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All sourcesNASA NTRSDOE OSTI
DOE OSTI2022

First-principles study of L -shell iron and chromium opacity at stellar interior temperatures

Recently developed free-energy density functional theory (DFT)-based methodology for optical property calculations of warm dense matter has been applied for studying L-shell opacity of iron and chromium at T = 182 eV. We use Mermin–Kohn–Sham density functional theory with a ground-state and a fully temperature-dependent generalized gradient approximation exchange-correlation (XC) functionals. It is demonstrated that the role of XC at such a high-T is negligible due to the total free-energy of interacting system being dominated by the noninteracting free-energy term in agreement with estimations for the homogeneous electron gas. Furthermore, our DFT predictions are compared to the radiative emissivity and opacity of dense plasmas model, to the real-space Greens function method, and to experimental measurements. Good agreement is found between all three theoretical methods, and in the bound–continuum region for Cr when compared to the experiment, while the discrepancy between direct DFT calculations and the experiment for Fe remains essentially the same as for plasma-physics models.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Orbital-selective Mott phase and non-Fermi liquid in FePS 3

The layered metal phosphorous trisulfide FePS 3 is reported to be a Mott insulator at ambient conditions and to undergo structural and insulator-metal phase transitions under pressure. However, the character of the resulting metallic states has not been understood clearly so far. Here, we theoretically study the phase transitions of FePS 3 using first-principles methods based on density functional theory and embedded dynamical mean field theory. We find that the Mott transition in FePS 3 can be orbital selective, with t 2g states undergoing a correlation-induced insulator-to-metal transition while e g states remain gapped. We show that this orbital-selective Mott phase, which occurs only when nonhydrostatic pressure is used, is a bad metal (or non-Fermi liquid) with large fluctuating moments due to Hund's coupling. Further application of pressure increases the crystal-field splitting and converts the system to a conventional Fermi liquid with low-spin configurations dominant. Finally, our results show that FePS 3 is an example of a system that realizes an orbital-selective Mott phase, allowing tuning between correlated and uncorrelated metallic properties in an accessible pressure range (≤ 18 GPa).

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2023

Assessing the Role of Interfacial and Metal Sites in Pt/TiO2-Catalyzed Acetic Acid Hydrodeoxygenation

Thermochemical conversion of biomass to produce drop-in quality biofuels typically involves hydrodeoxygenation (HDO) steps following catalytic fast pyrolysis (CFP) to remove excess oxygen and create a more-stable bio-oil product. HDO reactions are performed by co-feeding the CFP vapor-phase product and H2 gas over a bi-functional catalyst. Noble-metal catalysts supported on reducible metal oxides (e.g., Pt/TiO2) are active and selective toward these HDO reactions. Griffin and co-workers showed that Pt/TiO2 catalysts promote the desired deoxygenation steps for m-cresol HDO while mitigating undesired C-C bond-breaking steps that reduce the overall value/energy density of the biofuel. Such model-compound studies to inform the design of improved catalysts for HDO chemistry are necessary to improve the overall process economics/efficiencies for biofuels production. One important class of bio-derived compounds that has not been studied extensively with respect to HDO chemistry, particularly at the atomic level, is carboxylic acids. Past experimental work indicates that Pt/C and Pt/TiO2 catalysts are selective toward C-C and C-O bond-dissociation products for acetic acid HDO, respectively. To understand the role of Pt/TiO2 active sites in this observed change in selectivity and guide catalyst development, the work in this presentation focuses on modeling the role of Pt-metal and Pt-TiO2-interfacial sites in promoting key C-C bond-breaking, C-O bond-breaking, and (de)hydrogenation steps in acetic acid HDO. Density functional theory (DFT) calculations were performed using the Vienna Ab initio Simulation Package (VASP). The exchange-correlation functional was approximated by the Perdew-Burke-Ernzerhof functional. Dispersion interactions were captured using the D3 method. Projector augmented-wave potentials described electron-ion interactions, and electron wavefunctions were expanded via a planewave basis with an energy cutoff of 500 eV. Activation barriers were calculated using the climbing image nudged elastic band method. Results and Discussion: To discern the role of Pt-metal and Pt-TiO2 interface sites in promoting acetic acid HDO chemistry, Pt(111) slab and anatase TiO2(101)-supported Pt-nanowire (PtNW/OH-TiO2) surface models were constructed, respectively. Because H2 is co-fed in HDO reactions, the anatase support was terminated with OH groups. Interfacial vacancies have been shown to facilitate Ru/TiO2-catalyzed phenol HDO; thus, an interfacial model with an OH vacancy was also considered (PtNW/OHv-TiO2). Pt-TiO2-interface sites stabilize adsorption of all studied acetic acid HDO surface intermediates relative to terrace Pt-metal sites, particularly when an interfacial-OH vacancy is present. Oxygenated species prefer to bind at the OH vacancy through the O atom, suggesting a preference for C-O over C-C bond cleavage at these sites. This hypothesis is supported by net-negative and net-positive average shifts in the reaction energy and activation energy barriers for C-O and C-C bond-breaking steps, respectively, at Pt-TiO2-interface sites relative to Pt-metal sites. Using the calculated energetics, the predicted minimum-energy pathway was determined for each surface. Pt(111) is predicted to follow decarboxylation, producing undesired methane and carbon dioxide. Conversely, PtNW/OH-TiO2 and PtNW/OHv-TiO2 are both predicted to produce desired acetaldehyde and ethane. The interfacial vacancy may also play a key role in facilitating the first C-O bond-breaking step in acetic acid HDO, lowering the barrier by 0.6 eV relative to the defect-free interface model. These results demonstrate the critical role of the Pt-TiO2 interface in the shift in acetic acid HDO selectivity experimentally observed on Pt/C and Pt/TiO2 catalysts. The results herein demonstrate the important role of the Pt-TiO2 interface and interfacial oxygen vacancies in improving the carbon efficiency for HDO reactions in CFP upgrading.

BIOMASS FUELS,INORGANIC, ORGANIC, PHYSICAL, AND AN↗
DOE OSTI2020

Comprehensive Studies on Hot Compaction and Vibration-Assisted Compaction Tests of Aluminum Powder

Abstract Aluminum powder compaction was studied using both test and simulation. Cold compaction, hot compaction, and vibration-assisted (cold) compaction tests were conducted to achieve different density ratios. First, the hot compaction test (at 300 °C, compression pressure 140 MPa) improved about 6% compared with cold compaction under the same compression pressure. Second, although the relative density ratio does not obviously improve at a vibration-assisted (cold) compaction, the strength of the specimens made under vibration loading is much better than those of cold compaction. Additionally, finite element models with well-calibrated Drucker–Prager Cap (DPC) material constitutive model were built in abaqus/standard to simulate the powder compaction process. The results of the finite element model have very good correlations with test results up to the tested range, and this finite element model further predicts the loading conditions needed to achieve the higher density ratios. Two exponential equations of the predicted density ratio were obtained by combining the test data and the simulation results. A new analytical solution was developed to predict the axial pressure versus the density ratio for the powder compaction according to DPC material model. The results between the analytical solution and the simulation model have a very good match.

Engineering↗
DOE OSTI2022

Assessing MP2 frozen natural orbitals in relativistic correlated electronic structure calculations

The high computational scaling with the basis set size and the number of correlated electrons is a bottleneck limiting applications of coupled cluster algorithms, in particular for calculations based on two- or four-component relativistic Hamiltonians, which often employ uncontracted basis sets. This problem may be alleviated by replacing canonical Hartree–Fock virtual orbitals by natural orbitals (NOs). Here, in this paper, we describe the implementation of a module for generating NOs for correlated wavefunctions and, in particular, second order Møller–Plesset perturbation frozen natural orbitals (MP2FNOs) as a component of our novel implementation of relativistic coupled cluster theory for massively parallel architectures [Pototschnig et al. J. Chem. Theory Comput. 17, 5509, (2021)]. Our implementation can manipulate complex or quaternion density matrices, thus allowing for the generation of both Kramers-restricted and Kramers-unrestricted MP2FNOs. Furthermore, NOs are re-expressed in the parent atomic orbital (AO) basis, allowing for generating coupled cluster singles and doubles NOs in the AO basis for further analysis. By investigating the truncation errors of MP2FNOs for both the correlation energy and molecular properties—electric field gradients at the nuclei, electric dipole and quadrupole moments for hydrogen halides HX (X = F–Ts), and parity-violating energy differences for H 2 Z 2 (Z = O–Se)—we find MP2FNOs accelerate the convergence of the correlation energy in a roughly uniform manner across the Periodic Table. It is possible to obtain reliable estimates for both energies and the molecular properties considered with virtual molecular orbital spaces truncated to about half the size of the full spaces.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Site-selective correlations in interacting flat-band quasicrystals

Model lattices such as the kagome and Lieb lattices have been widely investigated to elucidate the properties of interacting flat-band systems. While a quasicrystal does not have proper bands, the noninteracting density of states of several of them displays the typical signature of a flat band pinned at the Fermi level: a delta-function zero-energy peak. Here, we employ quantum Monte Carlo simulations to determine the effect of onsite repulsion on these quasicrystals. While global properties such as the antiferromagnetic structure factor and the specific heat behave similarly as in the case of periodic lattices undergoing a Mott transition, the behavior of the local density of states depends on the coordination number of the site. In particular, sites with the smallest coordination number, which give the dominant spectral-weight contribution to the zero-energy peak, are the ones most strongly impacted by the interaction. Besides establishing site-selective correlations in quasicrystals, our work also points to the importance of the real-space structure of flat bands in interacting systems.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2025

Flat band driven itinerant magnetism in the Co-pnictides CaCo 2⁢ As 2 and LaCo 2 ⁢P 2

Flat bands can induce strong electron correlation effects that help stabilize both magnetic and superconducting states. Here, in this study, we carry out angle-resolved photoemission spectroscopy and density functional theory calculations to study the electronic structure of the Co-pnictides CaCo 2 ⁢As 2 and LaCo 2 ⁢P 2 . We find that the 𝑘 𝑧 Fermi topology of ferromagnetic LaCo 2 ⁢P 2 is markedly two-dimensional, while the antiferromagnetic CaCo 2 ⁢As 2 develops a 3D Fermi surface described by a zigzaglike band dispersion perpendicular to the Co-As plane. Furthermore, the magnetism is driven by the electronic correlations of the flat bands with 𝑑 𝑥⁢𝑦 and 𝑑 𝑧 2 orbital character at the Fermi level. Our results link the electronic dimensionality and the magnetic order and emphasize the critical role of the As-As and P-P bond strength along the 𝑐 direction to understand the electronic band structure and the rich phase diagram of transition metal pnictides.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Revisiting the structures and phase transitions of Ba 2 NaNb 5 O 15

The room-temperature and low-temperature structure(s) of Ba 2 NaNb 5 O 15 (BNN) have been debated since the structure was proposed in the 1960s. This work revisits the structures and phase transitions of BNN, combining high-resolution X-ray and neutron powder diffraction with density functional theory calculations. Temperature-dependent high-resolution X-ray powder diffraction patterns are collected from 4 to 918 K, and sequential batch Rietveld refinement using a symmetry mode approach to describe the structure is used to extract the main structural changes as a function of temperature. The data show that the average structure of BNN is best described by the Ama2 space group, and no other structural phase transitions were observed below the ferroelastic transition. The symmetry mode analysis, combining results from diffraction and density functional theory, shows significant octahedral tilting and corrugations of both the A1 and A2 sites along the c direction. A strong correlation between the spontaneous strain and the octahedral tilting was observed, and a potential connection with emerging microstructure at low temperatures is proposed, all enabled by the symmetry mode approach used in this work.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2022

Modification of LiMn 2 O 4 surfaces by controlling the Acid–Base surface chemistry of atomic layer deposition

In this combined theoretical and experimental study, we report the effects of Al precursor selection on the growth chemistry, interfacial structure, and electrochemistry of Al 2 O 3 coatings on spinel LiMn 2 O 4 (LMO) surfaces by atomic layer deposition (ALD). Five Al precursors, exhibiting a range of Lewis acid-base properties, were used to establish trends in the ALD Al 2 O 3 growth chemistry on LMO powders in comparison to redox-inactive planar silicon substrates. We show that the Lewis acid-base properties of the Al precursor ligands can be used to tune both the Mn oxidation states and the Al 2 O 3 coverage on the LMO surface. Density functional theory calculations are used to examine the reaction mechanism of each precursor on LMO surfaces, elucidating a correlation between the Lewis acidity of the ligands and the decomposition thermochemistry. In-depth X-ray photoelectron spectroscopy and in situ Fourier transform infrared spectroscopy measurements support these theoretical predictions and further reveal how different Al precursors modify the atomic and electronic structure near the LMO surface during ALD. While the Mn oxidation state is strongly influenced by the Lewis acidity of the precursor ligand, the surface coverage and thickness of the Al 2 O 3 coating are a more representative descriptor of the electrochemical performance measured in coin cell experiments. Finally, we show how the ligand acid-base properties of ALD precursors can be used to rationally tailor atomic layer growth mechanisms, which may enable atomic level control over the structure of functionalized interfaces for various applications related to catalysis, semiconductors, and energy storage.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Facet-Dependent Doping and Dopant-Dependent Faceting in Si-Doped GaAsSb Nanowires

Here, we correlate the spatial distributions of Si, Sb, and rotational twins in Si-doped GaAs 1-x Sb x nanowires. GaAs 1-x Sb x nanowires were grown epitaxially on Si(111) substrates by tuning process conditions to achieve repeated nucleation of rotational twins and growth along the [111]B direction; dilute Sb and Si fluxes were chosen to create a sufficient twin density to achieve high yield while avoiding growth of the wurtzite phase. While the impact of Si and Sb on twin density and nanowire growth rate has been previously reported, the facet-dependent incorporation of these species has not been established. Scanning transmission electron microscopy was used to confirm that Sb incorporates preferentially on the (111)B facets relative to {1 ̅1 ̅0} facets prior to nucleation of a rotational twin. With periodic twinning, this facet dependence leads to alternating regions of enriched and depleted Sb concentration attributed to a growth rate-dependent Sb-As-exchange mechanism. Atom probe tomography measurements establish that while Si doping is not perturbed by twinning on (111)B facets, Si and Sb concentrations are anti-correlated for growth on non-(111)B facets. Density functional theory calculations underpin a thermodynamic model that explains the observed anisotropies in dopant incorporation.

36 MATERIALS SCIENCE↗
DOE OSTI2022

Stochastic exciton-scattering theory of optical line shapes: Renormalized many-body contributions

Spectral line shapes provide a window into the local environment coupled to a quantum transition in the condensed phase. In this paper, we build upon a stochastic model to account for non-stationary background processes produced by broad-band pulsed laser stimulation, as distinguished from those for stationary phonon bath. In particular, we consider the contribution of pair-fluctuations arising from the full bosonic many-body Hamiltonian within a mean-field approximation, treating the coupling to the system as a stochastic noise term. Herein, using the Itô transformation, we consider two limiting cases for our model, which lead to a connection between the observed spectral fluctuations and the spectral density of the environment. In the first case, we consider a Brownian environment and show that this produces spectral dynamics that relax to form dressed excitonic states and recover an Anderson–Kubo-like form for the spectral correlations. In the second case, we assume that the spectrum is Anderson–Kubo like and invert to determine the corresponding background. Using the Jensen inequality, we obtain an upper limit for the spectral density for the background. The results presented here provide the technical tools for applying the stochastic model to a broad range of problems.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2026

A comprehensive first-principles study of the effects of the exchange-correlation functional and magnetism on defect and diffusion properties of the CoCrNi medium-entropy alloy

The present work is a novel, systematic study of the effect of density functional theory input parameters on the vacancy formation energy (VFE), migration barrier for diffusion, and electronic structure for each element in the CoCrNi medium-entropy alloy (MEA). In particular, the novelties include: (1) calculating the aforementioned properties of Co, Cr, or Ni, in the CoCrNi MEA using magnetic and non-magnetic states, and two versions of the generalized gradient approximation: Perdew, Burke, and Ernzerhof (PBE) and the PBE version for solids (PBEsol), and (2) a detailed comparison of 0 K activation energy to experimental creep activation energies. First-principles calculations at 0 K are performed using the Vienna ab-initio simulation package. Special quasirandom structures (SQS) and Widom-type substitution are employed. For each element, Co, Cr, or Ni, non-magnetic calculations result in a higher VFE and larger range of calculated values for the configurations studied. The averaged migration barrier is the highest for Co in the CoCrNi for three of four sets of calculation parameters in the configurations studied. Finally, the results indicate that the average 0 K activation energy for diffusion makes up 70–80% of the experimental creep activation energy, depending on the exchange-correlation functional employed.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Nearest neighbour distributions: New statistical measures for cosmological clustering

ABSTRACT The use of summary statistics beyond the two-point correlation function to analyse the non-Gaussian clustering on small scales, and thereby, increasing the sensitivity to the underlying cosmological parameters, is an active field of research in cosmology. In this paper, we explore a set of new summary statistics – the k-Nearest Neighbour Cumulative Distribution Functions (kNN-CDF). This is the empirical cumulative distribution function of distances from a set of volume-filling, Poisson distributed random points to the k-nearest data points, and is sensitive to all connected N-point correlations in the data. The kNN-CDF can be used to measure counts in cell, void probability distributions, and higher N-point correlation functions, all using the same formalism exploiting fast searches with spatial tree data structures. We demonstrate how it can be computed efficiently from various data sets – both discrete points, and the generalization for continuous fields. We use data from a large suite of N-body simulations to explore the sensitivity of this new statistic to various cosmological parameters, compared to the two-point correlation function, while using the same range of scales. We demonstrate that the use of kNN-CDF improves the constraints on the cosmological parameters by more than a factor of 2 when applied to the clustering of dark matter in the range of scales between 10 and $40\, h^{-1}\, {\rm Mpc}$. We also show that relative improvement is even greater when applied on the same scales to the clustering of haloes in the simulations at a fixed number density, both in real space, as well as in redshift space. Since the kNN-CDF are sensitive to all higher order connected correlation functions in the data, the gains over traditional two-point analyses are expected to grow as progressively smaller scales are included in the analysis of cosmological data, provided the higher order correlation functions are sensitive to cosmology on the scales of interest.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2024

A numerical study of bounds in the correlations of fractional quantum Hall states

We numerically compute the guiding center static structure factor \bar S(k) S ‾ ( k ) of various fractional quantum Hall (FQH) states to \mathcal{O}(k\ell)^6 𝒪 ( k ℓ ) 6 where k k is the wavenumber and \ell ℓ is the magnetic length. Employing density matrix renormalization group on an infinite cylinder of circumference L_y L y , we study the two-dimensional limit using L_y/\xi \gg 1 L y / ξ ≫ 1 , where \xi ξ is the correlation length. The main findings of our work are: 1) the ground states that deviate away from the ideal conformal block wavefunctions, do not saturate the Haldane bound, and 2) the coefficient of O(k\ell)^6 O ( k ℓ ) 6 term appears to be bounded above by a value predicted by field theories proposed in the literature. The first finding implies that the graviton mode is not maximally chiral for experimentally relevant FQH states.

Kumar, Prashant (ORCID:0000000258002768)↗
DOE OSTI2023

Imposing correct jellium response is key to predict the density response by orbital-free DFT

Orbital-free density functional theory constitutes a computationally highly effective tool for modeling electronic structures of systems ranging from room-temperature materials to warm dense matter. Its accuracy critically depends on the employed kinetic energy (KE) density functional, which has to be supplied as an external input. In this work we consider several nonlocal and Laplacian-level KE functionals and use an external harmonic perturbation to compute the static density response at T = 0 K in the linear and beyond-linear response regimes. We test for the satisfaction of exact conditions in the limit of uniform densities and for how approximate KE functionals reproduce the density response of realistic materials (e.g., Al and Si) against the Kohn-Sham DFT reference, which employs the exact KE. The results illustrate that several functionals violate exact conditions in the uniform electron gas (UEG) limit. We find a strong correlation between the accuracy of the KE functionals in the UEG limit and in the strongly inhomogeneous case. This empirically demonstrates the importance of imposing the limit of UEG response for uniform densities and validates the use of the Lindhard function in the formulation of kernels for nonlocal functionals. This conclusion is substantiated by additional calculations for bulk aluminum (Al) with a face-centered cubic (fcc) lattice and silicon (Si) with an fcc lattice, body-centered cubic (bcc) lattice, and semiconducting crystal diamond state. The analysis of fcc Al, and fcc as well as bcc Si data follows closely the conclusions drawn for the UEG, allowing us to extend our conclusions to realistic systems that are subject to density inhomogeneities induced by ions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2020

Effects of Fluorinated Diluents in Localized High‐Concentration Electrolytes for Lithium–Oxygen Batteries

Abstract A stable electrolyte is critical for practical application of lithium–oxygen batteries (LOBs). Although the ionic conductivity and electrochemical stability of the electrolytes have been extensively investigated before, their oxygen solubility, viscosity, volatility, and the stability against singlet oxygen ( 1 O 2 ) still need to be comprehensively investigated to provide a full picture of the electrolytes, especially for an open system such as LOBs. Herein, a systematic investigation is reported on the localized high‐concentration electrolytes (LHCEs) using different fluorinated diluents in comparison with those of conventional electrolytes. The physical properties and activation energies for reactions with singlet oxygen ( 1 O 2 ) of these electrolytes are calculated by density functional theory. The electrochemical performances of LOBs using these electrolytes are compared. This study reveals that the correlation between the stability of the electrolytes and their physical and electrochemical properties depends strongly on the diluents in LHCEs. Therefore, it shines light on the rational design of new electrolytes for LOBs.

Kwak, Won‐Jin↗
DOE OSTI2024

Cellular Phosphate Sensing and Anion Binding by an Azacrown‐Calixpyrrole Hybrid

A hybrid receptor-sensor for anions originating from the merging of positively charged ammonium moieties for electrostatic attraction/stronger binding of azacrowns with directionality of calixpyrrole hydrogen bond donors for selectivity is investigated. As demonstrated this hybrid receptor-sensor shows a remarkable selectivity for orthophosphate even in the presence of other phosphates and anions found in cellular materials (K assoc H 2 PO 4 − >H 2 P 2 O 7 2− >AMP − ≫ADP 2− or ATP 3− over halides, nitrate, or hydrogen sulfate; all Na + salts in water) but also cellular polyphosphate or phospholipids. This selectivity is harnessed in a real-time monitoring of cell lysis by lysozyme, which releases orthophosphate and other phosphates and anions from the cells. This sensitive (LOD 0.4 μM) fluorescence-based microscale method compares favorably with the state-of-the-art techniques but can easily be practiced in a high-throughput screening (HTS) manner. The anion binding and selectivity in aqueous solutions were investigated by NMR and put in context with phosphate binding of the parent calix[4]pyrrole. The microscopic understanding of anion binding by the hybrid receptor was then obtained from a combination of density functional theory (DFT), classical molecular dynamics (MD) with explicit water solvation, and ab initio MD (AIMD) simulations. Correlating the NMR and fluorescence binding data with studies of solvation of the receptor, phosphate anion, and the resulting complex confirms the binding is largely driven by entropic component (TΔS) associated with receptor and anion desolvation.

Anions↗
DOE OSTI2024

Numerical analysis of a time discretized method for nonlinear filtering problem with Lévy process observations

Abstract In this paper, we consider a nonlinear filtering model with observations driven by correlated Wiener processes and point processes. We first derive a Zakai equation whose solution is an unnormalized probability density function of the filter solution. Then, we apply a splitting-up technique to decompose the Zakai equation into three stochastic differential equations, based on which we construct a splitting-up approximate solution and prove its half-order convergence. Furthermore, we apply a finite difference method to construct a time semi-discrete approximate solution to the splitting-up system and prove its half-order convergence to the exact solution of the Zakai equation. Finally, we present some numerical experiments to demonstrate the theoretical analysis.

Mathematics↗