Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “graph states”

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 397 records · Page 22
All sourcesNASA NTRSDOE OSTI
NASA NTRS1991

The dynamics of particle disks. III - Dense and spinning particle disks

The kinetic theory of planetary rings developed by Araki and Tremaine (1986) and Araki (1988) is extended and refined, with a focus on the implications of finite particle size: (1) nonlocal collisions and (2) finite filling factors. Consideration is given to the derivation of the equations for the local steady state, the low-optical-depth limit, and the steady state at finite filling factors (including the effects of collision inelasticity, spin degrees of freedom, and self-gravity). Numerical results are presented in extensive graphs and characterized in detail. The importance of distinguishing effects (1) and (2) at low optical depths is stressed, and the existence of vertical density profiles with layered structures at high filling factors is demonstrated.

Araki, Suguru↗
DOE OSTI2024

Using Convex Optimization to Efficiently Apportion Tracer and Pollutant Sources From Point Concentration Observations

Abstract Rivers transport elements, minerals, chemicals, and pollutants produced in their upstream basins. A sample from a river is a mixture of all of its upstream sources, making it challenging to pinpoint the contribution from each individual source. Here, we show how a nested sample design and convex optimization can be used to efficiently unmix downstream samples of a well‐mixed, conservative tracer in a steady state system into the contributions of their upstream sources. Our approach is significantly faster than previous methods. We represent the river's sub‐catchments, defined by sampling sites, using a directed acyclic graph. This graph is used to build a convex optimization problem which, thanks to its convexity, can be quickly solved to global optimality—in under a second on desktop hardware for data sets of ∼100 samples or fewer. Uncertainties in the upstream predictions can be generated using Monte Carlo resampling. We provide an open‐source implementation of this approach in Python. The inputs required are straightforward: a table containing sample locations and observed tracer concentrations, along with a D8 flow‐direction raster map. As a case study, we use this method to map the elemental geochemistry of sediment sources for rivers draining the Cairngorms mountains, UK. This method could be extended to non‐conservative and non‐steady state tracers. We also show, theoretically, how multiple tracers could be simultaneously inverted to recover upstream run‐off or erosion rates as well as source concentrations. Overall, this approach can provide valuable insights to researchers in various fields, including water quality, geochemical exploration, geochemistry, hydrology, and wastewater epidemiology.

Barnes, Richard↗
DOE OSTI2024

Stochastic finite volume method for uncertainty quantification of transient flow in gas pipeline networks

We develop a weakly intrusive framework to simulate the propagation of uncertainty in solutions of generic hyperbolic partial differential equation systems on graph-connected domains with nodal coupling and boundary conditions. The method is based on the Stochastic Finite Volume (SFV) approach and can be applied for uncertainty quantification (UQ) of the dynamical state of fluid flow over actuated transport networks. The numerical scheme has specific advantages for modeling intertemporal uncertainty in time-varying boundary parameters, which cannot be characterized by strict upper and lower (interval) bounds. We describe the scheme for a single pipe, and then formulate the controlled junction Riemann problem (JRP) that enables the extension to general network structures. In conclusion, we demonstrate the method's capabilities and performance characteristics using a standard benchmark test network.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Traffic Signal Optimization by Integrating Reinforcement Learning and Digital Twins

Machine learning (ML) methods, especially reinforcement learning (RL), have been widely considered for traffic signal optimization in intelligent transportation systems. Most of these ML methods are centralized, lacking in scalability and adaptability in large traffic networks. Further, it is challenging to train such ML models due to the lack of training platforms and/or the cost of deploying and training in a real traffic networks. This paper presents an approach for the integration of decentralized graph-based multi-agent reinforcement learning (DGMARL) with a Digital Twin (DT) to optimize traffic signals for the reduction of traffic congestion and network-wide fuel consumption related to stopping. Specifically, the DGMARL agents learn traffic state patterns and make decisions regarding traffic signal control with assistance from a Digital Twin module, which simulates and replicates the traffic behaviors of a real traffic network. The proposed approach was evaluated using PTV-Vissim [1], a microscopic traffic simulation platform. PTV-Vissim is also the simulation engine of the DT, enabling emulation and optimization of the traffic signals on the MLK Smart Corridor in Chattanooga, Tennessee. Compared to an actuated signal control baseline approach, experiment results show that Eco_PI, a developed performance measure capturing the impact of stops on fuel consumption, was reduced by 44.27% in a 24-hour and an average of 29.88% in a PM-peak-hour scenario.

Multi-Agent Reinforcement Learning, Digital Twin, ↗
DOE OSTI2022

Tackling the Challenges in Scene Graph Generation With Local-to-Global Interactions

In this work, we seek new insights into the underlying challenges of the scene graph generation (SGG) task. Quantitative and qualitative analysis of the visual genome (VG) dataset implies: 1) ambiguity: even if interobject relationship contains the same object (or predicate), they may not be visually or semantically similar; 2) asymmetry: despite the nature of the relationship that embodied the direction, it was not well addressed in previous studies; and 3) higher-order contexts: leveraging the identities of certain graph elements can help generate accurate scene graphs. Motivated by the analysis, we design a novel SGG framework, Local-to-global interaction networks (LOGINs). Locally, interactions extract the essence between three instances of subject, object, and background, while baking direction awareness into the network by explicitly constraining the input order of subject and object. Globally, interactions encode the contexts between every graph component (i.e., nodes and edges). Finally, Attract and Repel loss is utilized to fine-tune the distribution of predicate embeddings. By design, our framework enables predicting the scene graph in a bottom-up manner, leveraging the possible complementariness. To quantify how much LOGIN is aware of relational direction, a new diagnostic task called Bidirectional Relationship Classification (BRC) is also proposed. Overall, experimental results demonstrate that LOGIN can successfully distinguish relational direction than existing methods (in BRC task), while showing state-of-the-art results on the VG benchmark (in SGG task).

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2020

Learning Hidden Structure in Multi-Fidelity Information Sources for Efficient Uncertainty Quantification (LDRD 218317)

This report summarizes the work done under the Laboratory Directed Research and Development (LDRD) project entitled "Learning Hidden Structure in Multi-Fidelity Information Sources for Efficient Uncertainty Quantification". In this project we investigated multi-fidelity strategies for fusing data from information sources of varying cost and accuracy. Most existing strategies exploit hierarchical relationships between models, for example that occur when different models are generated by refining a numerical discretization parameter. In this work we focused on encoding the relationships between information sources using directed acyclic graphs. The multi-fidelity networks can have general structure and represent a significantly greater variety of modeling relationships than recursive networks used in the current state literature. Numerical results show that a non-hierarchical multi-fidelity Monte Carlo strategy can reduce the cost of estimating uncertainty in predictions of a model of plasma expanding in a vacuum by almost two orders of magnitude.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
NASA NTRS1991

MPD thruster technology

Inhouse magnetoplasmadynamic (MPD) thruster technology is discussed. The study focussed on steady state thrusters at powers of less than 1 MW. Performance measurement and diagnostics technologies were developed for high power thrusters. Also developed was a MPD computer code. The stated goals of the program are to establish: performance and life limitation; influence of applied fields; propellant effects; and scaling laws. The presentation is mostly through graphs and charts.

Myers, Roger M.↗
NASA NTRS1991

Comparison of straight and 15 degree vectored nozzles using a six component thrust stand

This project compared the forces and moments produced by straight and 15 degree vectored nozzles. Using the six component thrust stand in the engines laboratory at California Polytechnic State University, several trials were performed. This data was then reduced using first a computer program and then later an electronic spreadsheet. This reduced data was graphed and compared. As a result of these comparisons some unexpected forces were discovered. Several more tests were run including a zero thrust test and a statistical comparison were done to discover the source of these discrepancies. As a direct result several nozzle changes were made and significant revisions to the thrust stand are being made.

Carpenter, Thomas W.↗
DOE OSTI2022

The holographic entropy cone from marginal independence

The holographic entropy cone characterizes the relations between entanglement entropies for a spatial partitioning of the boundary spacetime of a holographic CFT in any state describing a classical bulk geometry. We argue that the holographic entropy cone, for an arbitrary number of parties, can be reconstructed from more fundamental data determined solely by subadditivity of quantum entropy. We formulate certain conjectures about graph models of holographic entanglement, for which we provide strong evidence, and rigorously prove that they all imply that such a reconstruction is possible. Our conjectures (except only for the weakest) further imply that the necessary data is remarkably simple. In essence, all one needs to know to reconstruct the holographic entropy cone, is a certain subset of the extreme rays of this simpler “subadditivity cone”, namely those which can be realized in holography. This recasting of the bewildering entanglement structure of geometric states into primal building blocks paves the way to distilling the essence of holography for the emergence of a classical bulk spacetime.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
NASA NTRS2013

Dynamics and Control of a Disordered System in Space

In this paper, we present some ideas regarding the modeling, dynamics and control aspects of granular spacecraft. Granular spacecraft are complex multibody systems composed of a spatially disordered distribution of a large number of elements, for instance a cloud of N grains in orbit, with N greater than 10(exp 3). These grains can be large (Cubesat-size) or small (mm-size), and can be active, i.e., a fully equipped vehicle capable sensing their own position and attitude, and enabled with propulsion means, or entirely passive. The ultimate objective would be to study the behavior of the single grains and of large ensembles of grains in orbit and to identify ways to guide and control the shape of a cloud composed of these grains so that it can perform a useful function in space, for instance, as an element of an optical imaging system for astrophysical applications. This concept, in which the aperture does not need to be continuous and monolithic, would increase the aperture size several times compared to large NASA observatories such as ATLAST, allowing for a true Terrestrial Planet Imager that would be able to resolve exo-planet details and do meaningful spectroscopy on distant world. In the paper, we address the modeling and autonomous operation of a distributed assembly (the cloud) of large numbers of highly miniaturized space-borne elements (the grains). A multi-scale, multi-physics model is proposed of the dynamics of the cloud in orbit, as well as a control law for cloud shape maintenance, and preliminary simulation studies yield an estimate of the computational effort, indicating a scale factor of approximately N(exp 1.4) as a function of the number of grains. A granular spacecraft can be defined as a collection of a large number of space-borne elements (in the 1000s) designed and controlled such that a desirable collective behavior emerges, either from the interactions among neighboring grains, and/or between the grains and the environment. In this paper, each grain is considered to be a highly miniaturized spacecraft which has limited size and mass, hence it has limited actuation, limited propulsive capability, limited power, limited sensing, limited communication, limited computational resources, limited range of motion, limited lifetime, and may be expendable. The modeling and dynamics of clouds of vehicles is more challenging than with conventional vehicles because we are faced with a probabilistic vehicle composed of a large number of physically disconnected vehicles. First, different scales of motion occur simultaneously in a cloud: translations and rotations of the cloud as a whole (macro-dynamics), relative rotation and translation of one cloud member with respect to another (meso-dynamics), and individual cloud member dynamics (micro-dynamics). Second, the control design needs to be tolerant of the system complexity, of the system architecture (centralized vs. decentralized large scale system control) as well as robust to un-modeled dynamics and noise sources. Figure 1, top left, shows the kinematic parameters of a 1000 element cloud in orbit. The motion of the system is described with respect to a local vertical-local horizontal (LV-LH) orbiting reference frame (x,y,z)=F(sub ORF) of origin O(sub ORF) which rotates with mean motion omega and orbital semi-major axis R(sub 0). The orbital geometry at the initial time is defined in terms of its six orbital elements, and the orbital dynamics equation for point O(sub ORF) is propagated forward in time under the influence of the gravitational field of the primary and other external perturbations, described below. The origin of this frame coincides with the initial position of the center of mass of the system, and the coordinate axes are z along the local vertical, x toward the flight direction, and y in the orbit normal direction. The assumptions we used to model the dynamics are as follows: 1) The inertial frame is fixed at Earth's center. 2) The orbiting Frame ORF follows Keplerian orbit. 3) the cloud system dynamics is referred to ORF. 4) the attitude of each grain uses the principal body frame as body fixed frame. 5) the atmosphere is assumed to be rigidly rotating with the Earth. Regarding the grains forming the cloud: 1) each grain is modeled as a rigid body; 2) a simple attitude estimator provides attitude estimates, 3) a simple guidance logic commands the position and attitude of each grain, 4) a simple local feedback controller based on PD control of local states is used to stabilize the attitude of the vehicle. Regarding the cloud: 1) the cloud as a whole is modeled as an equivalent rigid body in orbit, and 2) an associated graph establishes agent connectivity and enables coupling between modes of motion at the micro and macro scales; 3) a simple guidance and estimation logic is modeled to estimate and command the attitude of this equivalent rigid body; 4) a cloud shape maintenance controller is based on the dynamics of a stable virtual truss in the orbiting frame. Regarding the environmental perturbations acting on the cloud: 1) a non-spherical gravity field including JO (Earth's spherical field) zonal component, J2 (Earth's oblateness) and J3 zonal components is implemented; 2) atmospheric drag is modeled with an exponential model; 3) solar pressure is modeled assuming the Sun is inertially fixed; and 4) the Earth's magnetic field is model using an equivalent dipole model. The equations of motion are written in a referential system with respect to the origin of the orbiting frame and the state is propagated forward in time using an incremental predictor-corrector scheme. A representative cloud with varying number of grains is simulated to identify the limitations in computation time as the number of grains grows. We derive a control law to track a desired surface in the ORF (equivalently to maintain a reference cloud shape) by defining an error from a desired surface shape, and designing a control law that is exponentially stable and reduces the tracking error to zero. Figure 1 (top right) shows a comparison of various requirements for simulation of single spacecraft vs. granular spacecraft, indicating the high degree of complexity that needs to be taken into consideration. The ORF components of control force required by one of the grains is, for this particular case, in the micro-Newton range. However, no attempt has been made yet to reconfigure (or re-orient) the cloud configuration internally, for which forces in the milli-Newton level are expected, depending on the time required to do the reconfiguration. Figure 1, bottom, shows the computation time as a function of the number of grains, indicating an order N(exp 1.43) scaling on a 8 Gb, 1067 MHz RAM MacOSX computer with a 3.06 GHz Intel Core 2 Duo processor. With this metric, the same simulation for a system of N=1000 grains would take 5.4 hours, and 146 hours (i.e., 6 days) for a system with N=10,000 grains. Therefore, efficient ways to simulate this complex system, where not only the time scales of natural system dynamics, but also the sampling times of the Guidance, Navigation, and Control are included, remain to be explored. Additional details on the cloud modeling, dynamics, and control will be described in the paper.

simulation↗
DOE OSTI2020

Optimized Lie–Trotter–Suzuki decompositions for two and three non-commuting terms

Lie–Trotter–Suzuki decompositions are an efficient way to approximate operator exponentials exp ( t H ) when H is a sum of n (non-commuting) terms which, individually, can be exponentiated easily. They are employed in time-evolution algorithms for tensor network states, digital quantum simulation protocols, path integral methods like quantum Monte Carlo, and splitting methods for symplectic integrators in classical Hamiltonian systems. Here, we provide optimized decompositions up to order t 6 . The leading error term is expanded in nested commutators (Hall bases) and we minimize the 1-norm of the coefficients. For n = 2 terms, several of the optima we find are close to those in McLachlan (1995). Generally, our results substantially improve over unoptimized decompositions by Forest, Ruth, Yoshida, and Suzuki. We explain why these decompositions are sufficient to efficiently simulate any one- or two-dimensional lattice model with finite-range interactions. This follows by solving a partitioning problem for the interaction graph.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

η-pairing in Hubbard models: From spectrum generating algebras to quantum many-body scars

In this work, we revisit the η-pairing states in Hubbard models and explore their connections to quantum many-body scars to discover a universal scars mechanism. η-pairing occurs due to an algebraic structure known as a spectrum generating algebra (SGA), giving rise to equally spaced towers of eigenstates in the spectrum. We generalize the original η-pairing construction and show that several Hubbard-like models on arbitrary graphs exhibit SGAs, including ones with disorder and spin-orbit coupling. We further define a restricted spectrum generating algebra (RSGA) and give examples of perturbations to the Hubbard-like models that preserve an equally spaced tower of the original model as eigenstates. The states of the surviving tower exhibit a subthermal entanglement entropy, and we analytically obtain parameter regimes for which they lie in the bulk of the spectrum, showing that they are exact quantum many-body scars. The RSGA framework also explains the equally spaced towers of eigenstates in several well-known models of quantum scars, including the Affleck-Kennedy-Lieb-Tasaki model.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Bounce-averaged theory in arbitrary multi-well plasmas: solution domains and the graph structure of their connections

Bounce-averaged theories provide a framework for simulating relatively slow processes, such as collisional transport and quasilinear diffusion, by averaging these processes over the fast periodic motions of a particle on a closed orbit. This procedure dramatically increases the characteristic time scale and reduces the dimensionality of the modelled system. The natural coordinates for such calculations are the constants of motion (COM) of the fast particle motion, which by definition do not change during an orbit. However, for sufficiently complicated fields – particularly in the presence of local maxima of the electric potential and magnetic field – the COM are not sufficient to specify the particle trajectory. In such cases, multiple domains in COM space must be used to solve the problem, with boundary conditions enforced between the domains to ensure continuity and particle conservation. Previously, these domains have been imposed by hand, or by recognising local maxima in the fields, limiting the flexibility of bounce-averaged simulations. Here, we present a general set of conditions for identifying consistent domains and the boundary condition connections between the domains, allowing the application of bounce-averaged theories in arbitrarily complicated and dynamically evolving electromagnetic field geometries. We also show how the connections between the domains can be represented by a directed graph, which can help to succinctly represent the trajectory bifurcation structure.

fusion plasma↗
DOE OSTI2020

Deep Learning-Based Queue-Aware Eco-Approach and Departure System for Plug-In Hybrid Electric Buses at Signalized Intersections: A Simulation Study

Eco-Approach and Departure (EAD) has been considered as a promising eco-driving strategy for vehicles traveling in an urban environment, where information such as signal phase and timing (SPaT) and geometric intersection description is well utilized to guide vehicles passing through intersections in the most energy-efficient manner. Previous studies formulated the optimal trajectory planning problem as finding the shortest path on a graphical model. While this method is effective in terms of energy saving, its computation efficiency can be further enhanced by adopting machine learning techniques. In this paper, we propose an innovative deep learning-based queue-aware eco-approach and departure (DLQ-EAD) system for a plug-in hybrid electric bus (PHEB), which is able to provide an online optimal trajectory for the vehicle considering both the downstream traffic condition (i.e. traffic lights, queues) and the vehicle powertrain efficiency. Based on optimal solutions obtained from the graph-based trajectory planning algorithm (GTPA), a deep neural network (DNN) is developed to learn the optimal vehicle speed for the next time step given its current state. It is demonstrated that the trained DNN can provide energy-efficient trajectories with high computational efficiency and high flexibility adopting to dynamic changes in the surrounding environment. To address the impact of downstream traffic, a queue prediction model is further developed using data from radars and connected vehicles (CVs), as well as signal timing data from SPaT messages. A comprehensive simulation study in the microscopic traffic modeling software PTV VISSIM shows that the proposed DLQ-EAD can achieve 18.7%-24.0% energy efficiency improvements for a single PHEB on various traffic congestion levels. The proposed queue prediction model can be of practical significance even at low penetration rates of CVs. Specifically, additional energy savings of 2.0%-8.2% can be further achieved with 20% vehicles in the network.

Ye, Fei↗
DOE OSTI2025

Application-level benchmarking of quantum computers using nonlocal game strategies

In a nonlocal game, two noncommunicating players cooperate to convince a referee that they possess a strategy that does not violate the rules of the game. Quantum strategies allow players to optimally win some games by performing joint measurements on a shared entangled state, but computing these strategies can be challenging. We present a variational quantum algorithm to compute quantum strategies for nonlocal games by encoding the rules of a nonlocal game into a Hamiltonian. We show how this algorithm can generate a short-depth optimal quantum strategy for a graph coloring game with a quantum advantage. This quantum strategy is then evaluated on fourteen different quantum hardware platforms to demonstrate its utility as a benchmark. Finally, we discuss potential sources of errors that can explain the observed decreased performance of the executed task and derive an expression for the number of samples required to accurately estimate the win rate in the presence of noise.

nonlocal games↗
DOE OSTI2023

CHGNet as a pretrained universal neural network potential for charge-informed atomistic modelling

Large-scale simulations with complex electron interactions remain one of the greatest challenges for atomistic modelling. Although classical force fields often fail to describe the coupling between electronic states and ionic rearrangements, the more accurate ab initio molecular dynamics suffers from computational complexity that prevents long-time and large-scale simulations, which are essential to study technologically relevant phenomena. Here we present the Crystal Hamiltonian Graph Neural Network (CHGNet), a graph neural network-based machine-learning interatomic potential (MLIP) that models the universal potential energy surface. CHGNet is pretrained on the energies, forces, stresses and magnetic moments from the Materials Project Trajectory Dataset, which consists of over 10 years of density functional theory calculations of more than 1.5 million inorganic structures. The explicit inclusion of magnetic moments enables CHGNet to learn and accurately represent the orbital occupancy of electrons, enhancing its capability to describe both atomic and electronic degrees of freedom. We demonstrate several applications of CHGNet in solid-state materials, including charge-informed molecular dynamics in Li x MnO 2 , the finite temperature phase diagram for Li x FePO 4 and Li diffusion in garnet conductors. We highlight the significance of charge information for capturing appropriate chemistry and provide insights into ionic systems with additional electronic degrees of freedom that cannot be observed by previous MLIPs.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Topology-Aware Reinforcement Learning for Voltage Control: Centralized and Decentralized Strategies

Volt-VAR control (VVC) methods based on deep reinforcement learning (DRL) can effectively control distribution grid voltage and minimize power loss by implementing corrective and preventive control measures on the reactive power output of inverter-based distributed energy resources (DERs). However, model-free DRL-based VVC approaches usually cannot capture the important topological feature of the power system since they use a fully-connected network (FCN) to deliver the action. Therefore, this paper proposes a graph convolutional network (GCN)-based DRL approach that can employ the topological information of the network to take better control action for regulating the voltage. Our implementation allows for both centralized and decentralized configurations, utilizing a single agent and multiple agents respectively. Although the centralized GCN-based DRL approach has its advantages of minimizing voltage fluctuation and power loss, it is not suitable for large scale power systems due to its challenges in terms of scalability, computation speed and potential single points of failure. Therefore, these problems can be resolved using the decentralized GCN-based DRL approach. Moreover, to ensure the safe operation of the model, our proposed approach incorporates an exponential barrier function while formulating the reward function for each agent. To validate performance of the proposed approaches, the proposed model is tested on modified IEEE test systems and the performances are measured in terms on voltage fluctuation reduction, minimization of power loss and computational speed. Finally, the results show that the proposed topology-aware approach outperforms the FCN-based DRL approach in terms of reducing voltage fluctuation and minimizing power loss of the network. Moreover, it is shown that the decentralized GCN-based DRL has faster computational speed than other approaches.

42 ENGINEERING↗
DOE OSTI2026

Risk-Aware Measurement Synchronization and Recovery for DSSE With Heterogeneous Data Sources

Power distribution systems are increasingly integrating heterogeneous sensors with varying data reporting rates and types, which pose challenges to achieving observability at the desired temporal resolution of distribution system state estimation (DSSE). Multisensor failures caused by extreme events exacerbate these issues, introducing substantial uncertainties into DSSE. This article proposes a novel solution to these challenges by ensuring high-resolution system observability despite heterogeneous data sources and multisensor failures. First, a deep learning architecture combining long short-term memory (LSTM) and graph convolutional network (GCN) is employed to synchronize meters with different reporting rates, aiming to achieve system observability. A random-walk-model-based approach is introduced to generate pseudo-measurements while properly characterizing their uncertainties under multisensor failures. Finally, a disaster-risk-informed observability metric (RiOM) is defined to quantify the uncertainty associated with state estimation results. The proposed framework offers deeper insights into the system observability on the fly compared with conventional analysis. The effectiveness of the framework is demonstrated on an IEEE standard test case and a large-scale real-world distribution feeder in mid-Minnesota in the U.S.

97 MATHEMATICS AND COMPUTING↗