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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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At least 55 records · Page 3
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Quantum Anomalies in Condensed Matter

Quantum materials provide a fertile ground in which to test and realize unusual phenomena such as quantum anomalies predicted by quantum field theory. There are three important symmetries that are broken when classical field theory is moved into the quantum regime, the scale anomaly, the axial (chiral) anomaly, and the parity anomaly. Several potential device applications may be realized by the discovery of quantum anomalies in condensed matter, enabled by the new physics they embody, including ultra‐sensitive dark matter detectors, far infrared optical modulators, micro‐bolometric detectors, low‐dissipation ballistic transporters, terahertz‐based qubits, terahertz polarization state controls, passive magnetic field sensors, stable topological superconductors that host Majorana fermions, and qubits topologically protected against decoherence. In this perspective article, the definition of these quantum anomalies is laid out, how little is known in the context of condensed matter, and how quantum anomalies are predicted to manifest as anomalous electronic, thermal, and magnetic behavior in experiments on topological quantum materials, including Weyl and Dirac semimetals. Furthermore, the importance that mechanical strain and defects will play in modifying signatures of quantum anomalies is discussed.

36 MATERIALS SCIENCE↗
DOE OSTI2024

Crosstalk Effects in Microwave SQUID Multiplexed TES Bolometer Readout

Transition-edge sensor (TES) bolometers are broadly used for background-limited astrophysical measurements from the far-infrared to mm-waves. Many planned future instruments require increasingly large detector arrays, but their scalability is limited by their cryogenic readout electronics. Microwave SQUID multiplexing offers a highly capable scaling solution through the use of inherently broadband circuitry, enabling readout of hundreds to thousands of channels per microwave line. As with any multiplexing technique, the channelization mechanism gives rise to electrical crosstalk which must be understood and controlled so as to not degrade the instrument sensitivity. Here, in this study, we explore implications relevant for TES bolometer array applications, focusing in particular on upcoming mm-wave observatories such as the Simons Observatory and AliCPT. We model the relative contributions of the various underlying crosstalk mechanisms, evaluate the difference between fixed-tone and tone-tracking readout systems, and discuss ways in which crosstalk nonlinearity will complicate on-sky measurements.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Three-pole wigglers at NSLS-II

The National Synchrotron Light Source-II (NSLS-II) at Brookhaven National Lab (BNL) is a third-generation medium-energy storage ring with 3 GeV and sub-nm-rad horizontal emittance, equipped with a top-off capability of 500 mA. It is designed to deliver an extremely intense photon beam across a wide spectral range, from the far-infrared to the very hard X-ray region. This outstanding performance is achieved through a combination of bending magnets (BM), three-pole wigglers (3 PW), and advanced insertion device (ID) sources. Six three-pole wigglers have been constructed at the NSLS-II IDs Lab. These devices are used to produce broadband radiation with lower angular power density and to monitor the electron beam emittance and energy spread. Here, this paper describes the R&D activities focused on developing the required magnetic and mechanical designs, along with magnetic field optimization and the final magnetic measurement results. It also covers the spectral brightness, flux, power density, spatial and angular output properties of the 3 PWs, including their commissioning results and their effects on the performance of the NSLS-II storage ring.

36 MATERIALS SCIENCE↗
DOE OSTI2025

Delineating the Effects of Counterions on the Structural and Vibrational Properties of U(IV) Lindqvist Polyoxometalate Complexes

Herein we conducted a full investigation into the fundamental structural and vibrational properties of uranium(IV) Peacock−Weakley-type lacunary Lindqvist (W 10 ) polyoxometalate (POM) complexes. We recently demonstrated the importance of the secondary lattice elements in tuning the distortion of the D 4d symmetry in W 10 POM complexes, and here, we synthesized eight UW 10 complexes with different alkali metal counterions and evaluated how the composition and packing of counterion species affected complex structural and vibrational properties. Single-crystal X-ray diffraction analysis on complexes 1−8 revealed changes in structural distortion parameters as a function of differences in counterion configurations, while far-infrared and Raman spectra for 1−8 also demonstrated that vibrational mode frequencies were sensitive to changes in counterion composition and packing. To more effectively compare different counterion configurations, we developed counterion effective ionic radius (eIR) as a new structural parameter, and comparisons between structural distortion parameters and eIR values strongly suggested that modulation by the secondary lattice elements can affect structural and vibrational manifolds within POM complexes. Partial least squares (PLS) analysis was used to quantitatively evaluate correlations observed within this investigation, and PLS statistical models showed a strong correlation between counterion eIR and both structural distortion parameters and vibrational mode frequencies.

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

Multi-Spectroscopic Determination of Exchange Coupling, Zero-Field Splitting, and g-Matrices in Radical-Bridged Dinuclear Fe(III) Complexes

When the energy gap, Δ, between the lowest-lying spin manifolds within a spin-exchange coupled molecule approaches Δ/k B ≈ 300 K, the traditional temperature-dependence (T < 400 K) of the molar magnetic susceptibility is not always a reliable way to obtain a good estimate of intramolecular exchange couplings. We develop a spectroscopic approach capable of accurately parametrizing complex magnetic Hamiltonians by exploiting the separation of the anisotropy and exchange energy scales in strongly coupled magnetic molecules. Specifically, we combine inelastic neutron scattering, high-frequency electron paramagnetic resonance, far-infrared magneto-spectroscopy and magnetometry, and obtain detailed information about the magnetic properties of a series of diiron complexes derived from [[Fe(cth)] 2 (dxbq)] 3+ (H 2 dxbq: 2,5-dihydroxy-1,4-benzoquinone (x = h) or 3,6-dichloro-2,5-dihydroxy-1,4-benzoquinone (x = c), cth: 5,5,7,12,12,14-hexamethyl-1,4,8,11-tetraazacyclotetradecane). Well-isolated S = 9/2 ground states emerge due to strong direct antiferromagnetic exchange between the Fe 3+ centers (S = 5/2) and the radical bridging benzoquinone ligand (S = 1/2). The specific sensitivities and transition selection rules of the applied methods allow us to determine the parameters of the microscopic Hamiltonian including exchange coupling, fourth-order Stevens operators and g-factors. Our methodology is directly portable to other strongly coupled molecular compounds.

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

Pressure-Induced Metal-like Transport and Magnetoresistance in a Au 2+ –Au 3+ Halide Perovskite

The Cs 4 Au II Au III 2 Cl 12 perovskite (1), featuring AuCl 4 trimers separated by vacancies, enables the first high-pressure study of Au 2+/3+ mixed-valence. Our computational analysis of the gold frontier orbitals suggests that the Au 2+ →Au 3+ intervalence charge transfer (IVCT) occurs across the vacancies. Computational structures indicate that these vacancies rapidly shrink with pressure and the Au 2+ and Au 3+ coordination spheres become very similar at the phase transition to nearly cubic symmetry at ca. 15 GPa─enabling facile IVCT. Although the activation energy of conductivity of 0.73(4) meV and far-infrared absorption indicate a small but nonzero bandgap, ambient thermal energy drives the IVCT, affording metallic properties: prominent infrared reflectivity and transport values of 10 2 S·cm –1 . This prompted us to perform the first high-pressure studies of magnetoresistance (MR) and Hall effect in halide perovskites. At 16 GPa, the MR increases by 9.3% at 2 K and 9 T; this value is maintained up to 27 GPa, when a local distortion drives electronic localization. By globally fitting the MR and Hall resistance to a two-carrier model we quantify how the carrier densities and mobilities evolve with pressure. Thus, metal-like transport and MR in 1 is driven by a pressure-induced transition from localized to partially delocalized mixed-valence.

Deschene, Christina R. [Stanford University, CA (U↗
DOE OSTI2024

Wafer-Scale MgB 2 Superconducting Devices

Progress in superconducting device and detector technologies over the past decade has realized practical applications in quantum computers, detectors for far-infrared telescopes, and optical communications. Superconducting thin-film materials, however, have remained largely unchanged, with aluminum still being the material of choice for superconducting qubits and niobium compounds for high-frequency/high kinetic inductance devices. Magnesium diboride (MgB 2 ), known for its highest transition temperature (T c = 39 K) among metallic superconductors, is a viable material for elevated temperature and higher frequency superconducting devices moving toward THz frequencies. However, difficulty in synthesizing wafer-scale thin films has prevented implementation of MgB 2 devices into the application base of superconducting electronics. Here, we report ultrasmooth (<0.5 nm root-mean-square roughness) and uniform MgB 2 thin (<100 nm) films over 100 mm in diameter and present prototype devices fabricated with these films demonstrating key superconducting properties including an internal quality factor over 10 4 at 4.5 K and high tunable kinetic inductance in the order of tens of pH/sq in a 40 nm thick film. This advancement will enable development of elevated temperature, high-frequency superconducting quantum circuits, and devices.

Boron↗
DOE OSTI2025

Roadmap for Photonics with 2D Materials

Triggered by advances in atomic-layer exfoliation and growth techniques, along with the identification of a wide range of extraordinary physical properties in self-standing films consisting of one or a few atomic layers, two-dimensional (2D) materials such as graphene, transition metal dichalcogenides (TMDs), and other van der Waals (vdW) crystals now constitute a broad research field expanding in multiple directions through the combination of layer stacking and twisting, nanofabrication, surface-science methods, and integration into nanostructured environments. Photonics encompasses a multidisciplinary subset of those directions, where 2D materials contribute remarkable nonlinearities, long-lived and ultraconfined polaritons, strong excitons, topological and chiral effects, susceptibility to external stimuli, accessibility, robustness, and a completely new range of photonic materials based on layer stacking, gating, and the formation of moiré patterns. These properties are being leveraged to develop applications in electro-optical modulation, light emission and detection, imaging and metasurfaces, integrated optics, sensing, and quantum physics across a broad spectral range extending from the far-infrared to the ultraviolet, as well as enabling hybridization with spin and momentum textures of electronic band structures and magnetic degrees of freedom. The rapid expansion of photonics with 2D materials as a dynamic research arena is yielding breakthroughs, which this Roadmap summarizes while identifying challenges and opportunities for future goals and how to meet them through a wide collection of topical sections prepared by leading practitioners.

2D materials↗
DOE OSTI2024

Coercive Fields Exceeding 30 T in the Mixed-Valence Single-Molecule Magnet (Cp iPr5 ) 2 Ho 2 I 3

Mixed-valence dilanthanide complexes of the type (Cp iPr5 ) 2 Ln 2 I 3 (Cp iPr5 = pentaisopropylcyclopentadienyl; Ln = Gd, Tb, Dy) featuring a direct Ln–Ln σ-bonding interaction have been shown to exhibit well-isolated high-spin ground states and, in the case of the Tb and Dy variants, a strong axial magnetic anisotropy that gives rise to a large magnetic coercivity. Here, we report the synthesis and characterization of two new mixed-valence dilanthanide compounds in this series, (Cp iPr5 ) 2 Ln 2 I 3 (1-Ln; Ln = Ho, Er). Both compounds feature a Ln–Ln bonding interaction, the first such interaction in any molecular compounds of Ho or Er. Like the Tb and Dy congeners, both complexes exhibit high-spin ground states arising from strong spin–spin coupling between the lanthanide 4f electrons and a single σ-type lanthanide–lanthanide bonding electron. Beyond these similarities, however, the magnetic properties of the two compounds diverge. In particular, 1-Er does not exhibit observable magnetic blocking or slow magnetic relaxation, while 1-Ho exhibits magnetic blocking below 28 K, which is the highest temperature among Ho-based single-molecule magnets, and a spin reversal barrier of 556(4) cm –1 . Additionally, variable-field magnetization data collected for 1-Ho reveal a coercive field of greater than 32 T below 8 K, more than 6-fold higher than observed for the bulk magnets SmCo 5 and Nd 2 Fe 14 B, and the highest coercive field reported to date for any single-molecule magnet or molecule-based magnetic material. Multiconfigurational calculations, supported by far-infrared magnetospectroscopy data, reveal that the stark differences in magnetic properties of 1-Ho and 1-Er arise from differences in the local magnetic anisotropy of the lanthanide centers.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Theory of optical responses in clean multi-band superconductors

Electromagnetic responses in superconductors provide valuable information on the pairing symmetry as well as physical quantities such as the superfluid density. However, at the superconducting gap energy scale, optical excitations of the Bogoliugov quasiparticles are forbidden in conventional Bardeen-Cooper-Schrieffer superconductors when momentum is conserved. Accordingly, far-infrared optical responses have been understood in the framework of a dirty-limit theory by Mattis and Bardeen for over 60 years. Here we show, by investigating the selection rules imposed by particle-hole symmetry and unitary symmetries, that intrinsic momentum-conserving optical excitations can occur in clean multi-band superconductors when one of the following three conditions is satisfied: (i) inversion symmetry breaking, (ii) symmetry protection of the Bogoliubov Fermi surfaces, or (iii) simply finite spin-orbit coupling with unbroken time reversal and inversion symmetries. This result indicates that clean-limit optical responses are common beyond the straightforward case of broken inversion symmetry. We apply our theory to optical responses in FeSe, a clean multi-band superconductor with inversion symmetry and significant spin-orbit coupling. This result paves the way for studying clean-limit superconductors through optical measurements.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2022

Analysis of vibronic coupling in a 4f molecular magnet with FIRMS

Vibronic coupling, the interaction between molecular vibrations and electronic states, is a fundamental effect that profoundly affects chemical processes. In the case of molecular magnetic materials, vibronic, or spin-phonon, coupling leads to magnetic relaxation, which equates to loss of magnetic memory and loss of phase coherence in molecular magnets and qubits, respectively. The study of vibronic coupling is challenging, and most experimental evidence is indirect. Here we employ far-infrared magnetospectroscopy to directly probe vibronic transitions in [Yb(trensal)] (where H 3 trensal = 2,2,2-tris(salicylideneimino)trimethylamine). We find intense signals near electronic states, which we show arise due to an “envelope effect” in the vibronic coupling Hamiltonian, which we calculate fully ab initio to simulate the spectra. We subsequently show that vibronic coupling is strongest for vibrational modes that simultaneously distort the first coordination sphere and break the C 3 symmetry of the molecule. With this knowledge, vibrational modes could be identified and engineered to shift their energy towards or away from particular electronic states to alter their impact. Hence, these findings provide new insights towards developing general guidelines for the control of vibronic coupling in molecules.

36 MATERIALS SCIENCE↗
DOE OSTI2023

Insulator–metal transition in CrSiTe 3 triggered by structural distortion under pressure

van der Waals solids are well known to host remarkable phase diagrams with competing phases, unusual energy transfer processes, and elusive states of matter. Among this class of materials, chalcogenides have emerged as the most flexible and relevant platforms for unraveling charge–structure–function relationships. In order to explore the properties of complex chalcogenides under external stimuli, we measured the far infrared spectroscopic response of CrSiTe 3 under extreme pressure–temperature conditions. Analysis of the 368 cm –1 Si–Te stretching mode and the manner in which it is screened by the closure of the indirect gap reveals that the insulator–metal transition takes place immediately after the structural phase transition—once the mixed phase aspect of the lattice distortion is resolved. At the same time, the two-phase region associated with the structural transition widens with decreasing temperature, and the slope of the insulator–metal transition under pressure is consistent with increasing entropy. These trends completely revise the character of the temperature–pressure phase diagram as well as the relationship between the structural and insulator–metal transitions, leading to a critical nexus of activity that may hide a quantum critical point and allow superconductivity to emerge.

36 MATERIALS SCIENCE↗
DOE OSTI2020

Single-ion magnetism in the extended solid-state: insights from X-ray absorption and emission spectroscopy

Large single-ion magnetic anisotropy is observed in lithium nitride doped with iron. The iron sites are two-coordinate, putting iron doped lithium nitride amongst a growing number of two coordinate transition metal single-ion magnets (SIMs). Uniquely, the relaxation times to magnetisation reversal are over two orders of magnitude longer in iron doped lithium nitride than other 3d-metal SIMs, and comparable with high-performance lanthanide-based SIMs. To understand the origin of these enhanced magnetic properties a detailed characterisation of electronic structure is presented. Access to dopant electronic structure calls for atomic specific techniques, hence a combination of detailed single-crystal X-ray absorption and emission spectroscopies are applied. Together K-edge, L 2,3 -edge and Kβ X-ray spectroscopies probe local geometry and electronic structure, identifying iron doped lithium nitride to be a prototype, solid-state SIM, clean of stoichiometric vacancies where Fe lattice sites are geometrically equivalent. Extended X-ray absorption fine structure and angular dependent single-crystal X-ray absorption near edge spectroscopy measurements determine Fe I dopant ions to be linearly coordinated, occupying a D 6h symmetry pocket. The dopant engages in strong 3dπ-bonding, resulting in an exceptionally short Fe–N bond length (1.873(7) Å) and rigorous linearity. It is proposed that this structure protects dopant sites from Renner–Teller vibronic coupling and pseudo Jahn–Teller distortions, enhancing magnetic properties with respect to molecular-based linear complexes. The Fe ligand field is quantified by L 2,3 -edge XAS from which the energy reduction of 3d z 2 due to strong 4s mixing is deduced. Quantification of magnetic anisotropy barriers in low concentration dopant sites is inhibited by many established methods, including far-infrared and neutron scattering. We deduce variable temperature L 3 -edge XAS can be applied to quantify the J = 7/2 magnetic anisotropy barrier, 34.80 meV (~280 cm –1 ), that corresponds with Orbach relaxation via the first excited, M J = ±5/2 doublet. The results demonstrate that dopant sites within solid-state host lattices could offer a viable alternative to rare-earth bulk magnets and high-performance SIMs, where the host matrix can be tailored to impose high symmetry and control lattice induced relaxation effects.

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

Two-dimensional talc as a natural abundant ultra-broadband hyperbolic material

Here, we demonstrate that two-dimensional talc, a naturally abundant phyllosilicate mineral, supports hyperbolic phonon-polaritons (HPhPs) across the mid- and far-infrared wavelengths. Using scattering scanning near-field optical microscopy (s-SNOM) and synchrotron infrared nano-spectroscopy (SINS), we reveal tunable HPhP modes in talc flakes with long lifetimes, high confinement, and quality factors of up to 5. We further observe Fabry–Pérot cavity modes in tapered flakes, confirmed by simulations and analytical modeling. Compared to synthetic crystals, talc offers an ultra-broadband, low-cost, and sustainable platform for infrared nanophotonics and optoelectronics.

36 MATERIALS SCIENCE↗
DOE OSTI2022

The Close AGN Reference Survey (CARS)

Active galactic nuclei (AGN) are thought to be responsible for the suppression of star formation in massive ~10 10 M ⊙ galaxies. While this process is a key feature in numerical simulations of galaxy formation, it has not been unambiguously confirmed in observational studies yet. The characterization of the star formation rate (SFR) in AGN host galaxies is challenging as AGN light contaminates most SFR tracers. Furthermore, the various SFR tracers are sensitive to different timescales of star formation from approximately a few to 100 Myr. We aim to obtain and compare SFR estimates from different tracers for AGN host galaxies in the Close AGN Reference Survey (CARS) to provide new observational insights into the recent SFR history of those systems. We constructed integrated panchromatic spectral energy distributions to measure the far infrared (FIR) luminosity as a tracer for the recent (< 100 Myr) SFR. In addition we used the integral-field unit observation of the CARS targets to employ the Hα luminosity decontaminated by AGN excitation as a proxy for the current (< 5 Myr) SFR. We find that significant differences in specific SFR of the AGN host galaxies as compared with the larger galaxy population disappear once cold gas mass, in addition to stellar mass, is used to predict the SFR for a specific AGN host. Only a tentative trend with the inclination of the host galaxy remains, such that SFR appears slightly lower than expected when the galaxies of unobscured AGN appear more edge-on along our line-of-sight, particular for dust-insensitive FIR-based SFRs. We identify individual galaxies with a significant difference in their SFR which can be related to a recent enhancement or decline in their SFR history that might be related to various processes including interactions, gas consumption, outflows, and AGN feedback. AGN can be present in various stages of galaxy evolution which makes it difficult to relate the SFR solely to the impact of the AGN. Our study shows that stellar mass alone is an insufficient parameter to estimate the expected SFR of an AGN host galaxy compared to the underlying non-AGN galaxy population. We do not find any strong evidence for a global positive or negative AGN feedback in the CARS sample. However, there is tentative evidence that (1) the relative orientation of the AGN engine with respect to the host galaxies might alter the efficiency of AGN feedback and that (2) the recent SFH is an additional tool to identify rapid changes in galaxy growth driven by the AGN or other processes.

79 ASTRONOMY AND ASTROPHYSICS↗
DOE OSTI2020

Trajectory of a flying plasma mirror traversing a target with density gradient

It has been proposed that laser-induced relativistic plasma mirror can accelerate if the plasma has a properly tailored density profile. Such accelerating plasma mirrors can serve as analog black holes to investigate Hawking evaporation and the associated information loss paradox. Here we reexamine the underlying dynamics of mirror motion in a graded-density plasma to provide an explicit trajectory as a function of the plasma density and its gradient. Specifically, a decreasing plasma density profile (down-ramp) along the direction of laser propagation would in general accelerate the mirror. In particular, a constant-plus-exponential density profile would generate the Davies–Fulling trajectory with a well-defined analog Hawking temperature, which is sensitive to the plasma density gradient but not to the density itself. Furthermore, we show that without invoking nano-fabricated thin-films, a much lower density gas target at, for example, ~1 × 10 17 cm -3 , would be able to induce an analog Hawking temperature, k B T H ~3.1 × 10 -2 eV, in the far-infrared region. We hope that this would help to better realize the experiment proposed by Chen and Mourou.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2022

Scattering-type scanning near-field optical microscopy with Akiyama piezo-probes

Recent developments of the scattering-type scanning near-field optical microscope at cryogenic temperatures (cryogenic s-SNOM or cryo-SNOM) have led to many breakthroughs in the studies of low energy excitations in quantum materials. However, the simultaneous demands on vibration isolation, low base temperature, precise nano-positioning, and optical access make the construction of a cryo-SNOM a daunting task. Adding to the overhead space required for a cryo-SNOM is the atomic force microscopy control, which predominantly utilizes a laser-based detection scheme for determining the cantilever tapping motion. In this work, we provide an alternative and straightforward route to performing s-SNOM using metal-coated Akiyama probes, where the cantilever tapping motion is detected through a piezoelectric signal. We show that the Akiyama-based cryo-SNOM attains high spatial resolution, good near-field contrast, and is able to perform imaging with a significantly more compact system compared to other cryo-SNOM implementations. Our results firmly establish the potential of s-SNOM based on self-sensing piezo-probes, which can easily accommodate far-infrared wavelengths and high magnetic fields in the future.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Ab initio generalized Langevin equation

We introduce a machine learning–based approach called ab initio generalized Langevin equation (AIGLE) to model the dynamics of slow collective variables (CVs) in materials and molecules. In this scheme, the parameters are learned from atomistic simulations based on ab initio quantum mechanical models. Force field, memory kernel, and noise generator are constructed in the context of the Mori–Zwanzig formalism, under the constraint of the fluctuation–dissipation theorem. Combined with deep potential molecular dynamics and electronic density functional theory, this approach opens the way to multiscale modeling in a variety of situations. Here, we demonstrate this capability with a study of two mesoscale processes in crystalline lead titanate, namely the field-driven dynamics of a planar ferroelectric domain wall, and the dynamics of an extensive lattice of coarse-grained electric dipoles. In the first case, AIGLE extends the reach of ab initio simulations to a regime of noise-driven motions not accessible to molecular dynamics. In the second case, AIGLE deals with an extensive set of CVs by adopting a local approximation for the memory kernel and retaining only short-range noise correlations. The scheme is computationally more efficient than molecular dynamics by several orders of magnitude and mimics the microscopic dynamics at low frequencies where it reproduces accurately the dominant far-infrared absorption frequency.

97 MATHEMATICS AND COMPUTING↗