Order Perturbation Research Articles

Is word order considered by foundation models? A comparative task-oriented analysis

Word order, a linguistic concept essential for conveying accurate meaning, is seemingly not that necessary in language models based on the existing works. Contrary to this prevailing notion, our paper delves into the impacts of word order by employing carefully selected tasks that demand distinct abilities. Using three large language model families (ChatGPT, Claude, LLaMA), three controllable word order perturbation strategies, one novel perturbation qualification metric, four well-chosen tasks, and three languages, we conduct experiments to shed light on this topic. Empirical findings demonstrate that Foundation models take word order into consideration during generation. Moreover, tasks emphasizing reasoning abilities exhibit a greater reliance on word order compared to those primarily based on world knowledge.

High-order mixed vortex beam generator

A mixed multi-order vortex beam generator, based on a Reuleaux triangle core fiber chiral grating (RCFG), is proposed. The triangular perturbation and off-axis effects induced by core shape, result in the simultaneous coupling of the core mode with the 1st- and 3rd-order vortex modes. To the best of our knowledge, this is the first time that a mixed vortex beam was generated in a single chiral fiber. The phase matching conditions required for the co-coupling of multi-order vortex beams are analyzed based on the coupled mode theory. Additionally, a cladding shrinkage method is proposed to flexibly adjust the co-coupling wavelength. We found that the key to co-coupling lies in balancing the different order perturbations of the Reuleaux triangle core fiber (RTF). The proposed method offers a new approach for the design of mixed multi-order vortex beam generators, with potential applications in fields such as fiber OAM communications, optical tweezers, and super-resolution imaging.

Molecular dynamics simulations reveal methylation in Me-GDGTs as a microbial low-temperature adaptation

Tetraether membrane lipid glycerol dialkyl glycerol tetraether (GDGT) is a characteristic lipid of archaea that contributes to adaptation to a variety of environments and has been widely used in ecological and geochemical studies. Alkyl-chain methyl-modified GDGTs (Me-GDGTs) as one important component have been detected in both environmental and culture samples. However, the effect of methyl modification on archaeal membrane lipids remains unclear, hampering the development of environmental proxies based on the exact understanding of the underlying mechanism. Here, we reveal the perturbation of molecular order by methyl modification through molecular dynamics simulations, showing that methyl modification can increase membrane fluidity, explaining that this is a form of low-temperature acclimation. This hypothesis was also validated in environmental samples. This study reveals the role of methyl modifications in low-temperature adaptation in archaea, expanding that hydrophobic carbon chain methylation has unified physiological significance in three domain life. Our study highlights the role of molecular dynamics simulations in the development of lipid biomarkers, provides new insights into the mechanisms of temperature adaptation in microorganisms, and lays the groundwork for the development of new environmental proxies.

Entropy-current for dynamical black holes in Chern-Simons theories of gravity

We construct an entropy current and establish a local version of the classical second law of thermodynamics for dynamical black holes in Chern-Simons (CS) theories of gravity. We work in a chosen set of Gaussian null coordinates and assume the dynamics to be small perturbations around the Killing horizon. In explicit examples of both purely gravitational and mixed gauge gravity CS theories in (2 + 1) and (4 + 1)-dimensions, the entropy current is obtained by studying the off-shell structure of the equations of motion evaluated on the horizon. For the CS theory in (2 + 1) dimensions, we argue that the second law holds to quadratic order in perturbations by considering it as a low energy effective field theory with the leading piece given by Einstein gravity. In all such examples, we show that the construction of entropy current is invariant under the reparameterization of the null horizon coordinates. Finally, extending an existing formalism for diffeomorphism invariant theories, we construct an abstract proof for the linearised second law in arbitrary Chern-Simons theories in any given odd dimensions by studying the off-shell equations of motion. As a check of consistency, we verify that the outcome of this algorithmic proof matches precisely with the results obtained in explicit examples.

Introducing the embedded random phase approximation: H2 dissociative adsorption on Cu(111) as an exemplar

The random phase approximation (RPA) as a means of treating electron correlation recently has been shown to outperform standard density functional theory (DFT) approximations in a variety of cases. However, the computational cost of the RPA is substantially more than DFT, especially when aiming to study extended surfaces. Properly accounting for sufficient surface ensemble size, Brillouin zone sampling, and vacuum separation of periodic images in standard periodic-planewave-based DFT code raises the cost to achieve converged results. Here, we show that sub-system embedding schemes enable use of the RPA for modeling heterogeneous reactions at reduced computational cost. We explore two different embedded RPA (emb-RPA) approaches, periodic emb-RPA and cluster emb-RPA. We use the (experimentally and theoretically) well-studied H2 dissociative adsorption on Cu(111) as our exemplar, and first perform full periodic RPA calculations as a benchmark. The full RPA results match well the semi-empirical barrier fit to experimental observables and others derived from high-level computations, e.g., from recent embedded n-electron valence second order perturbation theory [Zhao et al., J. Chem. Theory Comput. 16(11), 7078–7088 (2020)] and quantum Monte Carlo [Doblhoff-Dier et al., J. Chem. Theory Comput. 13(7), 3208–3219 (2017)] simulations. Among the two emb-RPA approaches tested, the cluster emb-RPA accurately reproduces the energy profile (maximum error of 50 meV along the reaction pathway) while reducing the computational cost by approximately two orders of magnitude. We therefore expect that the embedded cluster approach will enable wider RPA implementation in heterogeneous catalysis.

Formula omitted] and TQQ within QCD Laplace sum rules

We present a review of our results for the masses and couplings of the scalar fully heavy four-quarks and TQQq¯q¯′(JP=0±,1±) tetraquarks states from (inverse) QCD Laplace sum rule (LSR), their ratios R and double ratio of sum rules (DRSR) within stability criteria and including Factorized Next-to-Leading Order (FNLO) Perturbative (PT) corrections. As the Operator Product Expansion (OPE) usually converges for d⩽6−8, we evaluated the QCD spectral functions at Lowest Order (LO) of PT QCD and up to 〈G3〉. Our results for the 0++ fully heavy four-quark states may explain the LHCb broad structure around (6.2−6.7)GeV which can be due to ηc‾ηc,χc1‾χc1 and J/ψ‾J/ψ molecules or/and their analogue ScSc,AcAc and VcVc tetraquarks. The peak at (6.8−6.9)GeV can be identified to the χc0‾χc0 molecule or/and the PcPc tetraquark state. Then, combining R and DRSR we focus on the analysis of the four-quark nature of Tccq¯q¯′1± and 0± states. We show that combining R and DRSR can provide more precise results: MTcc1+=3886(6)MeV and MTcc0+=3883(3)MeV. In a similar way, one obtain the improved mass predictions: MTccs¯u¯1+=3931(7)MeV, MTccs¯u¯0+=3983(7)MeV and MTccs¯s¯0+=3993(11)MeV. From our estimates of the masses of the Pseudoscalar and Vector Tccq¯q¯′ states, we observe that the interpolating currents lead to two classes: Class H (Heavy) states with masses around 6GeV and Class L (Light) states around (3.8−4.4)GeV where the pseudoscalar (resp. all vector states) are below the D‾D0,D‾sDs0 (resp. D‾D1,D‾sDs1) open charm thresholds. Finally, we extend the whole study to the bottom sector and confront our results with the ones from different LSR predictions and some other approaches (lattice, potential models, ⋯) in the literature.

When do tripdoublet states fluoresce? A theoretical study of copper(II) porphyrin.

Open-shell molecules rarely fluoresce, due to their typically faster non-radiative relaxation rates compared to closed-shell ones. Even rarer is the fluorescence from states that have two more unpaired electrons than the open-shell ground state, since they involve excitations from closed-shell orbitals to vacant-shell orbitals, which are typically higher in energy compared to excitations from or out of open-shell orbitals. States that are dominated by the former type of excitations are known as tripdoublet states when they can be described as a triplet excitation antiferromagnetically coupled to a doublet state, and their description by unrestricted single-reference methods (e.g., U-TDDFT) is notoriously inaccurate due to large spin contamination. In this work, we applied our spin-adapted TDDFT method, X-TDDFT, and the efficient and accurate static-dynamic-static second order perturbation theory (SDSPT2), to the study of the excited states as well as their relaxation pathways of copper(II) porphyrin; previous experimental works suggested that the photoluminescence of some substituted copper(II) porphyrins originate from a tripdoublet state, formed by a triplet ligand π → π* excitation antiferromagnetically coupled with the unpaired d electron. Our results demonstrated favorable agreement between the X-TDDFT, SDSPT2 and experimental excitation energies, and revealed noticeable improvements of X-TDDFT compared to U-TDDFT, not only for vertical excitation energies but also for adiabatic energy differences. These suggest that X-TDDFT is a reliable tool for the study of tripdoublet state fluorescence. Intriguingly, we showed that the aforementioned tripdoublet state is only slightly above the lowest doublet excited state and lies only slightly higher than the lowest quartet state, which suggests that the tripdoublet of copper(II) porphyrin is long-lived enough to fluoresce due to a lack of efficient non-radiative relaxation pathways; an explanation for this unusual state ordering is given. Indeed, thermal vibration correlation function (TVCF)-based calculations of internal conversion, intersystem crossing, and radiative transition rates confirm that copper(II) porphyrin emits thermally activated delayed fluorescence (TADF) and a small amount of phosphorescence at low temperature (83K), in accordance with experiment. The present contribution is concluded by a few possible approaches of designing new molecules that fluoresce from tripdoublet states.

A revised analysis towards accurate estimation of isothermal temperature coefficients in fast reactors having different power levels and core sizes

The accurate prediction of safety coefficient such as isothermal temperature coefficient of reactivity is an important requisite in the safe operation of any reactor and also in optimizing the reactor theoretically. At IGCAR, a multi-purpose safety analysis code PREDIS is being used for this purpose by employing the spatial distribution of first order perturbation worth as input. Estimated isothermal temperature coefficient is validated against the measured value of a small 40 MWt carbide core reactor FBTR and 400 MWt FFTF. Though the results are conservative, PREDIS is found to be under predicting the safety coefficients. A detailed parametric study showed that isothermal temperature coefficient is under predicted through PREDIS analysis because of neglecting the contributions of removal worth from the non-fuelled regions surrounding the core. Relative contributions of non-fuelled regions to isothermal temperature coefficient have been systematically quantified in different fast reactor cores such as 40 MWt carbide core of FBTR, 400 MWt FFTF, 1250 MWt oxide core of PFBR and 1500 MWt oxide core of FBR 1&2. From the study, the leakage contribution to the surrounding non-fuelled region is found to be significant in a smaller core, hence the removal worth of these regions should not be ignored while estimating the safety coefficients. Conservative assumptions of considering only the core region for the prediction of safety coefficient is assumed to be good, only for the medium sized reactors, where the leakage contribution to the isothermal temperature coefficient is not that prominent.

Self-consistent dynamical field theory of kernel evolution in wide neural networks *

We analyze feature learning in infinite-width neural networks trained with gradient flow through a self-consistent dynamical field theory. We construct a collection of deterministic dynamical order parameters which are inner-product kernels for hidden unit activations and gradients in each layer at pairs of time points, providing a reduced description of network activity through training. These kernel order parameters collectively define the hidden layer activation distribution, the evolution of the neural tangent kernel (NTK), and consequently, output predictions. We show that the field theory derivation recovers the recursive stochastic process of infinite-width feature learning networks obtained by Yang and Hu with tensor programs. For deep linear networks, these kernels satisfy a set of algebraic matrix equations. For nonlinear networks, we provide an alternating sampling procedure to self-consistently solve for the kernel order parameters. We provide comparisons of the self-consistent solution to various approximation schemes including the static NTK approximation, gradient independence assumption, and leading order perturbation theory, showing that each of these approximations can break down in regimes where general self-consistent solutions still provide an accurate description. Lastly, we provide experiments in more realistic settings which demonstrate that the loss and kernel dynamics of convolutional neural networks at fixed feature learning strength are preserved across different widths on a image classification task.

WIMP decay as a possible Warm Dark Matter model

Abstract The Weakly Interacting Massive Particles (WIMPs) have long been the favoured Cold Dark Matter (CDM) candidate in the standard ΛCDM model. However, owing to great improvement in the experimental sensitivity in the past decade, some parameter space of the Supersymmetric (SUSY)-based WIMP model is ruled out. In addition, a massive stable WIMP as the CDM particle is also at variance with other astrophysical observables at small scales. We consider a model that addresses both these issues. In the model, the WIMP decays into a massive particle and radiation. We study the background evolution and the first order perturbation theory (coupled Einstein-Boltzmann equations) for this model and show that the dynamics can be captured by a single parameter r = mL/q, which is the ratio of the lighter mass and the comoving momentum of the decay particle. We incorporate the relevant equations in the existing Boltzmann code CLASS to compute the matter power spectra and Cosmic Microwave Background (CMB) angular power spectra. The decaying WIMP model is akin to a non-thermal Warm Dark Matter (WDM) model and suppresses matter power at small scales, which could alleviate several issues that plague the CDM model at small scales. We compare the predictions of the model with CMB and galaxy clustering data. As the model deviates from the ΛCDM model at small scales, the evolution of the collapse fraction of matter in the universe is compared with the high-redshift Sloan Digital Sky Survey (SDSS) HI data. Both these data sets yield r ≳ 106, which can be translated into the bounds on other parameters. In particular, we obtain the following lower bounds on the thermally-averaged self-annihilation cross-section of WIMPs, ⟨σv⟩, and the lighter mass: ⟨σv⟩ ≳ 4.9 × 10-34 cm3 sec-1 and mL ≳ 2.4 keV. The lower limit on mL is comparable to constraints on the mass of thermally-produced WDM particle. The limit on the self-annihilation cross-section greatly expands the available parameter space as compared to the stable WIMP scenario.

Experimental and Theoretical Studies of Hyperthermal N + O2 Collisions.

The dynamics of hyperthermal N(4S) + O2 collisions were investigated both experimentally and theoretically. Crossed molecular beams experiments were performed at an average center-of-mass (c.m.) collision energy of ⟨Ecoll⟩ = 77.5 kcal mol-1, with velocity- and angle-resolved product detection by a rotatable mass spectrometer detector. Nonreactive (N + O2) and reactive (NO + O) product channels were identified. In the c.m. reference frame, the nonreactively scattered N atoms and reactively scattered NO molecules were both directed into the forward direction with respect to the initial direction of the reagent N atoms. On average, more than 90% of the available energy (⟨Eavl⟩ = 77.5 kcal mol-1) was retained in translation of the nonreactive products (N + O2), whereas a much smaller fraction of the available energy for the reactive pathway (⟨Eavl⟩ = 109.5 kcal mol-1) went into translation of the NO + O products, and the distribution of translational energies for this channel was broad, indicating extensive internal excitation in the nascent NO molecules. The experimentally derived c.m. translational energy and angular distributions of the reactive products suggested at least two dynamical pathways to the formation of NO + O. Quasiclassical trajectory (QCT) calculations were performed with a collision energy of Ecoll = 77 kcal mol-1 using two sets of potential energy surfaces, denoted as PES-I and PES-II, and these theoretical results were compared to each other and to the experimental results. PES-I is a reproducing kernel Hilbert space (RKHS) representation of multireference configurational interaction (MRCI) energies, while PES-II is a many-body permutation invariant polynomial (MB-PIP) fit of complete active space second order perturbation (CASPT2) points. The theoretical investigations were both consistent with the experimental suggestion of two dynamical pathways to produce NO + O, where reactive collisions may proceed on the doublet (12A') and quartet (14A') surfaces. When analyzed with this theoretical insight, the experimental c.m. translational energy and angular distributions were in reasonably good agreement with those predicted by the QCT calculations, although minor differences were observed which are discussed. Theoretical translational energy and angular distributions for the nonreactive N + O2 products matched the experimental translational energy and angular distributions almost quantitatively. Finally, relative yields for the nonreactive and reactive scattering channels were determined from the experiment and from both theoretical methods, and all results are in reasonable agreement.

Unraveling the impact of polar solvation on the molecular geometry, spectroscopy (FT-IR, UV, NMR), reactivity (ELF, NBO, HOMO-LUMO) and antiviral inhibitory potential of Cissampeline by molecular docking approach

Cissampeline, a highly promising natural substance derived from medicinal plants of the Cissampelos genus, has recently garnered significant interest due to its potent antiviral properties against a broad spectrum of viral infections. In this comprehensive study, we employed gd3bj-B3LYP/def2svp level of theory to investigate the impact of polar solvation on the molecular structure, dynamical stability, spectroscopy, nature of bonding, and antiviral inhibitory potential of Cissampeline. Our results demonstrated excellent agreement between the theoretically characterized structure and the experimentally determined one. Interestingly, we observed that in the absence of a solvent environment, the gas phase exhibited shorter bond angles compared to when different solvents were utilized, indicating reduced solvent interactions. Regarding solvation dynamics, we found that the total energy of the structure, when optimized in different solvents, followed the order DMSO > MeOH > Water > Gas, with corresponding total final energies of 1736.599 > 867.932 > 837.760 > 413.989 kcal/mol, respectively. Furthermore, NBO analysis revealed the strength of electron delocalization, with the order of perturbation energies being DMSO > MeOH > H2O > Gas phase, measured at 626.07 > 241.40 > 238.65 > 72.93 kcal/mol, respectively. Particularly noteworthy was the σ-σ* transition in the DMSO solvent phase, displaying the highest perturbation energy of 626.07 kcal/mol. FMO analysis provided insights into the energy levels of the studied species, with values of 4.5432 eV for Gas, 4.5250 eV for MeOH, 4.5247 eV for H2O, and 4.5242 eV for DMSO, respectively. Regarding the interaction of Cissampeline with amino acid residues, we found that the ligand exhibited the highest binding affinity with 3MX2 at -7.7 kcal/mol, followed by CMPL + 3T5N at -7.3 kcal/mol, and CMPL + 3MX5 at -6.0 kcal/mol. In comparison, the standard drug RIBAV only displayed successful interaction with 3MX2, showing the least binding affinity at -5.8 kcal/mol. This study showed highlights the remarkable potential of Cissampeline as an effective antiviral agent and sheds light on the importance of considering solvation effects in molecular investigations.

Cotton-Mouton Effect Using Algebraic Diagrammatic Construction Schemes in the Intermediate State Representation Formalism.

The Cotton-Mouton effect is theoretically investigated for a selected set of molecules by using a novel computational methodology based on algebraic diagrammatic construction (ADC) schemes in the intermediate state representation (ISR) formulation. Therefore, the electronic contributions to the frequency-dependent polarizabilities and, for the first time, to the magnetizabilities as well as mixed electric and magnetic hypermagnetizabilities have been computed in the ADC/ISR framework. In addition to calculation of the Cotton-Mouton constant and the birefringence, the gauge origin dependence of the computed tensors and the applied methodology are thoroughly investigated. The new ADC/ISR methodology, employing the recently presented responsefun package, is applied to a test set of Ne and small molecules (H2, HF, O2, CO2, and benzene) and compared to data from the experiment as well as other ab initio methods. The presented theoretical ab initio ADC/ISR approach is a substantial extension of the available computational methods for the investigation of complex nonlinear properties, however, with a gauge origin dependence inherent to the method that decreases with increasing perturbation order.

Adjoint-based high-order spectral method of wave simulation for coastal bathymetry reconstruction

Bathymetry is an important factor affecting wave propagation in coastal environments but is often challenging to measure in practice. We propose a method for inferring coastal bathymetry from spatial variations in surface waves by combining a high-order spectral method for wave simulation and an adjoint-based variational data assimilation method. Recursion-formed adjoint equations are derived to obtain the sensitivity of the wave surface elevation to the underlying bottom topography to any desired order of nonlinear perturbation. We also develop a multiscale optimisation method to eliminate spurious high-wavenumber fluctuations in the reconstructed bathymetry data caused by sensitivity variations over the different length scales of surface waves. The proposed bottom detection method is validated with a realistic coastal wave environment involving complex two-dimensional bathymetry features, non-periodic incident waves and nonlinear broadband multidirectional waves. In numerical experiments at both laboratory and field scales, the bathymetry reconstructed from our method agrees well with the ground truth. We also show that our method is robust against imperfect surface wave data in the presence of limited sampling frequency and noise.

Bjorken Sum Rule with Analytic Coupling at Low Q2 Values

The experimental data obtained for the polarized Bjorken sum rule Gamma _{1}^{{p - n}}({{Q}^{2}}) for small values of Q2 are approximated by the predictions obtained in the framework of analytic QCD up to the fifth order perturbation theory, whose coupling constant does not contain the Landau pole. We found an excellent agreement between the experimental data and the predictions of analytic QCD, as well as a strong difference between these data and the results obtained in the framework of perturbative QCD.

Efficient algorithm to calculate the optical properties of breast tumors by high-order perturbation theory.

An efficient algorithm to obtain the solutions for n-th order terms of perturbation expansions in absorption, scattering, and cross-coupling for light propagating in human tissue is presented. The proposed solution is free of any approximations and makes possible fast and efficient estimates of mammographic, optical tomographic, and fluorescent images, applying a perturbation order of 30 and more. The presented analysis sets the general limits for the applicability of the perturbation approach as a function of tumor size and optical properties of the human tissue. The convergence tests of the efficient calculations for large absorbing objects show excellent agreement with the reference data from finite element method calculations. The applicability of the theory is demonstrated in experiments on breast-like phantoms with high absorbing and low-scattering lesions.

Multi-bubble Bourgain-Wang solutions to nonlinear Schrödinger equations

We study a general class of focusing L 2 L^2 -critical nonlinear Schrödinger equations with lower order perturbations, in the possible absence of the pseudo-conformal symmetry and the conservation law of energy. In dimensions one and two, we construct multi-bubble Bourgain-Wang type blow-up solutions, which behave like a sum of pseudo-conformal blow-up solutions that concentrate at K K distinct singularities, 1 ≤ K > ∞ 1\leq K>{\infty } , and a regular profile. Moreover, we obtain the uniqueness in the energy class where the convergence rate is within the order ( T − t ) 4 + (T-t)^{4+} , for t t close to the blow-up time T T . These results in particular apply to the canonical nonlinear Schrödinger equations and, through the pseudo-conformal transform, yield the existence and conditional uniqueness of non-pure multi-solitons, which behave asymptotically as a sum of multi-solitons and a dispersive part. Thus, the results provide new examples of the mass quantization conjecture and the soliton resolution conjecture. Furthermore, through a Doss-Sussman type transform, we obtain multi-bubble Bourgain-Wang solutions for stochastic nonlinear Schrödinger equations, where the driving noise is taken in the sense of controlled rough path.

Theoretical and experimental study of the 6S-8S two-photon absorption cross-section in cesium atoms.

The probability of a two-photon absorption (TPA) process is quantified by means of the TPA cross-section. In this work the TPA cross-section for the 6S1/2 → 8S1/2 transition in cesium (Cs) atoms is obtained from a theoretical and experimental approach. It is calculated using second order perturbation theory including hyperfine structure and Doppler broadening for a thermal atomic vapor. The theoretical values for the TPA cross-sections of the two allowed hyperfine transitions are σ~3,3=3.10×10-24 cm4/W and σ~4,4=3.65×10-24 cm4/W. An experiment is set up to measure these cross-sections based on the quadratic dependence between the TPA transition rate and the intensity of the laser light. Values of σ~3,3=(3.6±1.1)×10-24 cm4/W and σ~4,4=(4.6±1.1)×10-24 cm4/W are obtained from the measurements. The results show an agreement between theory and experiment. To the best of our knowledge, this is the first time that these cross-sections are reported.

Effect of α-Substitution on the Reactivity of C(sp3)-H Bonds in Pd0-Catalyzed C-H Arylation.

We report mechanistic studies on the reactivity of different α-substituted C(sp3)-H bonds, -CHnR (R = H, Me, CO2Me, CONMe2, OMe, and Ph, as well as the cyclopropyl and isopropyl derivatives -CH(CH2)2 and -CHMe2) in the context of Pd0-catalyzed C(sp3)-H arylation. Primary kinetic isotope effects, kH/kD, were determined experimentally for R = H (3.2) and Me (3.5), and these, along with the determination of reaction orders and computational studies, indicate rate-limiting C-H activation for all substituents except when R = CO2Me. This last result was confirmed experimentally (kH/kD ∼ 1). A reactivity scale for C(sp3)-H activation was then determined: CH2CO2Me > CH(CH2)2 ≥ CH2CONMe2 > CH3 ≫ CH2Ph > CH2Me > CH2OMe ≫ CHMe2. C-H activation involves AMLA/CMD transition states featuring intramolecular O → H-C H-bonding assisted by C-H → Pd agostic bonding. The "AMLA coefficient", χ, is introduced to quantify the energies associated with these interactions via natural bond orbital 2nd order perturbation theory analysis. Higher barriers correlate with lower χ values, which in turn signal a greater agostic interaction in the transition state. We believe that this reactivity scale and the underlying factors that determine this will be of use for future studies in transition-metal-catalyzed C(sp3)-H activation proceeding via the AMLA/CMD mechanism.

(Slow-)twisting inflationary attractors

We explore in detail the dynamics of multi-field inflationary models. We first revisit the two-field case and rederive the coordinate independent expression for the attractor solution with either small or large turn rate, emphasizing the role of isometries for the existence of rapid-turn solutions.Then, for three fields in the slow-twist regime we provide elegant expressions for the attractor solution for generic field-space geometries and potentials and study the behaviour of first order perturbations. For generic 𝒩-field models, our method quickly grows in algebraic complexity. We observe that field-space isometries are common in the literature and are able to obtain the attractor solutions and deduce stability for some isometry classes of 𝒩-field models. Finally, we apply our discussion to concrete supergravity models. These analyses conclusively demonstrate the existence of 𝒩 > 2 dynamical attractors distinct from the two-field case, and provide tools useful for future studies of their phenomenology in the cosmic microwave background and stochastic gravitational wave spectrum.