains characteristic points
UNIQUENESS THEOREM↗
A Rodrigues' formula for the Laguerre polynomials.
Rodrigues type formula for generalized Laguerre polynomials based upon semigroup property for solutions of heat equation and generating function related to source solution
HEAT EQUATION↗
Symbolic-numeric interface: A review
A survey of the use of a combination of symbolic and numerical calculations is presented. Symbolic calculations primarily refer to the computer processing of procedures from classical algebra, analysis, and calculus. Numerical calculations refer to both numerical mathematics research and scientific computation. This survey is intended to point out a large number of problem areas where a cooperation of symbolic and numerical methods is likely to bear many fruits. These areas include such classical operations as differentiation and integration, such diverse activities as function approximations and qualitative analysis, and such contemporary topics as finite element calculations and computation complexity. It is contended that other less obvious topics such as the fast Fourier transform, linear algebra, nonlinear analysis and error analysis would also benefit from a synergistic approach.
Ng, E. W.↗
Determination of eigenvalues of dynamical systems by symbolic computation
A symbolic computation technique for determining the eigenvalues of dynamical systems is described wherein algebraic operations, symbolic differentiation, matrix formulation and inversion, etc., can be performed on a digital computer equipped with a formula-manipulation compiler. An example is included that demonstrates the facility with which the system dynamics matrix and the control distribution matrix from the state space formulation of the equations of motion can be processed to obtain eigenvalue loci as a function of a system parameter. The example chosen to demonstrate the technique is a fourth-order system representing the longitudinal response of a DC 8 aircraft to elevator inputs. This simplified system has two dominant modes, one of which is lightly damped and the other well damped. The loci may be used to determine the value of the controlling parameter that satisfied design requirements. The results were obtained using the MACSYMA symbolic manipulation system.
Howard, J. C.↗
Pristine lunar glasses - Criteria, data, and implications
The examination of glasses from all of the Apollo landing sites has led to the identification of 25 groups of pristine glass. The nickel found in these glasses is shown to be indigenous, not meteoritic contamination, and to be correlated with Mg. Chemical data indicate that these glasses are more suitable for primary magma than most crystalline mare basalts. In addition, these pristine glasses support the view that assimilative processes involving two cumulative systems in the differentiated mantle operated during mare petrogenesis. Two linear arrays found among the chemistries of the glasses attest to the existence of these interactions. Data suggest that these cumulate components in the mantle continue for 1000 km (laterally) and therefore are likely to be products of a magma ocean that existed early in lunar history.
Delano, J. W.↗