SEARCH · Search NASA

Results for “Closed form solution”

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.

SourceResearch connectionFrom yearThrough year

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

At least 181 records · Page 10

All sources NASA NTRS DOE OSTI

DOE OSTI2020

A Metal-on-Metal Growth Approach to Metal–Metal Oxide Core–Shell Nanostructures with Plasmonic Properties

Hybrid core–shell nanoparticles integrate the material properties from individual components; however, the synthesis of core–shell nanoparticles with dissimilar materials remains challenging. Herein, we applied a metal-on-metal thin film growth approach to control the conformal deposition of non-precious metal shells on the Cu-based metal cores to form core–shell structures of metal–metal oxide hybrids (M–M'O x , where M = AuCu 3 or Cu, M' = Fe, Mn, and Ni). The deposition kinetics were controlled by a temperature-regulated thermal decomposition of zerovalent transition metal complexes. It is predicted that the conformal deposition can be promoted by keeping the initial deposition temperature close to the thermal decomposition temperature of the zerovalent precursors. The mechanisms of strain reduction and interdiffusion facilitate conformal deposition over island growth in the synthesis with slow reaction kinetics. This study provides insightful guidance to metal-on-metal growth in solution at the nanoscale and thus the seed-mediated approach to hybrid core–shell structures. The optical properties of these metal–metal oxide hybrids were investigated experimentally and interpreted by theoretical simulations. Despite damping effects, the plasmonic properties of these Cu-based core-metal oxide shell structures may have the potential to enable plasmon-enhanced applications.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗

DOE OSTI2025

A framework for challenges and solutions in biodesign research

The bioeconomy represents an advanced economic paradigm that builds upon previous agricultural, industrial, and digital economic models. It seeks to tackle critical global challenges such as resource scarcity, escalating healthcare demands, and environmental degradation. At the heart of the bioeconomy is biomanufacturing, which uses natural or engineered enzymes or cell factories built from biological components like promoters, terminators, regulatory sequences, reporters, and functional genes into various chassis hosts (including animal, microbial, plant, and de novo systems) to create products such as food, energy, medicine, materials, chemicals, and engineered tissue/organs. An enabler of biomanufacturing is biodesign – also known as biosystems design and closely related to synthetic biology or engineering biology. This interdisciplinary field aims to understand and predictably modify existing life forms or create entirely new biological entities/systems using rational engineering strategies and automated design tools. Through these capabilities, biodesign supports the discovery, optimization, and creation of efficient platforms for biomanufacturing.

59 BASIC BIOLOGICAL SCIENCES↗

DOE OSTI2025

Foundations of magnetohydrodynamics

In this tutorial, a derivation of magnetohydrodynamics (MHD) valid beyond the usual ideal gas approximation is presented. Non-equilibrium thermodynamics is used to obtain conservation equations and linear constitutive relations. When coupled with Maxwell's equations, this provides closed fluid equations in terms of material properties of the plasma, described by the equation of state and transport coefficients. These properties are connected to microscopic dynamics using the Irving–Kirkwood procedure and Green–Kubo relations. Symmetry arguments and the Onsager–Casimir relations allow one to vastly simplify the number of independent coefficients. Importantly, expressions for current density, heat flux, and stress (conventionally Ohm's law, Fourier's law, and Newton's law) take different forms in systems with a non-ideal equation of state. The traditional form of the MHD equations, which is usually obtained from a Chapman–Enskog solution of the Boltzmann equation, corresponds to the ideal gas limit of the general equations.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2023

Investigation of Theoretical Solutions to a Bottom-Raised Oscillating Surge Wave Energy Converter (OSWEC) Through Experimental and Parametric Studies

Experiments were conducted on a wave tank model of a bottom raised oscillating surge wave energy converter (OSWEC) model in regular waves. The OSWEC model shape was a thin rectangular flap, which was allowed to pitch in response to incident waves about a hinge located at the intersection of the flap and the top of the supporting foundation. Torsion springs were added to the hinge in order to position the pitch natural frequency at the center of the wave frequency range of the wave maker. The flap motion as well as the loads at the base of the foundation were measured. The OSWEC was modeled analytically using elliptic functions in order to obtain closed form expressions for added mass and radiation damping coefficients, along with the excitation force and torque. These formulations were derived and reported in a previous publication by the authors. While analytical predictions of the foundation loads agree very well with experiments, large discrepancies are seen in the pitch response close to resonance. These differences are analyzed by conducting a sensitivity study, in which system parameters, including damping and added mass values, are varied. The likely contributors to the differences between predictions and experiments are attributed to tank reflections, standing waves that can occur in long, narrow wave tanks, as well as the thin plate assumption employed in the analytical approach.

analytical↗

DOE OSTI2021

Topological order in Mott insulators

Here in this commemorative issue for Phil Anderson, I reproduce parts of a joint unpublished paper that first got me thinking about topological order in Mott insulators. With Phil, we explored aspects of the momentum distribution function and the non-classical rotational inertia in the Bose Hubbard model close to the quantum phase transition. My interest on this question continued, and later in the article I take up the question of topological order in the Kitaev Mott insulator on the honeycomb lattice with bond-dependent interactions in a finite magnetic field. Kitaev obtained the exact solution for zero field that demonstrated an exotic Z 2 quantum spin liquid phase with topological order. In a finite magnetic field, we discovered a new intermediate U(1) gapless quantum spin liquid phase, sandwiched between the known gapped Kitaev spin liquid and a polarized phase, that harbors fractionalized neutral fermionic excitations, dubbed spinons, that remarkably form a Fermi surface in a Mott insulator.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2021

Manifold Sampling for Optimizing Nonsmooth Nonconvex Compositions

Here we propose a manifold sampling algorithm for minimizing a nonsmooth composition $f= h\circ F$, where we assume $h$ is nonsmooth and may be inexpensively computed in closed form and $F$ is smooth but its Jacobian may not be available. We additionally assume that the composition $h\circ F$ defines a continuous selection. Manifold sampling algorithms can be classified as model-based derivative-free methods, in that models of $F$ are combined with particularly sampled information about $h$ to yield local models for use within a trust-region framework. We demonstrate that cluster points of the sequence of iterates generated by the manifold sampling algorithm are Clarke stationary. We consider the tractability of three particular subproblems generated by the manifold sampling algorithm and the extent to which inexact solutions to these subproblems may be tolerated. Numerical results demonstrate that manifold sampling as a derivative-free algorithm is competitive with state-of-the-art algorithms for nonsmooth optimization that utilize first-order information about $f$.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Combined Precursor Engineering and Grain Anchoring Leading to MA‐Free, Phase‐Pure, and Stable α‐Formamidinium Lead Iodide Perovskites for Efficient Solar Cells

Abstract α‐Formamidinium lead iodide (α‐FAPbI 3 ) is one of the most promising candidate materials for high‐efficiency and thermally stable perovskite solar cells (PSCs) owing to its outstanding optoelectrical properties and high thermal stability. However, achieving a stable form of α‐FAPbI 3 where both the composition and the phase are pure is very challenging. Herein, we report on a combined strategy of precursor engineering and grain anchoring to successfully prepare methylammonium (MA)‐free and phase‐pure stable α‐FAPbI 3 films. The incorporation of volatile FA‐based additives in the precursor solutions completely suppresses the formation of non‐perovskite δ‐FAPbI 3 during film crystallization. Grains of the desired α‐phase are anchored together and stabilized when 4‐ tert ‐butylbenzylammonium iodide is permeated into the α‐FAPbI 3 film interior via grain boundaries. This cooperative scheme leads to a significantly increased efficiency close to 21 % for FAPbI 3 perovskite solar cells. Moreover, the stabilized PSCs exhibit improved thermal stability and maintained ≈90 % of their initial efficiency after storage at 50 °C for over 1600 hours.

Ling, Xufeng↗

DOE OSTI2022

High entropy alloys with hexagonal close-packed structure derived from thin film combinatorial approach

A thin film combinatorial synthesis approach was utilized to develop a compositional and structural library for the OsRuWCo alloy system. The survey of this alloy space produced 24 unique compositions and their corresponding crystal structures, which represent a phase map where amorphous and single-phase hexagonal close-packed alloys were identified. Based on the identification of a promising candidate film region, a new high entropy alloy based on OsRuWCo, but with non-equiatomic composition, was synthesized in bulk form. The alloy exhibited a single-phase hexagonal close-packed structure in the as-cast state. Three derivatives from this system were also developed by considering the heat of mixing, atomic size, and binary solubility. These new alloys are based on OsRuWCoIr and OsRuWCoFe, both of which exhibit single-phase hexagonal close-packed as-cast structures, as determined by X-ray diffraction and electron microscopy. Additionally, this large compositional space was utilized to evaluate conventional parameters that describe high entropy alloys (ΔS mix , ΔH mix , Ω, δ, and VEC), and these were found to elucidate the criteria for solid solution formation in this system. Additionally, trends illustrating the evolution from amorphous to crystalline phases are demonstrated and discussed. It is concluded that in this particular case, the dominant factors promoting crystallinity are thermodynamic or electronic, rather than geometric.

36 MATERIALS SCIENCE↗

DOE OSTI2026

Spatial Correlations of the Poisson Model for Radiation Transport

Characterizing the relationship between bulk physical properties and mixing in randomly heterogeneous media is a central challenge across many areas of science and engineering. A benchmark model for such studies is the Poisson model, a random tessellation of space by a Poisson process of hyperplanes. In radiation transport studies, the lack of exact expressions for the Poisson model’s spatial multipoint functions has led to approximate methods being used, introducing unquantified sources of error. Here, we recently introduced an exact solution for the Poisson model’s multipoint functions and closely related conditional probability functions (CPFs), providing a new opportunity to understand and reduce these sources of error. In this paper, we enable a more rigorous investigation of radiation transport in stochastic media by applying the recently introduced exact solution for the Poisson model’s CPFs. This paper consists of three main contributions. First, we introduce a unified framework for CPFs of the Poisson model, encompassing the recently introduced exact CPFs as well as the previously introduced atomic mix, nearest-neighbor, and combination CPFs. This framework also includes existing pruning techniques for the approximate CPFs, such as angular exclusion, as well as a novel form of angular exclusion suitable for the exact CPFs. Second, we use the exact CPFs to characterize the spatial regions where each approximate three-point CPF is most accurate, thereby explaining the observed hierarchy of accuracy among the approximate models. Finally, we evaluate material transmittance, reflectance, and flux in a three-dimensional test problem using conditional point sampling, demonstrating the relationship between CPF accuracy and transport simulation accuracy.

Poisson model↗

DOE OSTI2023

Fitting Matérn smoothness parameters using automatic differentiation

The Mat$\acute{e}$rn covariance function is ubiquitous in the application of Gaussian processes to spatial statistics and beyond. Perhaps the most important reason for this is that the smoothness parameter $\nu$ gives complete control over the mean-square differentiability of the process, which has significant implications for the behavior of estimated quantities such as interpolants and forecasts. Unfortunately, derivatives of the Mat$\acute{e}$rn covariance function with respect to $\nu$ require derivatives of the modified second-kind Bessel function $K$ $\nu$ with respect to $\nu$. While closed form expressions of these derivatives do exist, they are prohibitively difficult and expensive to compute. For this reason, many software packages require fixing $\nu$ as opposed to estimating it, and all existing software packages that attempt to offer the functionality of estimating $\nu$ use finite difference estimates for $\partial$ $\nu$ $K$ $\nu$ . In this work, we introduce a new implementation of $K$$\nu$ that has been designed to provide derivatives via automatic differentiation (AD), and whose resulting derivatives are significantly faster and more accurate than those computed using finite differences. Here, we provide comprehensive testing for both speed and accuracy and show that our AD solution can be used to build accurate Hessian matrices for second-order maximum likelihood estimation in settings where Hessians built with finite difference approximations completely fail.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Semi-transparent p-type barium copper sulfide as a back contact interface layer for cadmium telluride solar cells

Optically transparent p-type materials play a critical role in transparent electronics including photovoltaic (PV) devices. P-type sulfide materials offer an alternative to oxides for PV application due to improved hole transport properties. Here, we report the solution-based synthesis of earth-abundant p-type transparent conducting barium copper sulfide (α-BaCu 4 S 3 , BCS) thin films. These films were characterized using scanning electron microscopy, X-ray diffraction, UV–Vis–NIR spectrophotometry, Raman spectroscopy, and spectroscopic ellipsometry. BCS films of ~100 nm thickness transmit >70% of visible light. We report on tests of the hole transport properties of these BCS films for cadmium telluride (CdTe) photovoltaics, finding that the BCS deposition process forms a beneficial tellurium (Te) rich surface on CdTe by selectively removing Cd from the surface. Based on our study, the BCS interface layer plays dual functions for CdTe PV devices as a hole transport material and as an etchant, enhancing the resulting device performance. We observed a significant increase in open-circuit voltage of CdTe solar cells with the BCS buffer layer. Furthermore, we discuss semitransparent CdTe solar cells with BCS as a hole transport layer and indium tin oxide as a finishing electrode. Semitransparent CdTe solar cells shows 13.3% conversion efficiency for the front side illumination and 1.2% efficiency for back side illumination, indicating high recombination of charge carriers generated close to the rear CdTe/BCS/ITO contact.

36 MATERIALS SCIENCE↗

DOE OSTI2023

Solvent Dynamics Are Critical to Understanding Carbon Dioxide Dissolution and Hydration in Water

Simulations of carbon dioxide (CO 2 ) in water may aid in understanding the impact of its accumulation in aquatic environments and help advance technologies for carbon capture and utilization (via, e.g., mineralization). Quantum mechanical (QM) simulations based on static molecular models with polarizable continuum solvation poorly reproduce the energetics of CO 2 hydration to form carbonic acid in water, independent of the level of QM theory employed. Only with density-functional-theory-based molecular dynamics and rare-event sampling, followed by energy corrections based on embedded correlated wavefunction theory (in conjunction with density functional embedding theory), can a close agreement between theory and experiment be achieved. Such multilevel simulations can serve as benchmarks for simpler, less costly models, giving insight into potential errors of the latter. The strong influence of sampling/averaging over dynamical solvent configurations on the energetics stems from the difference in polarity of both the transition state and product (both polar) versus the reactant (nonpolar). When a solute undergoes a change in polarity during reaction, affecting its interaction with the solvent, careful assessment of the energetic contribution of the solvent response to this change is critical. Here, we show that static models (without structural sampling) that incorporate three explicit water molecules can yield far superior results than models with more explicit water molecules because fewer water molecules yield less configurational artifacts. Static models intelligently incorporating both explicit (molecules directly participating in the reaction) and implicit solvation, along with a proper QM theory, e.g., CCSD(T) for closed-shell systems, can close the accuracy gap between static and dynamic models.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

DOE OSTI2025

Model-based analysis of solute transport and potential carbon mineralization in the active layer of a hillslope underlain by permafrost with seasonal variability and climate change

Permafrost carbon, stored in frozen organic matter across vast Arctic and sub-Arctic regions, represents a substantial and increasingly vulnerable carbon reservoir. As global temperatures rise, the accelerated thawing of permafrost releases greenhouse gases, exacerbating climate change. However, freshly thawed permafrost carbon may also experience lateral transport by groundwater flow to surface water recipients such as rivers and lakes, increasing the terrestrial-to-aquatic transfer of permafrost carbon. Mobilization and subsurface transport mechanisms are poorly understood and not accounted for in global climate models, leading to high uncertainties in the predictions of the permafrost carbon feedback. Here, we focus on a hillslope in Endalen Valley, Svalbard, as a representative example of a high-Arctic hillslope underlain by continuous permafrost. We analyze solute transport in the form of a non-reactive tracer representing dissolved organic carbon (DOC) using a physics-based numerical model with the objective to study governing cryotic and hydrodynamic transport mechanisms relevant for warming permafrost regions. We first analyze transport times for DOC pools at different locations within the active layer under present-day climatic conditions and proceed to study susceptibility for deeper ancient carbon release in the upper permafrost due to thaw under different warming scenarios. Results suggest that DOC in the active layer near the permafrost table experiences rapid lateral transport upon thaw due to saturated conditions and lateral flow, while DOC close to the ground surface experiences slower transport due to flow in unsaturated soil. Deeper permafrost carbon release exhibits vastly different transport behaviors depending on warming and thaw rate. Gradual warming leads to small fractions of DOC being mobilized every year, while the majority moves vertically through percolation and cryosuction. Abrupt thaw resulting from a single very warm year leads to faster lateral transport times, similar to active layer DOC released in saturated conditions. Lastly, we analyze the potential susceptibility of DOC to mineralization to CO 2 prior to export due to soil moisture and temperature conditions. We find that high liquid saturation during transport coincides with very low mineralization rates and potentially inhibits mineralization into CO 2 before export. Overall, the results highlight the importance of subsurface hydrologic and thermal conditions for the retention and lateral export of permafrost carbon by subsurface flow.

Hamm, Alexandra [Stockholm Univ. (Sweden)] (ORCID:↗

DOE OSTI2025

Streptococcus pneumoniae HtrA is a dynamic and monomeric virulence factor capable of forming larger oligomeric complexes

Abstract High‐temperature requirement A (HtrA) proteases are a conserved family of serine proteases central to protein quality control and bacterial virulence. While Gram‐negative and human HtrAs are structurally well studied, Gram‐positive homologs remain essentially uncharacterized. Here, we present the first integrated structural and mechanistic analysis of a Gram‐positive HtrA, from Streptococcus pneumoniae , a virulence factor essential for adhesion and infection in vivo. Proteomic profiling of an htrA knockout and cleavage assays demonstrate that S. pneumoniae HtrA is required for protein quality control, with the PDZ domain mediating substrate recognition. Biochemically, S. pneumoniae HtrA exists exclusively as a monomer in solution, a striking divergence from canonical trimeric HtrAs that we show is shared with other Gram‐positive homologs. NMR analyses reveal that the monomer dynamically samples open and closed conformations, while cryo‐EM of a catalytic mutant identifies a hexamer stabilized by a unique LoopA–PDZ interaction. Together, these findings define S. pneumoniae HtrA as a dynamic monomer with interdomain coupling between its protease and PDZ domains, establishing Gram‐positive HtrAs as a mechanistically divergent subgroup within the HtrA family.

Lee, Eunjeong [Department of Biochemistry and Mole↗

DOE OSTI2020

On the use of charged particles to characterize precipitation in irradiated reactor pressure vessel steels with a wide range of compositions

Nuclear reactor lifetimes may be limited by nano-scale Cu-Mn-Ni-Si precipitates (CRPs and MNSPs) that form under neutron irradiation (NI) of pressure vessel (RPV) steels, resulting in hardening and ductile to brittle transition temperature increases (embrittlement). Physical models of embrittlement must be based on characterization of precipitation as a function of the combination of metallurgical and irradiation variables. Here we focus on rapid and convenient charged particle irradiations (CPI) to both: a) compare to precipitates formed in NI; and, b) use CPI to efficiently explore precipitation in steels with a very wide range of compositions. Atom probe tomography (APT) comparisons show NI and CPI for similar bulk steel solute contents yield nearly the same precipitate compositions, albeit with some differences in their number density, size and volume fraction (f) dose (dpa) dependence. Yet, the overall precipitate evolutions are very similar. Advanced high Ni (>3 wt%) RPV steels, with superior unirradiated properties, were also investigated at high CPI dpa. For typical Mn contents, MNSPs have Ni 16 Mn 6 Si 7 or Ni 3 Mn 2 Si phase type compositions, with f values that are close to the equilibrium phase separated values. In contrast, in steels with very low Mn and high Ni, Ni 2-3 Si silicide phase type precipitate compositions are observed; and when Ni is low, the precipitate compositions are close to the MnSi phase field. Low Mn significantly reduces, but does not eliminate, precipitation in high Ni steels. A comparison of dispersed barrier model predictions with measured hardening data suggests that the Ni-Si dominated precipitates are weaker dislocation obstacles than the G phase type MNSPs.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

DOE OSTI2021

Combined Precursor Engineering and Grain Anchoring Leading to MA-Free, Phase-Pure, and Stable α-Formamidinium Lead Iodide Perovskites for Efficient Solar Cells

a-Formamidinium lead iodide (a-FAPbI 3 ) is one of the most promising candidate materials for high-efficiency and thermally stable perovskite solar cells (PSCs) owing to its outstanding optoelectrical properties and high thermal stability. However, achieving a stable form of a-FAPbI 3 where both the composition and the phase are pure is very challenging. In this work, we report on a combined strategy of precursor engineering and grain anchoring to successfully prepare methylammonium (MA)-free and phase-pure stable a-FAPbI 3 films. The incorporation of volatile FA-based additives in the precursor solutions completely suppresses the formation of non-perovskite d-FAPbI 3 during film crystallization. Grains of the desired a-phase are anchored together and stabilized when 4-tert-butylbenzylammonium iodide is permeated into the a-FAPbI 3 film interior via grain boundaries. This cooperative scheme leads to a significantly increased efficiency close to 21% for FAPbI 3 perovskite solar cells. Moreover, the stabilized PSCs exhibit improved thermal stability and maintained ~90 % of their initial efficiency after storage at 50 degrees C for over 1600 hours.

14 SOLAR ENERGY↗

DOE OSTI2021

Large gyro-orbit model of ion velocity distribution in plasma near a wall in a grazing-angle magnetic field

A model is presented for the ion distribution function in a plasma at a solid target with a magnetic field $\boldsymbol {B}$ inclined at a small angle, $\alpha \ll 1$ (in radians), to the target. Adiabatic electrons are assumed, requiring $\alpha \gg \sqrt {Zm_{e}/m_{i}}$ , where $m_{e}$ and $m_{i}$ are the electron and ion mass, respectively, and $Z$ is the charge state of the ion. An electric field $\boldsymbol {E}$ is present to repel electrons, and so the characteristic size of the electrostatic potential $\phi$ is set by the electron temperature $T_{e}$ , $e\phi \sim T_{e}$ , where $e$ is the proton charge. An asymptotic scale separation between the Debye length $\lambda _{D} = \sqrt {\epsilon _0 T_{{e}} / e^{2} n_{{e}} }$ , the ion sound gyro-radius $\rho _{s} = \sqrt { m_{i} ( ZT_{e} + T_{i} ) } / (ZeB)$ and the size of the collisional region $d_{c} = \alpha \lambda _{\textrm {mfp}}$ is assumed, $\lambda _{D} \ll \rho _{s} \ll d_{c}$ . Here $\epsilon _0$ is the permittivity of free space, $n_{e}$ is the electron density, $T_{i}$ is the ion temperature, $B= |\boldsymbol {B}|$ and $\lambda _{\textrm {mfp}}$ is the collisional mean free path of an ion. The form of the ion distribution function is assumed at distances $x$ from the wall such that $\rho _{s} \ll x \ll d_{c}$ , that is, collisions are not treated. A self-consistent solution of the electrostatic potential for $x \sim \rho _{s}$ is required to solve for the quasi-periodic ion trajectories and for the ion distribution function at the target. The large gyro-orbit model presented here allows to bypass the numerical solution of $\phi (x)$ and results in an analytical expression for the ion distribution function at the target. It assumes that $\tau =T_{i}/(ZT_{e})\gg 1$ , and ignores the electric force on the quasi-periodic ion trajectory until close to the target. For $\tau \gtrsim 1$ , the model provides an extremely fast approximation to energy–angle distributions of ions at the target. These can be used to make sputtering predictions.

Physics↗

DOE OSTI2023

Bootstrapping the String Kawai-Lewellen-Tye Kernel

We show that a generalized version of the 4-point string theory KLT double-copy map is the most general solution to the minimal-rank double-copy bootstrap in effective field theory. This follows from significant restrictions of the 4-point map resulting from the 6-point bootstrap analysis. The generalized 4-point double-copy map is defined by a function with only two parameters times a simple function that is symmetric in s,t,u. The two parameters can be interpreted as independent choices of α', one for each of the two sets of amplitudes double-copied with the map. Specifically, each of those two sets of amplitudes must obey either the string monodromy relations or the field theory KK & BCJ relations; there are no other options. We propose a closed form of the new double-copy map that interpolates between the original KLT string double-copy and the open & closed string period integrals. The construction clarifies the "single-valued projection" property of the Riemann zeta-function values for the 4-point string theory double copy.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗