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Results for “commuting projections”

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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All sourcesNASA NTRSDOE OSTI
DOE OSTI2013

Commuting projections on graphs

For a given (connected) graph, we consider vector spaces of (discrete) functions defined on its vertices and its edges. These two spaces are related by a discrete gradient operator, Grad and its adjoint, ₋Div, referred to as (negative) discrete divergence. We also consider a coarse graph obtained by aggregation of vertices of the original one. Then a coarse vertex space is identified with the subspace of piecewise constant functions over the aggregates. We consider the ℓ 2 -projection Q H onto the space of these piecewise constants. In the present paper, our main result is the construction of a projection π H from the original edge-space onto a properly constructed coarse edge-space associated with the edges of the coarse graph. The projections π H and Q H commute with the discrete divergence operator, i.e., we have div π H = Q H div. The respective pair of coarse edge-space and coarse vertexspace offer the potential to construct two-level, and by recursion, multilevel methods for the mixed formulation of the graph Laplacian which utilizes the discrete divergence operator. The performance of one two-level method with overlapping Schwarz smoothing and correction based on the constructed coarse spaces for solving such mixed graph Laplacian systems is illustrated on a number of graph examples.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2017

Projected Commutator DIIS Method for Accelerating Hybrid Functional Electronic Structure Calculations

The commutator direct inversion of the iterative subspace (commutator DIIS or C-DIIS) method developed by Pulay is an efficient and the most widely used scheme in quantum chemistry to accelerate the convergence of self-consistent field (SCF) iterations in Hartree–Fock theory and Kohn–Sham density functional theory. The C-DIIS method requires the explicit storage of the density matrix, the Fock matrix, and the commutator matrix. Hence, the method can only be used for systems with a relatively small basis set, such as the Gaussian basis set. We develop a new method that enables the C-DIIS method to be efficiently employed in electronic structure calculations with a large basis set such as planewaves for the first time. The key ingredient is the projection of both the density matrix and the commutator matrix to an auxiliary matrix called the gauge-fixing matrix. The resulting projected commutator-DIIS method (PC-DIIS) only operates on matrices of the same dimension as that consists of Kohn–Sham orbitals. The cost of the method is comparable to that of standard charge mixing schemes used in large basis set calculations. The PC-DIIS method is gauge-invariant, which guarantees that its performance is invariant with respect to any unitary transformation of the Kohn–Sham orbitals. We demonstrate that the PC-DIIS method can be viewed as an extension of an iterative eigensolver for nonlinear problems. We use the PC-DIIS method for accelerating Kohn–Sham density functional theory calculations with hybrid exchange–correlation functionals, and demonstrate its superior performance compared to the commonly used nested two-level SCF iteration procedure. Furthermore, we demonstrate that in the context of ab initio molecular dynamics (MD) simulation with hybrid functionals one can extrapolate the gauge-fixing matrix to achieve the goal of extrapolating the entire density matrix implicitly along the MD trajectory. Here, numerical results indicate that the new method significantly reduces the number of SCF iterations per MD step, compared to the commonly used strategy of extrapolating the electron density.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2014

Groups of homeomorphisms of the line. Criteria for the existence of invariant and projectively invariant measures in terms of the commutator subgroup

Existence criteria for invariant and projectively invariant measures are obtained for a group G of homeomorphisms of the line. These criteria are formulated in terms of the commutator subgroup [G,G]. For the special (but very important) case of groups of homeomorphisms of the line containing a freely acting element we obtain a criterion for the existence of a projectively invariant measure in the form of the absence of a special subgroup with two generators in which one of the generating elements is a freely acting element. Bibliography: 20 titles.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2013

Noncommutative Common Cause Principles in algebraic quantum field theory

States in algebraic quantum field theory 'typically' establish correlation between spacelike separated events. Reichenbach's Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions V{sub A} and V{sub B}, respectively, there is a local projection C not necessarily commuting with A and B such that C is supported within the union of the backward light cones of V{sub A} and V{sub B} and the set {l_brace}C, C{sup Up-Tack }{r_brace} screens off the correlation between A and B.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2014

Maximal quantum mechanical symmetry: Projective representations of the inhomogeneous symplectic group

A symmetry in quantum mechanics is described by the projective representations of a Lie symmetry group that transforms between physical quantum states such that the square of the modulus of the states is invariant. The Heisenberg commutation relations that are fundamental to quantum mechanics must be valid in all of these physical states. This paper shows that the maximal quantum symmetry group, whose projective representations preserve the Heisenberg commutation relations in this manner, is the inhomogeneous symplectic group. The projective representations are equivalent to the unitary representations of the central extension of the inhomogeneous symplectic group. This centrally extended group is the semidirect product of the cover of the symplectic group and the Weyl-Heisenberg group. Its unitary irreducible representations are computed explicitly using the Mackey representation theorems for semidirect product groups.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

What the complex joint probabilities observed in weak measurements can tell us about quantum physics

Quantummechanics does not permit joint measurements of non-commuting observables. However, it is possible to measure the weak value of a projection operator, followed by the precise measurement of a different property. The results can be interpreted as complex joint probabilities of the two non-commuting measurement outcomes. Significantly, it is possible to predict the outcome of completely different measurements by combining the joint probabilities of the initial state with complex conditional probabilities relating the new measurement to the possible combinations of measurement outcomes used in the characterization of the quantum state. We can therefore conclude that the complex conditional probabilities observed in weak measurements describe fundamental state-independent relations between non-commuting properties that represent the most fundamental form of universal laws in quantum physics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2014

Curvature and geometric modules of noncommutative spheres and tori

When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connection from Euclidean space, and we compute its scalar curvature.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Unification of finite symmetries in the simulation of many-body systems on quantum computers

Symmetry is fundamental in the description and simulation of quantum systems. Leveraging symmetries in classical simulations of many-body quantum systems can result in significant overhead due to the exponentially growing size of some symmetry groups as the number of particles increases. Quantum computers hold the promise of achieving exponential speedup in simulating quantum many-body systems; however, a general method for utilizing symmetries in quantum simulations has not yet been established. In this work, we present a unified framework for incorporating symmetry group transforms on quantum computers to simulate many-body systems. The core of our approach lies in the development of efficient quantum circuits for symmetry-adapted projection onto irreducible representations of a group or pairs of commuting groups. We provide resource estimations for common groups, including the cyclic and permutation groups. Our algorithms demonstrate the capability to prepare coherent superpositions of symmetry-adapted states and to perform quantum evolution across a wide range of models in condensed-matter physics and ab initio electronic structure in quantum chemistry. Specifically, we execute a symmetry-adapted quantum subroutine for small molecules in first-quantization on noisy hardware and demonstrate the emulation of symmetry-adapted quantum phase estimation for preparing coherent superpositions of quantum states in various irreducible representations of a symmetry group. In addition, we present a discussion of open problems regarding treating symmetries in digital quantum simulations of many-body systems, paving the way for future systematic investigations into leveraging symmetries quantumly for practical quantum advantage. The broad applicability and rigorous resource estimation for symmetry transformations make our framework appealing for achieving provable quantum advantage on fault-tolerant quantum computers, especially for symmetry-related properties.

quantum algorithms↗
DOE OSTI2024

Integrated Motor and Drive (IMD) for Traction Applications Project (Final Technical Report)

This paper proposes the concept of the topological equivalence between the commutation cells of a voltage source inverter (VSI) and a current source inverter (CSI) using duality and equivalent circuit transformation techniques. A switching resonance of CSI is analyzed and an effective three-phase RC damper is derived from the dc-link RC damper of VSIs based on the topological equivalence of the VSI and CSI commutation cells. Furthermore, a new three-phase RC damper design framework utilizing the topological equivalence is proposed. Experimental results verify that the proposed three-phase RC damper design for CSI significantly reduces the switching resonance and improves the CSI’s output waveform total harmonics distortions (THDs).

42 ENGINEERING↗
DOE OSTI2021

Commuting again to work on-site? Lab’s first drivers faced their own challenges decades ago

After a long pause in commuting, more and more Lab staff are filling travel mugs, topping off their tanks, and heading up the hill for on-site work. As long as the Lab has existed, workers have snaked through tight mountain roads to get to their jobs. And, back in the early 1940s, drivers faced even more challenges. In the Lab’s earliest years, drivers likely worried about getting stuck in the mud, navigating intense switchbacks, or making sure important equipment, such as a disassembled power plant, didn’t fall off the back of the truck that was hauling it. Creative solutions could go a long way to get out of a jam. As many of us return onsite and get reacquainted with commuting, we look back at how transportation has changed since the Manhattan Project, the U.S. government’s top secret effort at Los Alamos to create the first atomic bombs to help end World War II. These stories and photos are part of the collections in the National Security Research Center, which is the Lab’s classified library and also houses unclassified artifacts from our past.

96 KNOWLEDGE MANAGEMENT AND PRESERVATION↗
DOE OSTI2014

A New Kind of Shift Operators for Infinite Circular and Spherical Wells

A new kind of shift operators for infinite circular and spherical wells is identified. These shift operators depend on all spatial variables of quantum systems and connect some eigenstates of confined systems of different radii R sharing energy levels with a common eigenvalue. In circular well, the momentum operators P ± = P x ± i P y play the role of shift operators. The P x and P y operators, the third projection of the orbital angular momentum operator L z , and the Hamiltonian H form a complete set of commuting operators with the SO(2) symmetry. In spherical well, the shift operators establish a novel relation between ψ l m ( r ) and ψ ( l ± 1 ) ( m ± 1 ) ( r ) .

Sun, Guo-Hua↗
DOE OSTI2021

New Jersey Transit Grid Distributed Generation Program. Cybersecurity Design Assurance Assessment

Superstorm Sandy caused a major disruption to passenger-rail and other commuter systems throughout New York and New Jersey. To address this issue, New Jersey Transit (NJT) established the NJ TRANSITGRID project, an effort designed to power bus, ferry, and limited passenger-rail service during natural or man-made disasters. Given the importance of these transportation systems, NJT partnered with Sandia National Laboratories (Sandia) to assess the cyber-resilience of the information systems that monitor and control the electrical systems within the microgrid. The Sandia “tabletop” assessment is based on the most recent 20% design packages. From this assessment, the Sandia team identified several security areas that were undefined or did not implement industry best practices. Finally, the Sandia team presented possible follow-on assessment activities and recommended investigating multiple hardening technologies. Addressing these findings and adding state-of-the-art detection and mitigation technologies will help ensure the NJ TRANSITGRID is built with more comprehensive cyber-resilience features.

24 POWER TRANSMISSION AND DISTRIBUTION↗
DOE OSTI2014

Fast Charging Electric Vehicle Research & Development Project

The research and development project supported the engineering, design and implementation of on-road Electric Vehicle (“EV”) charging technologies. It included development of potential solutions for DC fast chargers (“DCFC”) capable of converting high voltage AC power to the DC power required by EVs. Additional development evaluated solutions related to the packaging of power electronic components and enclosure design, as well as for the design and evaluation of EV charging stations. Research compared different charging technologies to identify optimum applications in a municipal fleet. This project collected EV usage data and generated a report demonstrating that EVs, when supported by adequate charging infrastructure, are capable of replacing traditional internal combustion vehicles in many municipal applications. The project’s period of performance has demonstrated various methods of incorporating EVs into a municipal environment, and has identified three general categories for EV applications: Short Commute: Defined as EVs performing in limited duration, routine commutes. - Long Commute: Defined as tasks that require EVs to operate in longer daily mileage patterns. - Critical Needs: Defined as the need for EVs to be ready at every moment for indefinite periods. Together, the City of Charlottesville, VA (the “City”) and Aker Wade Power Technologies, LLC (“Aker Wade”) concluded that the EV has a viable position in many municipal fleets but with limited recommendation for use in Critical Needs applications such as Police fleets. The report also documented that, compared to internal combustion vehicles, BEVs have lower vehicle-related greenhouse gas (“GHG”) emissions and contribute to a reduction of air pollution in urban areas. The enhanced integration of EVs in a municipal fleet can result in reduced demand for imported oil and reduced municipal operating costs. The conclusions indicated in the project’s Engineering Report (see Attachment A) are intended to assist future implementation of electric vehicle technology. They are based on the cited research and on the empirical data collected and presented. The report is not expected to represent the entire operating conditions of any of the equipment under consideration within this project, and tested equipment may operate differently under other conditions.

42 ENGINEERING↗
DOE OSTI2026

Projective Representations, Bogomolov Multiplier, and Their Applications in Physics

We present a pedagogical review of projective representations of finite groups and their physical applications in quantum many-body systems. Some of our physical results are new. We begin with a self-contained introduction to projective representations, highlighting the role of group cohomology, representation theory, and classification of irreducible projective representations. We then focus on a special subset of cohomology classes, known as the Bogomolov multiplier, which consists of cocycles that are symmetric on commuting pairs but remain nontrivial in group cohomology. Such cocycles have important physical implications: they characterize (1+1)D SPT phases that cannot be detected by string order parameters and give rise, upon gauging, to distinct gapped phases with completely broken non-invertible Rep(G) symmetry. We construct explicit lattice models for these phases and demonstrate how they are distinguished by the fusion rules of local order parameters. We show that a pair of completely broken Rep(G) SSB phases host nontrivial interface modes at their domain walls. As an example, we construct a lattice model where the ground state degeneracy on a ring increases from 32 without interfaces to 56 with interfaces.

Bogomolov multiplier↗
DOE OSTI2021

Manifestly causal in-in perturbation theory about the interacting vacuum

In-In perturbation theory is a vital tool for cosmology and nonequilibrium physics. Here, we reconcile an apparent conflict between two of its important aspects with particular relevance to De Sitter/inflationary contexts: (i) the need to slightly deform unitary time evolution with an iϵ prescription that projects the free (“Bunch-Davies”) vacuum onto the interacting vacuum and renders vertex integrals well-defined, and (ii) Weinberg’s “nested commutator” reformulation of in-in perturbation theory which makes manifest the constraints of causality within expectation values of local operators, assuming exact unitarity. We show that a modified iϵ prescription maintains the exact unitarity on which the derivation of (ii) rests, while nontrivially agreeing with (i) to all orders of perturbation theory.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2023

Integrating Microtransit with Public Transit for Coordinated Multi-Modal Movement of People (Final Technical Report)

This project focused on potential benefits of mobility service providers and transit agencies cooperating to offer fully integrated, seamless multi-modal mobility services for commuters. Specifically, we aimed at developing a first/last mile microtransit and micromobility services tightly integrated with fixed-route transit to achieve the “point-to-point” service capability. Our goal was to provide fixed-route transit customers with two convenient and compelling options for accessing public transit that are also operational efficient.

29 ENERGY PLANNING, POLICY, AND ECONOMY↗
DOE OSTI2018

On Construction of Anticliques for Noncommutative Operator Graphs

In this paper, we construct anticliques for noncommutative operator graphs generated by generalized Pauli matrices. It is shown that application of entangled states for the construction of the code space K allows one to substantially increase the dimension of a noncommutative operator graph for which the projection on K is an anticlique.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2015

Hamiltonian purification

The problem of Hamiltonian purification introduced by Burgarth et al. [Nat. Commun. 5, 5173 (2014)] is formalized and discussed. Specifically, given a set of non-commuting Hamiltonians (h{sub 1}, …, h{sub m}) operating on a d-dimensional quantum system ℋ{sub d}, the problem consists in identifying a set of commuting Hamiltonians (H{sub 1}, …, H{sub m}) operating on a larger d{sub E}-dimensional system ℋ{sub d{sub E}} which embeds ℋ{sub d} as a proper subspace, such that h{sub j} = PH{sub j}P with P being the projection which allows one to recover ℋ{sub d} from ℋ{sub d{sub E}}. The notions of spanning-set purification and generator purification of an algebra are also introduced and optimal solutions for u(d) are provided.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗